Mathematical Physics, Analysis and Geometry - Volume 6

Mathematical Physics, Analysis and Geometry 6: 1–8, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

1

Hyper-Kähler Metrics Conformal to Left InvariantMetrics on Four-Dimensional Lie Groups

MARÍA LAURA BARBERIS�

FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000-Córdoba, Argentina.e-mail: [email protected]

(Received: 3 April 2001; in final form: 4 March 2002)

Abstract. Let g be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold(M,H). We show that when the isometry group I (M,g) contains a subgroup G acting simplytransitively on M by hypercomplex isometries, then the metric g is conformal to a hyper-Kählermetric. We describe explicitely the corresponding hyper-Kähler metrics, which are of cohomegeneityone with respect to a 3-dimensional normal subgroup of G. It follows that, in four dimensions, theseare the only hyper-Kähler metrics containing a homogeneous metric in its conformal class.

Mathematics Subject Classifications (2000): 53C15, 53C25, 53C30.

Key words: hyper-Hermitian metric, hypercomplex manifold, conformally hyper-Kähler metric.

1. Preliminaries

A hypercomplex structure on a 4n-dimensional manifold M is a family H ={Jα}α=1,2,3 of fibrewise endomorphisms of the tangent bundle TM of M satisfying:

J 2α = −I, α = 1, 2, 3, J1J2 = −J2J1 = J3, (1.1)

Nα ≡ 0, α = 1, 2, 3, (1.2)

where I is the identity on the tangent space TpM of M at p for all p in M and Nα

is the Nijenhuis tensor corresponding to Jα:

Nα(X, Y ) = [JαX, JαY ] − [X,Y ] − Jα([X, JαY ] + [JαX, Y ])for all X,Y vector fields on M. A differentiable map f : M → M is said to behypercomplex if it is holomorphic with respect to Jα , α = 1, 2, 3. The group ofhypercomplex diffeomorphisms on (M,H) will be denoted by Aut(H).

A Riemannian metric g on a hypercomplex manifold (M,H) is called hyper-Hermitian when g(JαX, JαY ) = g(X, Y ) for all vectors fields X,Y on M, α =1, 2, 3.

� The author was partially supported by CONICET, ESI (Vienna) and FOMEC (Argentina).

2 MARIA LAURA BARBERIS

Given a manifold M with a hypercomplex structure H = {Jα}α=1,2,3 and ahyper-Hermitian metric g consider the 2-forms ωα, α = 1, 2, 3, defined by

ωα(X, Y ) = g(X, JαY ). (1.3)

The metric g is said to be hyper-Kähler when dωα = 0 for α = 1, 2, 3.It is well known that a hyper-Hermitian metric g is conformal to a hyper-Kähler

metric g if and only if there exists an exact 1-form θ ∈ �1M such that

dωα = θ ∧ ωα, α = 1, 2, 3, (1.4)

where, if g = ef g for some f ∈ C∞(M), then θ = df .The reason for considering homogeneous conformally hyper-Kähler and not

homogeneous hyper-Kähler metrics is the following. Any hyper-Kähler metric isRicci flat, so that if we add the homogeneity condition we obtain a flat homoge-neous metric and therefore the corresponding Riemannian manifold is the productof a torus by a Euclidean space with their natural metrics. The classification ofcompact homogeneous locally conformal hyper-Kähler manifolds was carried outin [9]. There are no general results for the noncompact case. It is the aim of thisnote to study the situation when a given hyper-Kähler metric on a simply connectedfour-dimensional manifold admits a homogeneous metric in its conformal class(see Corollary 3.1). We prove the following result:

THEOREM 1.1. Let (M,H , g) be a simply connected hyper-Hermitian 4-mani-fold. Assume that there exists a Lie group G ⊂ I (M, g) ∩ Aut(H) acting simplytransitively on M. Then g is conformally hyper-Kähler.

We conclude that one of the hyper-Kähler metrics constructed by the Gibbons–Hawking ansatz [5] contains a homogeneous hyper-Hermitian metric in its confor-mal class. This hyper-Hermitian metric is not symmetric and has negative sectionalcurvature [1].

As a consequence of Theorem 1.1 and the results in [1] we obtain that thefollowing symmetric Riemannian metrics are conformally hyper-Kähler:

• the Riemannian product of the canonical metrics on R × S3;• the Riemannian product of the canonical metrics on R × RH 3, where RH 3

denotes the real hyperbolic space;• the canonical metric on the real hyperbolic space RH 4.

2. Proof of the Main Theorem

Proof of Theorem 1.1. Since G acts simply transitively on M, then M is dif-feomorphic to G and therefore the hypercomplex structure and hyper-Hermitianmetric can be transferred to G and will also be denoted by {Jα}α=1,2,3 and g, re-spectively. Since G acts by hypercomplex isometries it follows that both {Jα}α=1,2,3

and g are left invariant on G. All such simply connected Lie groups were classified

HYPER-KÄHLER METRICS CONFORMAL TO LEFT INVARIANT METRICS 3

in [1], where it is shown that the Lie algebra g of G is either Abelian or isomor-phic to one of the following Lie algebras (we fix an orthonormal basis {ej }j=1,...,4

of g):

(1) [e2, e3] = e4, [e3, e4] = e2, [e4, e2] = e3, e1 central;(2) [e1, e3] = e1, [e2, e3] = e2, [e1, e4] = e2, [e2, e4] = −e1;(3) [e1, ej ] = ej , j = 2, 3, 4;(4) [e3, e4] = 1

2e2, [e1, e2] = e2, [e1, ej ] = 12ej , j = 3, 4.

Observe that in case (1) above M is diffeomorphic to R×S3 while in the remainingcases it is diffeomorphic to R

4, therefore in all cases any closed form on M is exact.We now proceed by finding in each case a closed form θ ∈ �1g∗ satisfying (1.4).Note that we work on the Lie algebra level since g and ωα are all left invarianton G. Let {ej }j=1,...,4 ⊂ �1g∗ be the dual basis of {ej }j=1,...,4. From now on, wewill write eij ··· to denote ei ∧ ej ∧ · · ·. In all the cases below the 2-forms ωα aredetermined from (1.3) in terms of the hypercomplex structures constructed in [1].

Case 1. The 2-forms ωα are given as follows:

ω1 = −e12 − e34, ω2 = −e13 + e24, ω3 = −e14 − e23.

To calculate dωα we first obtain dej (recall that dσ (x, y) = −σ [x, y] for σ ∈�1g∗):

de1 = 0, de2 = −e34, de3 = e24, de4 = −e23. (2.1)

These equations and the fact that d(σ ∧τ) = dσ ∧τ +(−1)rσ ∧dτ for all σ ∈ �rg∗give the following formulas:

dω1 = −e134, dω2 = e124, dω3 = −e123

from which we conlude that (1.4) holds for θ = e1, which is closed and thereforeexact since G is diffeomorphic to R × S3. We conclude that this hyper-Hermitianmetric, which, as shown in [1], is homothetic to the Riemannian product of thecanonical metrics on R × S3, is conformal to a hyper-Kähler metric.

Case 2. In this case we have the following equations for ωα:

ω1 = e14 − e23, ω2 = −e12 + e34, ω3 = −e13 − e24.

and we calculate

de1 = −e13 + e24, de2 = −e23 − e14, de3 = 0, de4 = 0, (2.2)

dω1 = −2e134, dω2 = −2e123, dω3 = 2e234 (2.3)

so that (1.4) is satisfied for θ = 2e3, which again is closed, so this hyper-Hermitianmetric is also conformal to a hyper-Kähler metric. In this case the hyper-Hermitianmetric is homothetic to the Riemannian product of the canonical metrics on R ×RH 3, where RH 3 denotes the real hyperbolic space.


Case 3. In this case the 2-forms ωα are given as follows:

ω1 = −e12 − e34, ω2 = −e13 + e24, ω3 = −e14 − e23

and a calculation of exterior derivatives gives

de1 = 0, dej = −e1j , j = 2, 3, 4 (2.4)

dω1 = 2e134, dω2 = −2e124, dω3 = −2e123, (2.5)

so that (1.4) is satisfied for θ = −2e1. This hyper-Hermitian metric is homotheticto the canonical metric on the real hyperbolic space RH 4.

Case 4. In this case we have the following equations for ωα:

ω1 = −e12 + e34, ω2 = −e13 − e24, ω3 = e14 − e23

and we calculate

de1 = 0, de2 = −e12 − 12e

34, dej = − 12e

1j , j = 3, 4, (2.6)

dω1 = − 32e

134, dω2 = 32e

124, dω3 = 32e

123, (2.7)

so that (1.4) is satisfied for θ = − 32e

1. This hyper-Hermitian metric is not symmet-ric and has negative sectional curvature (cf. [1]).

Remark 2.1. All the hyper-Hermitian manifolds (M,H , g) considered aboveadmit a connection ∇ such that:

∇g = 0, ∇Jα = 0, α = 1, 2, 3

and the (3, 0) tensor c(X, Y,Z) = g(X, T (Y,Z)) is totally skew-symmetric, whereT is the torsion of ∇. Such a connection is called an HKT connection (cf. [7]). Incase M is diffeomorphic to R×S3 it can be shown that, moreover, the correspond-ing 3-form c is closed, that is, the HKT structure is strong.

3. Coordinate Description of the Hyper-Kähler Metrics

In this section we will use global coordinates on each of the Lie groups consideredin the previous section to describe the corresponding hyper-Kähler metrics. It turnsout that, although these metrics can no longer be G-invariant, they remain invariantunder the action of a codimension 1 normal subgroup of G, that is, they are ofcohomogeneity one. In the terminology of [6], all four hyper-Kähler metrics are ofBianchi type. We will also identify the hyper-Kähler metric in Section 2, Case 4,with one constructed by the Gibbons–Hawking ansatz [5].


Case 1. G = H∗ = GL(1,H) =

{(x −y −z −t

y x −t z

z t x −y

t −z y x

): (x, y, z, t) ∈ R

4 \ {0}}

.

We obtain a basis of left invariant 1-forms on G as follows. Set r2 = x2 + y2 +z2 + t2, r > 0, and & = g−1 dg for g ∈ G, that is,

if g =

x −y −z −t

y x −t z

z t x −y

t −z y x

, then & =

σ1 −σ2 −σ3 −σ4

σ2 σ1 −σ4 σ3

σ3 σ4 σ1 −σ2

σ4 −σ3 σ2 σ1

,

where

σ1

σ2

σ3

σ4

= 1

r2

x y z t

−y x t −z

−z −t x y

−t z −y x

dxdydzdt

.

Then σj , 1 � j � 4, is a basis of left invariant 1-forms on G and it follows fromd& + & ∧ & = 0 that

dσ1 = 0, dσ2 = −2σ3 ∧ σ4, dσ3 = 2σ2 ∧ σ4,

dσ4 = −2σ2 ∧ σ3.

Setting

e1 = 2σ1, e2 = 2σ2, e3 = 2σ3, e4 = 2σ4,

so that {ej }1�j�4 satisfy (2.1), the left-invariant hyper-Hermitian metric is

g = (e1)2 + (e2)2 + (e3)2 + (e4)2 = 4

r2(dx2 + dy2 + dz2 + dt2) (3.1)

that is, g is the standard conformally flat metric on R4 \ {0}, and since the Lee form

is θ = e1 = d(2 log r) the corresponding hyper-Kähler metric is g = e−2 log rg, thatis,

g = 4

r2

((dr)2

r2+ (σ2)

2 + (σ3)2 + (σ4)

2

)

= 4

r4(dx2 + dy2 + dz2 + dt2). (3.2)

Observe that g is the image of the canonical flat metric of R4 by an inversion

centered at the origin and the standard metric on any coordinate quaternionic Hopfsurface is locally conformally equivalent to g (cf. [2]). Moreover, g is of cohomo-geneity one with respect to SU(2).


Case 2. The Lie group G considered in Section 2, Case 2, is the universal cov-ering group of Aff(C), the semidirect product of C by the Abelian multiplicativegroup C

∗ acting on C by the standard representation. We can view Aff(C) as asubgroup of GL(4,R) in the following way:

Aff(C) =

a −b c −d

b a d c

0 0 1 00 0 0 1

: a2 + b2 �= 0

.

Define a product on R4 as follows:

(x, y, z, t)(x′, y′, z′, t ′)= (x + ez(x′ cos t − y′ sin t), y + ez(x′ sin t + y′ cos t), z + z′, t + t ′).

This defines a Lie group structure on R4 that makes it isomorphic to G. The

following 1-forms are left-invariant with respect to the above product:

e1 = e−z cos t dx + e−z sin t dy, e3 = −dz, (3.3)

e2 = −e−z sin t dx + e−z cos tdy, e4 = −dt. (3.4)

These forms satisfy relations (2.2). The hyper-Hermitian metric is therefore givenas follows:

g = (e1)2 + (e2)2 + (e3)2 + (e4)2 = e−2z(dx2 + dy2) + dz2 + dt2

and the Lee form is θ = 2e3 = −2 dz, so that the hyper-Kähler metric becomes

g = e2zg = (dx2 + dy2) + e2z(dz2 + dt2).

Observe that the change of coordinates s = ez gives the following simple form forg on R

+ × R3:

g = dx2 + dy2 + (ds2 + s2 dt2).

This allows us to identify g with the Riemannian product of two flat Kähler metrics:the Euclidean metric on R

2 and the warped product cone metric on R+×R (cf. [3]).

Observe that g is of cohomogeneity one with respect to E(2), that is, g is a Bianchitype VII0 metric.

Case 3. The Lie group G considered in Section 2, Case 3, is the semidirectproduct of R

3 by the aditive group R acting on R3 by t · v = etv, t ∈ R, v ∈ R

3.We endow R

4 with the following product:

(x, y, z, t)(x′, y′, z′, t ′) = (x + etx′, y + ety′, z + etz′, t + t ′)

thereby obtaining a Lie group isomorphic to G with corresponding left-invariant1-forms:

e1 = dt, e2 = e−t dx, e3 = e−t dy, e4 = e−t dz.


The hyper-Hermitian metric is therefore

g = (e1)2 + (e2)2 + (e3)2 + (e4)2 = e−2t (dx2 + dy2 + dz2) + dt2

with corresponding Lee form θ = −2e1 = −2 dt , yielding the following hyper-Kähler metric:

g = e2t g = dx2 + dy2 + dz2 + e2t dt2.

Setting s = et , g is the Euclidean metric ds2 + dx2 + dy2 + dz2 on R+ × R

3.Observe that g is incomplete and it is of cohomogeneity one with respect to R

3,that is, it is a Bianchi type I metric.

Case 4. Let H be the three-dimensional Heisenberg group, that is,

H ={( 1 a c

0 1 b

0 0 1

): a, b, c ∈ R

}.

The Lie group G considered in Section 2, Case 4, is the semidirect product of H

by the aditive group R acting on H by

t ·( 1 a c

0 1 b

0 0 1

)=( 1 e

t2 a etc

0 1 et2 b

0 0 1

).

This is the so-called Damek–Ricci extension of H (see [4]). Consider the followingproduct on R

4:

(x, y, z, t)(x′, y′, z′, t ′) = (x + et2 x′, y + e

t2 y′, z + etz′ + e

t2 xy′, t + t ′)

which yields the Lie group structure of G. It is easily checked that the followingleft-invariant 1-forms satisfy (2.6):

e1 = dt, e2 = 12e

−t (dz − x dy), e3 = e− t2 dx, e4 = e− t

2 dy.

The hyper-Hermitian metric is now obtained as in the above cases:

g = (e1)2 + (e2)2 + (e3)2 + (e4)2

= dt2 + e−t (dx2 + dy2) + 14e

−2t (dz − x dy)2

and the Lee form is θ = − 32 dt , from which we obtain the hyper-Kähler metric as

usual:

g = e32 tg = e

32 t dt2 + e

t2 (dx2 + dy2) + 1

4e− t

2 (dz − x dy)2.

Setting s = 2et2 , g becomes

g = s

2(ds2 + dx2 + dy2) + 1

2s(dz − x dy)2


on R+×R

3, which allows us to identify g with one of the hyper-Kähler metrics con-structed by the Gibbons–Hawking ansatz [5]. The identification is easily obtainedfrom [8], Proposition 1. Observe that g is of cohomogeneity one with respect to theHeisenberg group H , hence it is a Bianchi type II metric.

We can now rephrase Theorem 1.1 as follows, where [h] denotes the conformalclass of h:

COROLLARY 3.1. Let h be a hyper-Kähler metric on a simply connected hy-percomplex 4-manifold (M,H) such that there exist g ∈ [h] and a Lie groupG ⊂ I (M, g) ∩ Aut(H) acting simply transitively on M. Then (M, h) is homo-thetic to either R

4 with the Euclidean metric or one of the following Riemannianmanifolds:

(1) M = R4 \ {0}, h = r−4(dx2 + dy2 + dz2 + dt2),

(2) M = R2 × (R+ × R), h = (dx2 + dy2) + (ds2 + s2 dt2),

(3) M = R+ × R

3, h = ds2 + dx2 + dy2 + dz2,(4) M = R

+ × R3, h = s(ds2 + dx2 + dy2) + s−1(dz − x dy)2.

Acknowledgements

The author wishes to thank the organizers of the program ‘Holonomy Groups inDifferential Geometry’ for their kind invitation to visit the Erwin SchrödingerInstitute, Vienna. She is also grateful to D. Alekseevsky, I. Dotti, L. Ornea, andS. Salamon for useful conversations and the referee for drawing [6] to her attention.

References

1. Barberis, M. L.: Hypercomplex structures on 4-dimensional Lie groups, Proc. Amer. Math. Soc.125(4) (1997), 1043–1054.

2. Boyer, C. P.: A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102(1) (1988),157–164.

3. Boyer, C. P. and Galicki, K.: On Sasakian–Einstein geometry, Preprint.4. Damek, E. and Ricci, F.: Harmonic analysis on solvable extensions of H -type groups, J. Geom.

Anal. 2 (1992), 213–248.5. Gibbons, G. W. and Hawking, S. W.: Gravitational multi-instantons, Phys. Lett. B 78 (1978),

430–432.6. Gibbons, G. W. and Rychenkova, P.: Single-sided domain walls in M-theory, J. Geom. Phys.

32 (2000), 311–340.7. Grantcharov, G. and Poon, Y. S.: Geometry of hyper-Kähler connections with torsion, Comm.

Math. Phys. 213 (2000), 19–37, math.DG 9908015.8. LeBrun, C.: Explicit self-dual metrics on CP2# . . . #CP2, J. Differential Geom. 34(1) (1991),

223–253.9. Ornea, L. and Piccinni, P.: Locally conformal Kähler structures in quaternionic geometry,

Preprint.


9

Embedding Misner and Brill–LindquistInitial Data for Black-Hole Collisions

HSUNGROW CHANNational Pingtung Teachers College, Pingtung, 900-03, Taiwan, R.O.C.e-mail: [email protected]

(Received: 6 November 2001; in final form: 12 July 2002)

Abstract. In this article we consider the isometrical immersions into Euclidean three-space oftwo-dimensional slices of the Misner and the Brill–Lindquist initial data for black-hole collisions.We show negativity of curvature and deduce other geometric properties of the slices. Under theassumption that ends behave strongly like paraboloid of revolution, we prove that Misner and theBrill–Lindquist slices cannot be isometrically immersed into R3. This condition on an end is naturalin general relativity because it holds for each end of a slice of the Schwartzschild metric where it isembedded as a paraboloid of revolution.

Mathematics Subject Classifications (2000): Primary: 53C42; secondary: 83C05.

Key words: isometrical immersion, the Schwartzschild paraboloid end condition.

1. Introduction

Many problems arise in the study of solutions of the Einstein–Maxwell equa-tions of gravitation and electromagnetism in source-free space, because the so-lutions disclose the properties of black holes and cosmos. One problem involvestrying to visualize two-dimensional slices of a solution’s initial data. Considerthe Schwartzschild, Misner and Brill–Lindquist initial data. The initial data isfully specified by a two-dimensional surface. Each surface can be represented byM = (�, ds2), where � is a fundamental domain and ds2 is the metric on �. ThenM is a Riemannian surface. We consider the isometric embedding ofM into R3.

One well-known example is the initial data of the Schwartzschild metric MS =(R2\{(0, 0)}, ds2), where

ds2 =(

1 + ms√x2 + y2

)4

(dx2 + dy2)

and ms > 0. The singularities of the metric are p1 = (0, 0) and p∞ = ∞.ms is themass of the Schwartzschild black hole. MS has the following intrinsic properties.The Gauss curvature K is negative and the total curvature is

∫MSK = −4π . For

the points attending to the singularities p1 and p∞, K → 0. Let r = √x2 + y2

10 HSUNGROW CHAN

and γ = {r = ms} ⊂ MS be a closed geodesic curve that divides MS into twoparts U1, U∞. For each part,

∫K = −2π . MS has the topology of an annulus

and χ(MS) = 0, where χ(M) is the Euler–Poincaré characteristic and χ(M) = 0implies that M is homeomorphic to an annulus.MS can be isometrically embedded into R3. When r → 0 and r → ∞, the

embedding surfaces are two asymptotically flat sheets as a surface of revolution.From an extrinsic point of view, the Gauss map is one-to-one near each singularityand the Gauss image of a point tending toward each singularity converges to onepoint. Each singularity represents an end.

DEFINITION 1.0. We will refer to the Schwartzschild paraboloid end conditionif the Gauss map is one-to-one near each singularity and the Gauss image of a pointtending toward each singularity converges to one point.

The condition allows some perturbations on the shape of surfaces of revolu-tion at infinity and still keeps the phenomena of the asymptotically flatness. Canslices of various initial manifolds be isometrically immersed into R3 under theSchwartzschild paraboloid end condition?

Misner [11] constructed a class of solutions to the Einstein–Maxwell equations.Let

ϕ(x, y) :=n=∞∑n=−∞

1√cosh(x + 2µn)− cos y

, (1)

written in terms of the coordinates (x, y), where µ is a nonzero constant. Sinceϕ(x, y) = ϕ(x + 2k1µ, y + 2k2π), where k1 and k2 are integers, ϕ(x, y) is aperiodic function at x and y. Let T1 = R2/� be a torus which � = 2µZ ⊕ 2πZare the rectangular lattices. Let

�1 = T1\{p} = {(x, y) ∈ R2 | −µ � x � µ,−π � y � π}\{(0, 0)}be the fundamental domain of ϕ(x, y). It is clear that (0, 0) is a singularity inϕ(x, y). LetMMis

1 = (�1, ds2) be a Riemannian surface, where

ds2 = a2ϕ4{dx2 + dy2}, (2)

and a is a nonzero constant. ThenMMis1 is homeomorphic to a one-point punctured

torus.Let T2 = R2/� and � = 4µZ ⊕ 2πZ be another torus. Let

�2 = T2\{p1, p2}= {(x, y) ∈ R2 | −µ � x � 3µ,−π � y � π}\{(0, 0), (2µ, 0)}

be another fundamental domain for ϕ(x, y). Then MMis2 = (�2, ds2) is another

abstract surface and M2 is homeomorphic to a two points punctured torus. In fact,MMis

2 is a double covering surface for MMis1 . We can extend the same idea to have

MISNER AND BRILL–LINDQUIST INITIAL DATA 11

the surfaces MMisl , l > 0. We may call the MMis

l Misner surfaces. Clearly, theEuler–Poincaré characteristic χ(MMis

l ) = 0 − l.Brill and Lindquist (BL) [1, 2, 10] constructed another class of solutions to the

Einstein–Maxwell equations. LetMBLl = (R2, ds2

l ) be the BL surfaces, for which

ds2l = φ4

l {dx2 + dy2}, (3)

and

φl(x, y) := 1 +i=l∑i=1

mi√(x − xi)2 + (y − yi)2

, (4)

and l is a positive integer, mi > 0. The Euler–Poincaré characteristic χ(MBL1 ) =

1 − l. This model represents the collision of black holes with different masses.φl has l + 1 singularities on the plane, including the infinity point of R2. TheSchwartzschild surfaceMS is a special case of the BL surfaces such thatMBL

1 = MS.Price and Romano [13] attempted to propagate the Misner surfaces MMis

1 froma big circle of the Schwartzschild data into the interior. They numerically solvedthe hyperbolic Darboux equation. It turns out that the numerical procedure devel-ops a shock before too much of the surface is reconstructed. From this numericalevidence, we develop a theorem:

THEOREM 1.1. The Misner surfaces MMisl (l > 0) and the BL surfaces MBL

l

(l > 1) cannot be C3 isometrically immersed into R3 with the Schwartzschildparaboloid end condition.

In Sections 2 and 3, we compute the intrinsic properties of the Misner surfacesand the BL surfaces. Their curvatures are negative. Several criteria of isometricimmersions of complete noncompact Riemannian surfaces M into R3 are basedonly on the amount of Gauss curvature. In 1901, D. Hilbert [8] proved that acomplete constant negatively curved surface cannot be C2 isometrically immersedinto R3. N. V. Efimov generalized this theorem in 1964 by proving that no C2

isometric immersion of a complete surface withK � δ < 0 exists ([6]), and thatC2

immersions do not exist for certain surfaces with decaying negative curvature ([7]).Those nonisometric immersion theorems don’t apply on the Misner surfaces andthe BL surfaces, because their curvatures are negative and tending to zero at infinitywith a faster decaying rate.

Complete negatively curved surfaces do exist in R3. Thus, there are other rea-sons to rule out the possibility of C3 isometrical immersions of the Misner surfacesand the BL surfaces with the Schwartzschild paraboloid end condition into R3.In Section 4, we prove Theorem 1.1. The proof uses two restrictions that seemunnecessary, and we hope that someone will be able to eliminate them – especiallythat Gauss map is one-to-one near each singularity. We developed a theorem for aparticular Misner surface MMis

1 (l = 1).

12 HSUNGROW CHAN

Let |B| be the length of the second fundamental form in local orthonormalframe is given by |B| = ∑2

i,j=1 h2ij . Hence, |B|2 = 4|H |2 − 2K, where H =

12(h11 + h22) is the mean curvature.

Let N be a complete, oriented, nonpositively curved surface C2 immersedinto R3. White [18] proved that if

∫N

|B|2 da < ∞, then the Gauss map extendscontinuously to one point at infinity and N is properly immersed near the infinity.

THEOREM 1.2 [3]. Suppose M is a complete, oriented, one ended, nonpositivelycurved Riemannian surface with an isolated set of parabolic points

{p ∈ M : K(p) = 0}.ThenM cannot be C2 isometrically embedded into R3 when∫

M

|B|2 da <∞, (5)

and one end is embedded.

The proof of the theorem applies White’s result. The theorem holds under theassumption that the Gauss map extends continuously to one point at infinity insteadof (5). MMis

1 has only one end and its curvature is negative. The theorem impliesthatMMis

1 cannot be embedded into R3 with the Gauss map extending continuouslyto one point at infinity (one restriction). Theorem 1.2 was generated in [4].

We have a general mathematical theorem arising from the isometrical immer-sions of the Misner surfaces and the BL surfaces. Let M be a complete, non-compact, negatively curved C2 surface isometrically immersed into R3 with finitetotal curvature

∫ |K| da < ∞. Accordingly the Huber theorem [9], M is of finitetopological type, i.e., M is conformally equivalent to a compact Riemann surfacewith finitely many points {p1 · · ·pn} deleted. The complement of some compactset of M is the union of punctured disks corresponding to each pi . We define theend to be the punctured-disk neighborhood Ui of each pi .

DEFINITION 1.3. We classify an end as a bowl if there is a shortest length closedcurve at the end that loops around this end. A bowl whose shortest curve is a closedgeodesic ofM is called a strict bowl. The part between the shortest closed geodesicof a strict bowl and infinity is called the upper bowl. An end that is not a bowl iscalled a horn.

THEOREM 1.4. LetM be a complete, noncompact, negatively curved surface C3

isometrically immersed into R3 with∫ |B|2 da <∞. If the total curvature of each

upper bowl is 2π and the Gauss map is one-to-one near the infinity of each end,then χ(M) = 0.

In a way, Theorem 1.4. is related to Schoen’s theorem [15] from the conditionH ≡ 0 to the condition

∫ |B|2 da = ∫4|H |2 − 2K da <∞.


THEOREM 1.5 [15]. The only complete connected minimal immersions M ⊂ R3

that are regular at infinity and have two ends, are the catenoid.

The Gauss curvature of minimal surfaces are nonpositive. A complete minimalimmersion MH ⊂ R3 is said to be regular at infinity if there is a compact subsetK ⊂ MH such thatMH\K consists of n components U1, . . . , Un such that Ui is thegraph of a function zi with a bounded slope over the exterior of a bounded region insome plane Pi . According to this theorem, the two ends are embedded strict bowlsand χ(M) = 0. Perez and Ros [12] later generalized Schoen’s theorem. The planeand the catenoid are the only properly embedded minimal surfaces with finite totalcurvature and genus zero in R3.

2. Misner Surfaces

We compute the intrinsic geometry of the Misner surfaces in this section. Sinceeach surface MMis

l is a covering space of MMis1 , we can get the same properties for

MMisl as we obtain for MMis

1 . All computations are based on the surface MMis1 . We

quote Romano and Price’s curvature result:

LEMMA 2.1 [13]. The Gauss curvature of the Misner surface MMisl is

K = −1

2a−2ϕ−6

n=∞∑n=−∞

m=∞∑m=−∞

cosh(2µ(m− n))− 1

[m] 32 [n] 3

2

.

Therefore, K < 0 and K → 0 as (x, y) → (0, 0).

In Rn, Bε(p) is an ε Euclidean ball with the center at p. Let us estimate thefunction

ϕ(x, y) :=n=∞∑n=−∞

1√cosh(x + 2µn)− cos y

.

Using a series expansion, for (x, y) ∈ B 12((0, 0)) we obtain

ϕ(x, y) =√

2√x2 + y2

+ c0 −√

2

24

x2 − y2√x2 + y2

+ c1x2 + c2y

2 + (6)

+√

2

5760

7x4 + 38x2y2 + 7y4√x2 + y2

+ · · · , (7)

where

c0 = √2

∞∑n=1

1

sinh(µn),

14 HSUNGROW CHAN

c1 = 3

4√

2

∞∑n=1

cosh(µn)

sinh4(µn)− 3

4√

2

∞∑n=1

cosh2(µn)+ sinh2(µn)

sinh3(µn),

c2 = 1

2√

2

∞∑n=1

−1

sinh3(µn).

Since for |x| < 12

34 � 1√

1 + x � 32 , (8)

√2

2√x2 + y2

� ϕ(x, y) � c√x2 + y2

, (9)

where c > 0 is a constant. We now compute the decay rate of the Gauss curvatureas (x, y) → (0, 0).

LEMMA 2.2. Since |(x, y)| → (0, 0), the Gauss curvature of Misner surfaces isK = O((x2 + y2)

32 ).

Proof. We have the Gauss curvature formula for the conformal metric ds2 =a2ϕ4{dx2 + dy2} to the plane.

K = −+ log aϕ2

a2ϕ4= −2

a2ϕ4+ log ϕ = −2

a2ϕ6{ϕ+ϕ − |∇ϕ|2}.

We consider |(x, y)| → (0, 0) and compute ϕ6, ϕ+ϕ, and |∇ϕ|2 to get the majorterms.

ϕ6 = 1

(x2 + y2)3(8 + O((x2 + y2)

12 )),

ϕ+ϕ = 2

(x2 + y2)2+

√2c0

(x2 + y2)32

+ x2 − y2

6(x2 + y2)2+ 2

√2(c1 + c2)√x2 + y2

+

+ 2c0(c1 + c2)+ O((x2 + y2)12 ),

|∇ϕ|2 = 2

(x2 + y2)2+ x2 − y2

6(x2 + y2)2+ 1

72+ O((x2 + y2)

12 ).

Then,

ϕ+ϕ − |∇ϕ|2

=√

2c0

(x2 + y2)32

− 2√

2(c1 + c2)√x2 + y2

+ 2c0(c1 + c2)− 1

72+ O((x2 + y2)

12 ).

Thus, we have K = O((x2 + y2)32 ). The proof is complete. ✷


THEOREM 2.3 (Efimov’s linear growth criterion [14]). Let M be a complete C2

Riemannian surface. Let dist(x1, x2) be the distance function for x1, x2 ∈ M.If K < 0 and there are constants c1, c2 such that for any x1, x2 ∈ M∣∣∣∣ 1√−K(x1)

− 1√−K(x2)

∣∣∣∣ � c1 dist(x1, x2)+ c2,

then M cannot be C2 isometrically immersed into R3.

We show that Efimov’s linear growth criterion fails to rule out C2 isometric im-mersion of the Misner surfaces because the decay rate of K of the Misner surfacesis faster than the linear decay of the distance function.

We may call a subset S ⊂ M strongly convex if, for any two points q1, q2, in theclosure S of S, there exists an unique minimizing geodesic γ in M joining q1 withq2 whose interior is contained in S. Then, for any p ∈ M, there exists a numberε > 0 such that the geodesic ball Bε(p) is strongly convex ([5, p. 77]).

We show several geodesics inMMis1 .

LEMMA 2.4. Let

γ1 = {(x, π) | −µ � x � µ},γ2 = {(x,−π) | −µ � x � µ},γ3 = {(x, 0) | −µ � x < 0},γ4 = {(x, 0) | 0 < x � µ},γ5 = {(−µ, y) | −π � y � π},γ6 = {(µ, y) | −π � y � π},γ7 = {(0, y) | −π � y < 0},γ8 = {(0, y) | 0 < y � π}.

Then γi , i = 1, . . . , 8, is a geodesic.Proof. The surface is reflection symmetric across the curves γi . This property

implies that they are geodesics. For any point p ∈ γi , let Bε(p) be a stronglyconvex geodesic ball. Assume that γi is not a geodesic at p. Then there must existtwo points q1, q2 ∈ γi and in the ball Bε(p), such that the geodesic γ connectingthem is contained in one side of the γi . The surface is reflection symmetric, sothe reflection γ ∗ of γ is also a geodesic. Let the region between two geodesics beD and the external angles be θ1 and θ2. According to the Gauss–Bonnet theorem,∫DK dA + θ1 + θ2 = 2π . Since the two geodesics cannot be tangent to each

other, we have θi < π . On the other hand, K < 0, so this is a contradiction. Thiscompletes the proof. ✷LEMMA 2.5. Let γβ(t): [0, 2π ] → MMis

1 by γβ(t) = (β cos t, β sin t). Thenl(γβ(t)) = O(1/β).

16 HSUNGROW CHAN

Proof. The line element is aϕ2 dt . Now, |γ ′β(t)| = β and

l(γβ) =∫ 2π

0aϕ2β dt = aβ

∫ 2π

0ϕ2 dθ.

According to (9), we have

1

2(x2 + y2)� ϕ2(x, y) � c2

x2 + y2.

Then

aπ

β� l(γβ) � 2πac2(µ)

β.

Thus, l(γβ(t)) = O(1/β). This completes the proof. ✷We now check Efimov’s linear growth criterion. Along γ8 = {(0, y) | 0 < y

� π} we first fix y0 � π , then let ε→ 0. The distance dist((0, y0), (0, ε)) = O( 1ε).

However,∣∣∣∣ 1√−K((0, y0))− 1√−K((0, ε))

∣∣∣∣ = O

(1

ε1.5

).

This change in the function (√−K)−1 does not admit a linear estimate along

γ8. Specifically, the Gauss curvature decays to zero faster than linear. Therefore,we cannot rule out C3 isometric immersion of the Misner surfaces according to thebehavior of the Gauss curvature.

We now compute the total curvature of the Misner surfaces. Let

ϕ(x, y) = 1√x2 + y2

Y(x, y),

where

Y(x, y) =( √

2√1 + X(x, y)

)+ g(x, y)

√x2 + y2,

X(x, y) = 24 (x

2 − y2)+ 26(x

4 − x2y2 + y4)+ · · · ,and

g(x, y) =∞∑n=1

(1√

cosh(x + 2µn)− cos y+ 1√

cosh(x − 2µn)− cos y

).

g(x, y) is a bounded function such that

0 � g(x, y) � 4√

2

µ2. (10)


LEMMA 2.6. Let r = √x2 + y2. There then exists a positive number c such that∣∣∣∣∂ log Y(x, y)

∂r

∣∣∣∣ � c, (11)

for (x, y) ∈ B 12((0, 0)).

Proof.∣∣∣∣∂ log Y(x, y)

∂r

∣∣∣∣ = 1

Y(x, y)

{−

√2

2

(∂X(x, y)

∂r

)(1 + X(x, y))−

32 +

+ g(x, y) +√x2 + y2

(∂g(x, y)

∂r

)}.

For (x, y) ∈ B 12((0, 0)), by (8) and (10)∣∣∣∣ 1

Y(x, y)

∣∣∣∣ < 2√

2

3.

Also, for (x, y) ∈ B 12((0, 0)), (1 + X(x, y))−

32 and g(x, y) are bounded. Both

X(x, y) and g(x, y) are bounded analytic functions of x, y, so the derivative isalso bounded for (x, y) ∈ B 1

2((0, 0)). We prove (11). ✷

LEMMA 2.7. The total curvature ofMMis1 = (�1, ds2) is∫

MMis1

|K| = 4π.

Proof. Let �ε = �1\{Bε((0, 0))}. �ε the is homeomorphic to a torus with onedisc Bε((0, 0)) removed, then∫

�ε

K dσ =∫�ε

−+ log aϕ2 dx dy = −∫∂�ε

∇ log aϕ2 · ν ds,

where dσ = a2ϕ4 dx dy is the volume element. Now, ∂�ε = ∂Bε((0, 0)). Then,∫�ε

K dσ = −∫∂�ε

∇ log aϕ2 · ν ds

= −∫∂�ε

∂ log aϕ2

∂rds = −2

∫∂�ε

∂ log ϕ

∂rds. (12)

Let r = √x2 + y2. Then,

∂ log ϕ

∂r= ∂ log 1

r

∂r+ ∂ log Y(x, y)

∂r.

By (11) and (12), we have∣∣∣∣∫�ε

K dσ + 2∫∂�ε

∂ log 1r

∂rds

∣∣∣∣ � 2c∫∂�ε

ds.

18 HSUNGROW CHAN

This implies that∣∣∣∣∫�ε

K dσ + 2∫∂�ε

1

rds

∣∣∣∣ � 4cπε.

Since ε → 0 implies �ε → MMis1 , we have

∫MMis

1K = −4π . This completes the

proof. ✷MMisl is a covering space ofMMis

1 , so∫MMislK = −4πl.

LEMMA 2.8. There exists only one shortest closed geodesic curve looping aroundinfinity on MMis

1 such that γ does not go to the infinity point.

Proof. Let �β = MMis1 \{Bβ((0, 0))}. �β is homeomorphic to a torus with an

open disk removed. Then, π1(�β) �= ∅ and γβ = ∂�β is not contractible to a pointin MMis

1 . Let L be the set of all closed curves which are homotopic to γβ . Clearly,L is not empty.

Cartan proved the following theorem ([5, p. 255]). IfM is compact and L is nota constant class, then there exists a shortest closed geodesic of M in the class L.By the theorem, in �β , there is a shortest closed geodesic γ homotopic to γβ .

We now show that if β is small enough, the shortest geodesic cannot reach theboundary γβ . We use the notation �βi = MMis

1 \{Bβi ((0, 0))}, ∂�βi = γβi , andshow that γi is the shortest closed geodesic in �βi homotopic to γβi .

Assume that γi always reaches the boundary γβi on �βi . Let β1 = δβ2 andδ > 2c2(µ)(1+4πc2(µ)). Let γ2, the shortest closed geodesic in�β2, be containedin the region between γβ2 and γβ , where αβ2 = β and α � 1. We may estimate thelength of γ2. The line element is

ds = aϕ2 dt = aϕ2√

dr2 + r dθ2.

Then, by (9)

l(γ2) =∫γ2

ds = a∫γ2

ϕ2√

dr2 + r dθ2 � a

2

∫ β

β2

1

r2dr = a

2

(1

β2− 1

β

).

On the other hand,

l(γ2) � a

2

∫γ2

1

r|dθ | � a

2β

∫ 2π

0|dθ | � aπ

β.

Thus,

l(γ2) � max

{a

2

(1

β2− 1

β

),aπ

β

}.

Since �β1 ⊂ �β2, l(γ1) � l(γ2), γ1 is the shortest closed geodesic in �β1 , sol(γβ1) � l(γ1). By (9), we have

l(γβ1) � a∫ 2π

0

c2(µ)

r2dθ = 2πac2(µ)

β1.


Since l(γβ1) � l(γ1) � l(γ2),

2πac2(µ)

β1� aπ

β= aπ

αβ2= aπδ

αβ1.

Then, α � 2δc−2(µ). Also, we have

2πac2(µ)

β1� a

2

(1

β2− 1

β

)= a

2

(1

β2− 1

αβ

)= a(α − 1)

2αβ2� a(α − 1)

2β1.

Thus, 4πc2(µ) � α − 1 � 2δc−2(µ)− 1. Then

2c2(µ)(1 + 4πc2(µ)) � δ.

That is a contradiction. Thus, the shortest closed geodesic γ inMMis1 does not reach

the infinity. The uniqueness of the shortest closed geodesic γ inMMis1 follows from

the Gauss–Bonnet theorem and K < 0. This completes the proof. ✷Let the shortest closed geodesic curve on the surface MMis

1 be γ . Let U1 be theupper bowl, which is the subset of MMis

1 between γ and the infinity and let U c1 be

the complement of U1 in MMis1 .

LEMMA 2.9.∫U c

1

K =∫U1

K = −2π.

Proof. The closure U c1 is a compact set and is also the complement of U1 ⊂

MMis1 . By the Gauss–Bonnet theorem,∫

U c1

K +∫γ

kg = 2πχ(U c1 ) = −2π.

Then,∫U c

1

K = −2π.

Now,∫MMis

1K = −4π . Therefore,∫

U1

K = −2π.

This completes the proof. ✷Similarly, there is an upper bowl for each singularity ofMMis

l whose total curva-ture is −2π . ForMMis

l , there are l upper bowls and the total curvature of each upperbowl is −2π . Assume that for every positive integer l MMis

l admits C3 isometricimmersions in R3. The total curvatures ofMMis

l are finite.

20 HSUNGROW CHAN

3. Brill–Lindquist Surfaces

We now compute the intrinsic properties of

MBLl = (R2, ds2

l = φ4l {dx2 + dy2}).

Let p1, . . . , pl, p∞ be the singularities of

φl(x, y) := 1 +i=l∑i=1

mi√(x − xi)2 + (y − yi)2

.

LEMMA 3.1. For each MBLl , K < 0 and K → 0 as points tending to each

singularity pi .Proof.

K = −2φ−4[+ log φ] = −2φ−6[φ+φ − |∇φ|2].We show that φ+φ − |∇φ|2 > 0.

φ+φ =i=l∑i=1

mi

((x − xi)2 + (y − yi)2)3/2 +

+i=l∑i=1

j=l∑j=1

mimj

((x − xi)2 + (y − yi)2)1/2((x − xj )2 + (y − yj )2)3/2 .

|∇φ|2 =i=l∑i=1

j=l∑j=1

mimj ×

×(

(x − xi)(x − xj )((x − xi)2 + (y − yi)2)3/2((x − xj )2 + (y − yj )2)3/2 +

+ (y − yi)(y − yj )((x − xi)2 + (y − yi)2)3/2((x − xj )2 + (y − yj )2)3/2

).

Thus,

φ+− |∇φ|2 =i=l∑i=1

mi

((x − xi)2 + (y − yi)2)3/2 +

+∑i<j

mimj[(xi − xj )2 + (yi − yj )2

]> 0.

For each pi , K → 0 as (x, y) → pi . This completes the proof. ✷The local behavior of the φ of the BL surfaces near each singularity pi is similar

to the behavior of the ϕ of the Misner surfaces. By comparing {(1), (2), (6)} and{(3), (4)}, we find that the major terms of the two series are

c0 +√

2√x2 + y2

and 1 + mi√(x − xi)2 + (y − yi)2

.


Therefore, geometric properties of the ends of the Misner surfaces and the BLsurfaces are similar. On each end, the decay rate of K to 0 of the BL surfacesis faster than the linear decay of the distance function, so Efimov’s linear growthcriterion fails to apply this case.

According to the argument in Lemma 2.8, we have the following lemma:

LEMMA 3.2. For each pi , there exists a closed geodesic curve γi on MBLl that

loops around pi once, and does not reach pi , and γi ∩ γj = ∅.

The fundamental domain of MBLl is R2\{p1, . . . , pl}. Let Ui be the region be-

tween γi to pi . We use the same notation for curves on MBLl and on R2 where no

confusion arises. The area element of MBLl is dA = φ4 dx dy and the line element

ofMBLl is ds, and the line element of R2 is dt . For convenience, let

φ = φl = 1 +l∑i=1

mi

r − pi =(

l∏i=1

1

r − pi

)8,

where 8 is a polynomial of r. Now,

∂ log φ

∂r=

l∑i=1

1

r − pi + ∂ log8

∂r. (13)

LEMMA 3.3.∫Ui

K dA = −2π.

Proof. Let {ηj } be a sequence of closed curves looping around pi once andconverging to pi as j → ∞. Let �j be the region between γi and {ηj }.∫

�j

K dA = −2

{∫γi

∂ logφ

∂rds +

∫ηj

∂ logφ

∂rds

}. (14)

We estimate the first term of the right-hand part of the equation above. Since γiis a closed geodesic curve,

0 =∫γi

kg ds = −∫γi

k∗g dt − 2

∫γi

∂ logφ

∂rds,

where k∗g is the curvature of γi on R2. Then,

2∫γi

∂ log φ

∂rds = ±2π,

and the sign depends on the orientation. For γ∞,∫γ∞

∂ log φ

∂rds = π.

22 HSUNGROW CHAN

Let Ui = ⋃�j . As

{ηj } → p∞,∂ logφ

∂r→ 0,

then (14) becomes∫U∞ K dA = −2π .

For 1 � i � l,∫γi

∂ logφ

∂rds = −π.

If ηj is close enough to pi ,∫ηj

∂ log φ

∂rds =

∫ηj

1

r − pi +∫ηj

81,

where 81 is bounded. By taking the limit, we have

limj→∞

∫ηj

∂ log φ

∂rds = 2π.

Thus, as j → ∞ (14) becomes∫Ui

K dA = −2π.

This completes the proof. ✷By Lemma 3.3, each MBL

l has finite total curvature.

4. Proof of Theorem 1.1

We consider the relationship between the geometry and the topology for acomplete, noncompact, negatively curved surface M with finite total curvature∫ |K| da < ∞. We define the index of an end and apply the index formula toM asa Poincaré index formula for a compact surface. By following Verner’s paper [16],we define the index of the end corresponding to the direction field (line field). Letγ : I → M ⊂ R3 be a closed curve parameterized by arclength s ∈ I loopingaround the end, positively oriented relative to the end. Let T (s) = γ ′(s) be theunit tangent vector of γ . Let ν be a continuous nonvanishing direction field on Mand along γ (s). Let α be the angle from the vectors T (s) to ν(s), where ν(s) isa continuous vector field representing ν. Let +α be the increment of α(s) as γ istraversed around once.

DEFINITION 4.1. We define the index of an endU corresponding to the directionfield ν using I (U, ν) = 1 + (+α/π).


There are four natural directional fields for negatively curved surfaces. Two arethe principal directions and the other two are the asymptotic directions. The indexof any one of these is the same at each end. We call this common index the index ofthe end. The index of an end corresponding to the direction field η is independentof the loops chosen. Proofs of this can be found in Verner’s paper. Let N (s) be theunit normal vector to the surface at the point γ (s) ∈ M, and b(s) = T (s) × N (s)be the unit co-normal of γ . By differentiating the vectors {T , b,N }, we have theFrenet frame adapted to the surface:

T ′ = kgb + knN , b′ = −kgT + τgN , N ′ = −knT − τgb,where τg is the torsion of γ , kg is the geodesic curvature of γ , and kn is the co-normal curvature of γ . Let g be the Gauss map, and k∗

g be the geodesic curvatureof the image g(γ (s)) on S2. Let ρ be the arclength along the image g(γ (s)) on S2.

We find the next lemma in Verner’s paper [16] and call it the Verner lemma. Thepaper is in Russian, so for convenience we verify it here. The lemma connects thelocal geometry to topology as in the Gauss–Bonnet theorem. In the paper, Vernerclassified the number of possible horn and bowl ends for completely connectedsmoothly embedded surfaces into R3 with a global one-to-one Gauss map.

LEMMA 4.2 [16]. Let γ be a closed C3-curve on a C3-surface M which loopsaround an end U and is parameterized by arclength. If N ′(s) �= 0 at every pointon γ , then∫

g(γ (s))

k∗g dρ =

∫γ (s)

kg ds + 2π(I (U, η)− 1).

Proof. N ′(s) is a continuous nonvanishing vector field along γ . Let α(s) be theangle between T (s) and N ′(s) and+α be the increments of α(s) along γ (s) goingaround once.

We show that

dα

ds= knτ

′g − k′

nτg

k2g + τ 2

g

. (15)

Since N ′ = −knT −τgb, tanα(s) = τg/kn, and differentiating both sides, we have

sec2 αdα

ds= knτ

′g − k′

nτg

k2n

.

Since

sec2 α = (k2g + τ 2

g )

k2n

,

we have (15).

24 HSUNGROW CHAN

On S2, let

e1 = N ′(s)|N ′(s)| , e2 = N × e1.

Now, ρ is the arclength along the image g(γ (s)) on S2, so

dρ

ds= |N ′(s)| =

√k2n + τ 2

g .

For this setting, the line element on S2 for g(γ (s)) is

dρ = |N ′(s)|ds =√k2n + τ 2

g ds.

Now, along g(γ (s)),

k∗g = de1

dρ· e2 =

(de1

ds

ds

dρ

)· e2

= 1

|N ′(s)|(

N ′′(s)|N ′(s)| − dN ′(s)

ds

N ′(s)|N ′(s)|3

)· e2 = N ′′(s) · e2

|N ′(s)|

= N ′′(s) · N ′(s)|N ′(s)| × N

|N ′(s)|2 = |(N (s),N ′(s),N ′′(s))||N ′(s)|3 .

According to the Frenet frame,

|(N (s),N ′(s),N ′′(s))| = kg(k2n + τ 2

g )+ knτ ′g − k′

nτg.

Therefore,

k∗g = kg√

k2n + τ 2

g

+ knτ′n − k′

nτg

(k2n + τ 2

g )32

. (16)

Then, multiplying by√k2n + τ 2

g and taking the line integral of (16) along γ (s), we

have ∫g(γ (s))

k∗g dρ =

∫γ (s)

k∗g

√k2n + τ 2

g ds =∫γ (s)

kg ds +∫γ (s)

knτ′g − k′

nτg

k2n + τ 2

g

ds.

By (15), the second integral is∫γ (s)

knτ′g − k′

nτg

k2n + τ 2

g

ds =∫γ (s)

dα = 2π(I (U, ν)− 1).

This completes the proof. ✷Verner [17] applied the lemma to prove the Poincaré index formula for com-

plete noncompact surfaces in which K < 0 and∫ |K| da < ∞ which states the

following theorem:


THEOREM 4.3 [16]. If M is a complete, noncompact, negatively curved C2 sur-face with

∫ |K| da <∞, then

2χ(M) =n∑i=1

(I (Ui, η)− 1),

where I (Ui, η) is the index of the end Ui corresponding to a direction field η.

LEMMA 4.4. Let γ be the shortest closed geodesic and Dγ be the upper bowl ofa strict bowl U ⊂ R3. If

∫DγK = −2π , and the Gauss map of the upper bowl is

one-to-one near infinity and the Gauss image of the upper bowl converges to onepoint, then I (U, η) = 1.

Proof. Consider a sequence {γi} of closed disjoint curves that are homotopic toγ and γi → p (the infinity point) as i → ∞. Let the region between γ and γi beDi . Di → Dγ as i → ∞. According the Gauss–Bonnet theorem,∫

Di

K +∫γi

kg +∫γ

kg = 2πχ(Di) = 0.

Therefore,∫Di

K = −∫γi

kg. (17)

Thus,∫γikg → 2π as i → ∞.

Let k∗g be the geodesic curvature of the image of γi in the Gauss map g. Accord-

ing to the Verner formula,∫g(γi)

k∗g =

∫γi

kg + 2π(I (U, ν)− 1). (18)

Since the Gauss map ofU converges to a point at infinity and the Gauss map is one-to-one near infinity, as i increase, g(γi) is a closed curve homotopic to an equatorwith respect to the limiting Gauss image of infinity on S2. Let D∗

i be the regionon S2 from the equator to g(γi). By the Gauss–Bonnet theorem

Area(D∗i ) =

∫D∗i

dA∗ =∫g(γi)

k∗g. (19)

Then∫g(γi)

k∗g → 2π as i → ∞. Now combining (17), (18) and (19), we have∫

D∗i

dA∗ =∫g(γi)

k∗g =

∫γi

kg + 2π(I (U, ν)− 1).

I (U, ν) is an integer. As i → ∞,∫γikg → 2π and

∫g(γi)

k∗g → 2π . Therefore,

I (U, ν) = 1. This completes the proof. ✷

26 HSUNGROW CHAN

By way of contradiction, we prove Theorem 1.1. Assume that the Misner sur-faces MMis

l (l > 0) and the BL surfaces MBLl (l > 1) can be C3 isometrically

immersed into R3 with the Schwartzschild paraboloid end condition. In Sections 2and 3, we show that the curvature of each surface is negative and each surfacehas finite total curvature. For those surfaces, each singularity corresponds to anend U . There we also show that there exists a shortest closed geodesic curve γlooping around U . Let Dγ be the upper bowl of U . We show that

∫DγK = −2π .

By Lemma 4.4, for each end I (U) = 1 with the Schwartzschild paraboloid endcondition. By Theorem 4.3, the Euler–Poincaré characteristic of each surface iszero. They are contradictions to χ(MMis

l ) = 0 − l and χ(MBLl ) = 1 − l (l > 1).

Therefore, those surfaces cannot be C3 isometrically immersed into R3 with theSchwartzschild paraboloid end condition.

Now, we prove Theorem 1.4. Assume that M is a complete, noncompact, neg-atively curved surface C3 isometrically immersed into R3 with

∫ |B|2 da < ∞.The total curvature of each upper bowl is 2π and the Gauss map is one-to-one nearthe infinity of each end of M. It follows from a theorem of White [18] that M isproperly immersed near each end and the Gauss map ofM extends continuously toone point at infinity. For each upper bowl U of M,

∫UK = −2π . By Lemma 4.4,

the index of each end I (U) = 1.By Verner’s Theorem,

2χ(M) =n∑i=1

(I (Ui)− 1) = 0.

This completes the proof of Theorem 1.4.

Remark 4.5. It is interesting to know into what three-dimensional space formthe Misner surfaces MMis

l and the BL surfaces MBLl can be isometrically embed-

ded? What topological and geometric properties would be satisfied by the spaceform?

Does there exist a ‘slice’ in any initial data of solutions to the Einstein–Maxwellequations such that the curvature is negative, like curvatures of the slices of theSchwartzschild, the Misner and the Brill–Lindquist initial data?

Acknowledgement

It is a pleasure to thank Professor Andrejs Treibergs for discussions. This work waspartially supported by the National Science Council of Taiwan.

References

1. Abrahams, A. and Price, R.: Black hole collisions from Brill–Lindquist initial data: predictionsof perturbation theory, Phys. Rev. D 53 (1996), 1972–1976.


2. Brill, D. and Lindquist, R.: Interaction energy in geometrostatics, Phys. Rev. 131 (1963)471–476.

3. Chan, H.: Nonexistence of isometric immersions of surfaces with nonpositive curvature andone embedded end, Manuscripta Math. 102 (2000), 177–186.

4. Chan, H. and Treibergs, A.: Nonpositively curved surfaces in R3, J. Differential Geom. 57(2001), 389–407.

5. Do Carmo, M. P.: Riemannian Geometry, Birkhäuser, Boston, 1992.6. Efimov, V. N.: Impossibility of a complete regular surface in Euclidean 3-Space whose

Gaussian curvature has a negative upper bound (Russian), Dokl. Akad. Nauk SSSR 150 (1963),1206–1209; Engl. transl.: Soviet Math. Dokl. 4 (1963), 843–846.

7. Efimov, V. N.: Differential criteria for homeomorphism of certain mappings with application tothe theory of surfaces (Russian), Mat. Sb. Nov. Ser. 76 (1968); Engl. transl.: Sb. Math. USSR 5(1968), 475–488.

8. Hilbert, D.: Über Flächen von konstanter Gausscher Krümmung, Transl. Amer. Math. Soc. 2(1901), 87–99.

9. Huber, A.: On subharmonic functions and differential geometry in the large, Comment. Math.Helv. 32 (1957), 13–72.

10. Lindquist, R.: Initial-value problem on Einstein–Rosen manifold, J. Math. Phys. 4 (1963), 938–950.

11. Misner, C.: Wormhole initial conditions, Phys. Rev. 118 (1960), 1110–1111.12. Perez, J. and Ros, A.: Some uniqueness and nonexistence theorems for embedded minimal

surfaces, Math. Ann. 18 (1993), 513–525.13. Price, R. and Romano, J.: Embedding initial data for black-hole collisions, Classical Quantum

Gravity 12 (1995), 875–893.14. Rozendorn, E.: Surfaces of negative curvature, Encyclop. Math. Sci. 48 (1991), 89–180.15. Schoen, R.: Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential

Geom. 18 (1983), 791–809.16. Verner, A.: Topological structure of complete surfaces with nonpositive curvature which have

one to one spherical mappings, Vestn. LGU 20 (1965), 16–29 (Russian).17. Verner, A.: Tappering saddle surfaces, Sibirsk. Mat. Zh. 11 (1968), 567–581.18. White, B.: Complete surface of finite total curvature, J. Differential Geom. 26 (1987), 315–326.


29

Geometrical Aspects of Spectral Theory andValue Distribution for Herglotz Functions

S. V. BREIMESSER� and D. B. PEARSON��Department of Mathematics, University of Hull, Hull HU6 7RX, England.e-mail: {s.v.breimesser, d.b.pearson}@maths.hull.ac.uk

(Received: 14 December 2001)

Abstract. In this paper we show how spectral theory for Herglotz functions and differential operatorsis related to and dependent on the geometrical properties of the complex upper half-plane, viewedas a hyperbolic space. We establish a theory of value distribution for Lebesgue measurable functionsf : R → R and introduce the value distribution function associated with any given Herglotz func-tion F . We relate the theory of value distribution for boundary values of Herglotz functions to thedescription of asymptotics for solutions of the Schrödinger equation on the half-line. We establishtwo results which play a key role in understanding asymptotic value distribution for Schrödingeroperators with sparse potentials, and its implications for spectral theory.

Mathematics Subject Classifications (2000): 47E05, 34L05, 81Q10.

Key words: Herglotz functions, hyperbolic geometry, m-function, Schrödinger operator, spectraltheory, value distribution.

1. Introduction

A principal aim of this paper is to exhibit the degree to which spectral theoryis related to, and dependent on, the geometrical properties of the complex upperhalf-plane, viewed as a hyperbolic space, and mappings of this space by analyticfunctions.

A second aim is to establish the theory of value distribution for any real-valuedLebesgue measurable function f , treating as an important special case the valuedistribution function associated with any given Herglotz function F , and to de-rive estimates of boundary behaviour and associated limiting value distribution forHerglotz functions.

The present paper is a sequel to [1].We begin by establishing the notation and background for the study of spectral

theory within this context.Let a potential function V (x) on [0,∞) be given, with V real-valued and in-

tegrable over bounded subintervals of [0,∞). Associate with V the differential

� Work completed during the tenure of a University of Hull Open Scholarship.�� Partially supported by EPSRC.

30 S. V. BREIMESSER AND D. B. PEARSON

expression τ = −(d2/dx2) + V . Then τ may be used to define the self-adjointoperator T = −(d2/dx2) + V , acting in L2(0,∞), subject to Dirichlet boundarycondition at x = 0.

We shall normally denote by f (x, λ), in the case of real spectral parameter λ,and by f (x, z) where the spectral parameter z is complex with z ∈ C+, respectivesolutions of the Schrödinger equations

−d2f (x, λ)

dx2+ V (x)f (x, λ) = λf (x, λ), (1)

−d2f (x, z)

dx2+ V (x)f (x, z) = zf (x, z), (2)

in each case on the interval 0 � x < ∞.Define particular solutions u(x, λ), v(x, λ) for λ ∈ R, and correspondingly

u(x, z), v(x, z) in the case z ∈ C+, subject to initial conditions

u(0, λ) = 1, u′(0, λ) = 0,

v(0, λ) = 0, v′(0, λ) = 1.(3)

Assuming limit-point case at infinity (see [2]), the Weyl–Titchmarsh m-functionm(z) for all z ∈ C+, is defined by the condition that

u(·, z)+m(z)v(·, z) ∈ L2(0,∞). (4)

The Weyl–Titchmarsh m-function is a Herglotz function, i.e. analytic in the upperhalf-plane, with strictly positive imaginary part (see [3]).

In addition we will be interested in the m-function related to the differentialexpression τ = −(d2/dx2) + V , where V (x) is defined on the truncated interval[N,∞), for any N > 0. Taking for simplicity the case of Dirichlet boundary condi-tion at x = N , we may define the self-adjoint operator T N = −(d2/dx2)+V actingin L2(N,∞) subject to f (N) = 0. Corresponding solutions uN(x, z), vN(x, z) ofEquation (2), with z ∈ C+, may be defined subject to initial conditions

uN(N, z) = 1, (uN)′(N, z) = 0,

vN(N, z) = 0, (vN)′(N, z) = 1.

The m-function mN(z) for the interval (N,∞) with Dirichlet boundary conditionat x = N is determined by the condition that

uN(·, z)+mN(z)vN(·, z) ∈ L2(N,∞) (z ∈ C+).

Note that mN(z) is the standard m-function for the Dirichlet Schrödinger operator−(d2/dx2)+ V (x +N) acting in L2(0,∞).

Given any Herglotz function F , a corresponding right-continuous, nondecreas-ing function ρ may be defined uniquely up to an additive constant by the so-calledHerglotz representation (see [4])

F(z) = a + bz +∫ ∞

−∞

(1

t − z− t

t2 + 1

)dρ(t), (5)

SPECTRAL THEORY AND VALUE DISTRIBUTION 31

where ρ satisfies the integrability condition∫ ∞−∞ 1/(t2 + 1) dρ(t) < ∞. The func-

tion ρ may then be used to define a Lebesgue–Stieltjes measure µ = dρ(t).Such a measure may be defined in particular for the Weyl–Titchmarsh m-functionm(z), in which case the measure µ = dρ carries all of the spectral informationfor the Dirichlet operator T , in the sense that T is unitarily equivalent to themultiplication operator f (λ) → λf (λ) in the Hilbert space L2(R; dρ). In thatcase the Lebesgue–Stieltjes measure µ = dρ is called the spectral measure of theoperator T .

The boundary value F+(λ), for λ ∈ R, of a Herglotz function F(z) is definedby F+(λ) = limε→0+ F(λ+ iε) and exists for almost all λ ∈ R.

Define the angle θ(z, S) subtended at the point z ∈ C+ by the Borel subset Sof R by

θ(z, S) =∫S

Im1

t − zdt. (6)

For complex argument z ∈ C+, we define ω(·, S;F) by

ω(z, S;F) := 1

πθ(F (z), S),

and for real λ we have

ω(λ, S;F) = limδ→0+ ω(λ+ iδ, S;F). (7)

For almost all λ ∈ R, and given S, we have

ω(λ, S;F)= 1, if F+(λ) is real and F+(λ) ∈ S;

= 0, if F+(λ) is real and F+(λ) ∈ S;

= 1

πθ(F+(λ), S), if ImF+(λ) > 0.

The idea of value distribution for any real-valued Lebesgue measurable functionf (λ) is the following.

For Borel subsets S of R2, the measure M0 given by M0(S) = |{λ ∈ R;(λ, f (λ)) ∈ S}| (where |·| denotes Lebesgue measure) describes the distribution ofpoints (λ, f (λ)) of the graph of f . In the special case that S = A×S is the productof a pair of Borel subsets A, S of R, we shall write M0(A× S) = M(A, S). Then

M(A, S) = |{λ ∈ A;f (λ) ∈ S}| = |A ∩ f −1(S)|.Thus M: (A, S) → M(A, S) assigns an extended real nonnegative number topairs of Borel subsets A, S of R and has the properties

(i) A → M(A, S) defines a measure on Borel subsets of R, for fixed S;S → M(A, S) defines a measure on Borel subsets of R, for fixed A;


(ii) M(A,R) = |A|, hence the measure A → M(A, S) is absolutely continuouswith respect to Lebesgue measure;

In the following definition, we shall assume in addition that

(iii) the measure S → M(A, S) is absolutely continuous with respect to Lebesguemeasure.

DEFINITION 1. Any mapping (A, S) → M(A, S), where A, S are Borel subsetsof R, and satisfying properties (i)–(iii) above, will be called a value distributionfunction.

Not all value distribution functions M are of the form M(A, S) = |A∩f −1(S)|for some measurable function f : R → R, i.e. there may be no function f forwhich M(A, S) describes the distribution of values. However, the definition ofvalue distribution adopted here allows for the more general situation that M maydescribe a limiting value distribution for a sequence {fn} of functions, in the sensethat M(A, S) = limn→∞ |A ∩ f −1

n (S)|.We now consider value distribution for Herglotz functions. To any Herglotz

function F one may associate in a natural way a value distribution function Mdefined by

M(A, S) =∫A

ω(λ, S;F) dλ, (8)

where ω(λ, S;F) is given by (7) (for more detail, see [1]).

DEFINITION 2. We shall refer to the function M, which is defined by Equa-tion (8), with ω(λ, S;F) given by (7), as the associated value distribution functionfor the Herglotz function F .

Since M and related functions are dependent on F , we shall often indicate thisexplicitly, by writing M(A, S;F), and so on.

The theory of value distribution for boundary values of analytic functions can beused to describe asymptotics of solutions of the Schrödinger equation. If we con-sider the Dirichlet Schrödinger operator T = −(d2/dx2)+ V , acting in L2(0,∞),where V is an arbitrary locally integrable potential giving rise to absolutely contin-uous spectrum, the large x asymptotic behaviour of the solution v(x, λ) of Equa-tion (1), for λ in the support of the absolutely continuous part µa.c. of the spectralmeasure µ, is linked to the spectral properties of this measure. These in turn are de-termined by the boundary value of the Weyl–Titchmarsh m-function. The large Nvalue distribution for the logarithmic derivative v′(N, λ)/v(N, λ) of the solution vof (1) approaches the associated value distribution of the Herglotz function mN(z)

in the limit N → ∞. More precisely, if A is an arbitrary measurable subset of theessential support of µa.c., then

limN→∞

{∣∣∣∣{λ ∈ A; v′(N, λ)v(N, λ)

∈ S

}∣∣∣∣ − 1

π

∫A

θ(mN+(λ), S) dλ

}= 0. (9)


For a proof of this result, see [1].For sparse potentials (defined in Section 5), a comparison of the two expressions

in (9),∣∣∣∣{λ ∈ A; v′(N, λ)v(N, λ)

∈ S

}∣∣∣∣ and1

π

∫A

θ(mN+(λ), S) dλ,

with

1

π

∫A

θ(i√λ,−S) dλ and

1

π

∫A

θ(i√λ, S) dλ,

respectively (see Equations (37) and (38)) has important applications to spectraltheory. These equations imply the absence of absolutely continuous measure forλ < 0, and may also be used to prove, for various classes of sparse potentials, thatthe spectral measure for λ > 0 is purely singular. We shall provide further detailsin Section 5.

The paper is organised as follows: In Section 2, we exhibit the connection be-tween angle subtended and hyperbolic metric for C+. Rather than using hyperbolicmetric directly, we rely on an estimate of separation γ , defined by Equation (10),of points in C+. We relate γ to hyperbolic metric in Equation (13), and to anglesubtended in Equation (16). We show that γ is nonincreasing under any Herglotzmapping, and that any Möbius transformation mapping the upper half-plane intoitself leaves γ invariant (Equations (22) and (23)).

In Theorem 1 of Section 3, we prove for any Borel subset A of R havingfinite measure, that the associated value distribution function M(A, S;F) for aHerglotz function F is the limit, as δ approaches zero through positive values, ofthe associated value distribution functions M(A, S;Fδ) for the Herglotz functionsFδ(z) = F(z + iδ). For fixed A, we show that this limit is uniform over allHerglotz functions F and all subsets S of R. For the special case of A a finiteinterval (a, b) we give, in Corollary 1, a precise expression for EA(δ), where|M(A, S;F) − M(A, S;Fδ)| � EA(δ).

In Section 4 we show that the estimate EA(δ) of Theorem 1 in the general case isfinite if and only if either |A| < ∞ or |Ac| < ∞, whereAc denotes the complementof A.

In Section 5 we apply the theory to sparse potentials.Section 6 summarises some further extensions and developments of the ideas

presented in the paper.

2. Estimate of Separation γ (·, ·) of Points in the Upper Half-plane

The angle subtended by a Borel set S ⊂ R at a point z ∈ C+ was given in (6) by

θ(z, S) =∫S

Im1

t − zdt.


For z1, z2 ∈ C+, we can expect θ(z1, S) to be close to θ(z2, S) if z1 is close to z2,unless z1 or z2 approaches the real axis. To give a quantitative expression to thisstatement, define an estimate of separation γ (·, ·) of points in the upper half-planeC+ by

γ (z1, z2) := |z1 − z2|√Im z1

√Im z2

(z1, z2 ∈ C+). (10)

Although γ is positive-definite and symmetric, γ is not a metric because the trian-gle inequality is not satisfied (as a counterexample take z1 = 1 + i, z2 = 1 + 10i,z3 = 1 + 11i).

If r1, r2 are two points in the unit disc D = {r ∈ C; |r| < 1}, then the hyperbolicmetric (or non-Euclidean metric) d(r1, r2) in D is defined by (cf. [5])

dD(r1, r2) = minC

∫C

|dr|1 − |r|2 ,

where the minimum is taken over all curves C in D from r1 to r2. As a consequence,we have (cf. [5]):

tanh(dD(r1, r2)) = |r1 − r2||1 − r1r2| . (11)

If τ : D → C+ is a Möbius transformation, then the hyperbolic metric (or Poincarémetric) in C+ is defined by (cf. [5])

dC+(τ (r1), τ (r2)) = dD(r1, r2). (12)

PROPOSITION 1. The estimate of separation γ (·, ·), given by (10), is given interms of hyperbolic metric in C+ by

γ (z1, z2) = 2 sinh(dC+(z1, z2)). (13)

Proof. Let z1 = a + ib and z2 = c + id be two arbitrary points in C+.The function τ : D → C+ given by τ(r) = (r + 1)/(ir − i) is a Möbius

transformation, with inverse function τ−1: C+ → D given by τ−1(z) = (iz + 1)/(iz − 1). For z1, z2 ∈ C+ we thus have, using (12),

dC+(z1, z2) = dD(τ−1(z1), τ

−1(z2)). (14)

Therefore, using Equations (11) and (14), we may verify that

tanh(dC+(z1, z2)) = tanh(dD(τ−1(z1), τ

−1(z2))) = |τ−1(z1)− τ−1(z2)||1 − τ−1(z1)τ−1(z2)|

=√a2 − 2ac + c2 + b2 − 2bd + d2

√a2 − 2ac + c2 + b2 + 2bd + d2

. (15)


The numerator in (15) can be rewritten as |z1 − z2| and the denominator as√|z1 − z2|2 + 4 Im z1 Im z2. Hence

tanh(dC+(z1, z2)) = |z1 − z2|√|z1 − z2|2 + 4 Im z1 Im z2

= 1√1 + 4

γ 2(z1,z2)

=γ (z1,z2)

2√1 + γ 2(z1,z2)

4

,

⇔sinh(dC+(z1, z2))√

1 + sinh2(dC+(z1, z2))

=γ (z1,z2)

2√1 + (

γ (z1,z2)

2 )2.

Thus

γ (z1, z2) = 2 sinh(dC+(z1, z2)),

which verifies (13). ✷PROPOSITION 2. The estimate of separation γ (·, ·), given by (10), may be ex-pressed in terms of angle subtended, given by (6), as

γ (z1, z2) = supS

|θ(z1, S)− θ(z2, S)|√θ(z1, S)

√θ(z2, S)

, (16)

where the supremum is taken over all Borel subsets of R having positive Lebesguemeasure.

Proof. Using the definition of angle subtended we arrive at

|θ(z1, S)− θ(z2, S)|√θ(z1, S)

√θ(z2, S)

= | ∫S

Im( z1−z2(t−z1)(t−z2)

) dt|√∫S

Im 1t−z1

dt√∫

SIm 1

t−z2dt

= | ∫S

Im( z1−z2(t−z1)(t−z2)

) dt|√

Im z1

√∫S

1|t−z1|2 dt

√Im z2

√∫S

1|t−z2|2 dt

.

We have, in the numerator,∣∣∣∣∫S

Im

(z1 − z2

(t − z1)(t − z2)

)dt

∣∣∣∣�

∫S

∣∣∣∣Im(z1 − z2

(t − z1)(t − z2)

)∣∣∣∣ dt

�∫S

∣∣∣∣ z1 − z2

(t − z1)(t − z2)

∣∣∣∣ dt � |z1 − z2|√∫

S

1

|t − z1|2 dt

√∫S

1

|t − z2|2 dt ,


which leads to

|θ(z1, S)− θ(z2, S)|√θ(z1, S)

√θ(z2, S)

� γ (z1, z2). (17)

The second inequality above depended on∣∣∣∣Im(z1 − z2

(t − z1)(t − z2)

)∣∣∣∣ �∣∣∣∣ z1 − z2

(t − z1)(t − z2)

∣∣∣∣. (18)

To complete the proof of (16), we first show how to find a t0 ∈ R for which equalityholds in (18). Since | Im z| = |z| ⇔ Re z = 0, we need to consider the equation

Re

(z1 − z2

(t − z1)(t − z2)

)= 0. (19)

Setting z1 = a + ib, z2 = c + id, (19) can be rewritten as

t2(a − c)+ t (c2 + d2 − (a2 + b2))+ (c(a2 + b2)− a(c2 + d2)) = 0.

The discriminant of this quadratic equation in t can be evaluated as

(a − c)4 + (b2 − d2)2 + 2(b2 + d2)(a − c)2 � 0.

Therefore there exists a t0 ∈ R for which (19) is satisfied, so that equality holdsin (18).

Next, suppose a sequence (Sn) of Borel subsets of R can be found such that foreach z = z1, z2, with z1, z2 fixed in C+, we have

limn→∞

{θ(z, Sn)

Im 1t0−z |Sn|

}= 1, (20)

where again | · | denotes Lebesgue measure. In that case we have, on using equalityin (18), with t = t0,

limn→∞

|θ(z1, Sn)− θ(z2, Sn)|√θ(z1, Sn)

√θ(z2, Sn)

= limn→∞

| Im 1t0−z1

|Sn| − Im 1t0−z2

|Sn||√Im 1

t0−z1|Sn|

√Im 1

t0−z2|Sn|

= | Im z1−z2(t0−z1)(t0−z2)

|√Im 1

t0−z1

√Im 1

t0−z2

= | z1−z2(t0−z1)(t0−z2)

|√Im z1

|t0−z1|2√

Im z2|t0−z2|2

= |z1 − z2|√Im z1

√Im z2

= γ (z1, z2),

which, coupled with (17), gives

γ (z1, z2) = supS

|θ(z1, S)− θ(z2, S)|√θ(z1, S)

√θ(z2, S)

.


It remains to show that a sequence (Sn) with the property (20) can be found. Theidea is to construct a sequence of subsets of R shrinking to the point t0. We takeSn = (t0 − 1

n, t0 + 1

n) =: (t0 − δn, t0 + δn). Then

θ(z, Sn) =∫ t0+δn

t0−δnIm

1

t − zdt.

Expanding 1/(t − z) about t0, we have

θ(z, Sn) =∫ t0+δn

t0−δnIm

1

t0 − zdt +

∫ t0+δn

t0−δn

∞∑k=1

Im

((−1)k(t − t0)

k

(t0 − z)k+1

)dt

= Im1

t0 − z|Sn| + A, (21)

where

A =∫ t0+δn

t0−δn

∞∑k=1

Im

((−1)k(t − t0)

k

(t0 − z)k+1

)dt.

With |t − t0| < δn for t ∈ Sn we have

|A| �∞∑k=1

∫Sn

∣∣∣∣Im((−1)k(t − t0)

k

(t0 − z)k+1

)∣∣∣∣ dt

�∞∑k=1

∫Sn

∣∣∣∣(−1)k(t − t0)k

(t0 − z)k+1

∣∣∣∣ dt

�∞∑k=1

∫Sn

δkn

|t0 − z|k+1dt =

∞∑k=1

2δk+1n

|t0 − z|k+1

= O(δ2n) as δn → 0 for fixed z = z1, z2.

Using (21), we now have

θ(z, Sn)

Im 1t0−z |Sn|

= Im 1t0−z |Sn| + A

Im 1t0−z |Sn|

= 1 + A

(Im 1t0−z )2δn

→ 1 as n → ∞,

since the numerator is of order δ2n and the denominator of order δn. Hence, (20)

holds, for the sequence Sn, and the proposition is proved. ✷Finally in this section we consider the behaviour of the separation γ under

Möbius transformations and Herglotz mappings.


PROPOSITION 3. Let M be a Möbius transformation defined by

M(z) = az + b

cz + d(z ∈ C+),

with a, b, c, d ∈ R and ad − bc > 0. Then M leaves γ invariant, in the sense that

γ (M(z1),M(z2)) = γ (z1, z2). (22)

Moreover, if F is any Herglotz function, then F reduces the separation γ , in thesense that

γ (F (z1), F (z2)) � γ (z1, z2). (23)

Proof. Let M be a Möbius transformation with a, b, c, d ∈ R, ad − bc > 0.First we note that, for z ∈ C+,

ImM(z) = Imaz + b

cz + d= Im z(ad − bc)

|cz + d|2 > 0. (24)

Such Möbius transformations map the upper half-plane onto the upper half-plane,so that γ can be applied to M(z1) and M(z2).

Now using (24) we have

γ (M(z1),M(z2))

= |M(z1)−M(z2)|√ImM(z1)

√ImM(z2)

= | az1+bcz1+d − az2+b

cz2+d |√Im z1(ad−bc)

|cz1+d |2√

Im z2(ad−bc)|cz2+d |2

=|(ad−bc)(z1−z2)||cz1+d ||cz2+d |√

Im z1(ad−bc)|cz1+d |2

√Im z2(ad−bc)

|cz2+d |2

= |z1 − z2|√Im z1

√Im z2

= γ (z1, z2),

which proves (22).Next let F be a Herglotz function with Herglotz representation F(z) = A+Bz+∫ ∞

−∞(t − z)−1 − t (t2 + 1)−1 dρ(t), A,B ∈ R, B � 0. Note that, as F : C+ → C+,ImF(z1) > 0, ImF(z2) > 0 for z1, z2 ∈ C+, so that γ can be applied to F(z1)

and F(z2). We consider two cases, namely B = 0 and B > 0 in the Herglotzrepresentation of F .

Case 1: Let B = 0. Then

γ (F (z1), F (z2)) = |F(z1)− F(z2)|√ImF(z1)

√ImF(z2)


= | ∫ ∞−∞

z1−z2(t−z1)(t−z2)

dρ(t)|√Im z1

∫ ∞−∞

dρ(t)|t−z1|2

√Im z2

∫ ∞−∞

dρ(t)|t−z2|2

�∫ ∞−∞

|z1−z2||t−z1||t−z2| dρ(t)

√Im z1

√Im z2

√∫ ∞−∞

dρ(t)|t−z1|2

√∫ ∞−∞

dρ(t)|t−z2|2

,

and further, using the Cauchy–Schwarz inequality in the numerator,

γ (F (z1), F (z2)) � |z1 − z2|√Im z1

√Im z2

= γ (z1, z2),

so that (23) holds for the case that B = 0 in the Herglotz representation of F .Case 2: Let B > 0. We define a family {Fy}(y ∈ R) of Herglotz functions by

Fy(z) = F(z)/(1 − yF(z)). Then

Fy(z) = Ay + Byz +∫ ∞

−∞(t − z)−1 − t (t2 + 1)−1 dρy(t).

As B = 0, we have By = 0 for y = 0 (cf. [6]). From Case 1 we thus know that

γ (Fy(z1), Fy(z2)) � γ (z1, z2). (25)

However, Fy(z) is a Möbius transformation of F(z) with ad − bc = 1 > 0, soby (22),

γ (Fy(z1), Fy(z2)) = γ (F (z1), F (z2)). (26)

(25) combined with (26) now gives

γ (F (z1), F (z2)) � γ (z1, z2),

so (23) holds for the case that B > 0 in the Herglotz representation of F as well.This completes the proof of Proposition 3. ✷

3. Value Distribution for Herglotz Functions

Let F(z) be a Herglotz function with boundary values F+(λ) and F0(λ) defined asin Section 1. Let M(A, S;F) be the associated value distribution function for F asin Section 1.

In practice, it may be difficult to estimate M(A, S;F) through the integralformula (8). This is because the determination of ω(λ, S;F) through Equation (7)requires knowledge of the behaviour of the Herglotz function close to the real axis,where precise bounds are not easy to obtain. A useful technique is to consider thetranslation of λ by an increment δ off the real axis. Define first of all a translatedHerglotz function Fδ by Fδ(z) := F(z+ iδ), with δ > 0, and set

ωδ(λ, S;F) := ω(λ, S;Fδ) = 1

πθ(F (λ+ iδ), S).


In the following theorem we verify that

M(A, S;F) = limδ→0+ M(A, S;Fδ)

= limδ→0+

∫A

ωδ(λ, S;F) dλ =∫A

ω(λ, S;F) dλ,

and show that, for fixed A, this limit is uniform over all Borel sets S, and over allHerglotz functions F .

THEOREM 1. Let F(z) be an arbitrary Herglotz function, and let A be a set offinite measure. Let S be an arbitrary Borel subset of R. Then we have∣∣∣∣∫

A

ω(λ, S;F) dλ −∫A

ωδ(λ, S;F) dλ

∣∣∣∣ � EA(δ)

= 1

π

∫A

θ(λ+ iδ, Ac) dλ, (27)

where EA(δ) → 0 for δ → 0, and EA(δ) is a nondecreasing function of δ. SinceEA(δ) is independent of S and F , the bound is uniform over all sets S and allHerglotz functions F .

Proof. We make use of the result (cf. [6]):∫ ∞

−∞Im

1

y − zω(y, S;F) dy =

∫S

Im1

y − F(z)dy. (28)

Setting ωδ(λ, S;F) = 1πθ(F (λ+ iδ), S) we have∫

A

ωδ(λ, S;F) dλ =∫A

1

πθ(F (λ+ iδ), S) dλ

= 1

π

∫A

∫S

Im1

y − F(λ+ iδ)dy dλ

= 1

π

∫A

∫ ∞

−∞Im

1

y − λ− iδω(y, S;F) dy dλ.

Noting that

Im1

y − λ− iδ= Im

1

λ− y − iδ,

and using Fubini’s theorem to change the order of integration, we have∫A

ωδ(λ, S;F) dλ = 1

π

∫ ∞

−∞ω(y, S;F)

(∫A

Im1

λ− y − iδdλ

)dy

= 1

π

∫ ∞

−∞ω(y, S;F)θ(y + iδ, A) dy.


Hence∫A

ω(λ, S;F) dλ−∫A

ωδ(λ, S;F) dλ

=∫ ∞

−∞χA(λ)ω(λ, S;F) dλ−

∫ ∞

−∞1

πω(λ, S;F)θ(λ+ iδ, A) dλ

=∫ ∞

−∞

(χA(λ)− 1

πθ(λ+ iδ, A)

)ω(λ, S;F) dλ.

Since 0 � θ(λ + iδ, A) � π , the integrand is positive for λ ∈ A and negative forλ ∈ A. Noting that ω(λ, S;F) � 1, this implies∣∣∣∣∫

A


ωδ(λ, S;F) dλ

∣∣∣∣� max

{∫A

(1 − 1

πθ(λ+ iδ, A)

)dλ,

∫Ac

1

πθ(λ+ iδ, A) dλ

}= max

{|A| − 1

π

∫A

θ(λ+ iδ, A) dλ,1

π

∫Acθ(λ+ iδ, A) dλ

}=: max{E1

A(δ), E2A(δ)}.

It is straightforward to verify that

1

π

∫A

θ(λ+ iδ, A) dλ + 1

π

∫Acθ(λ+ iδ, A) dλ = |A|,

so that E1A(δ) = E2

A(δ) =: EA(δ). We also have∫C

θ(λ+ iδ,D) dλ =∫D

θ(λ+ iδ, C) dλ

for Borel sets C and D, from which (27) follows.To show that EA(δ) is a nondecreasing function of δ converging to zero in the

limit δ → 0, we will express EA(δ) in terms of Fourier transforms.Define the Fourier transform of f (x) by

f (k) = 1√2π

∫ ∞

−∞f (x)eikx dx,

with inverse transform given by

f (x) = 1√2π

∫ ∞

−∞f (k)e−ikx dk.

The convolution f ∗ g of f and g is defined by

(f ∗ g)(λ) = 1√2π

∫ ∞

−∞f (y)g(λ− y) dy.


Define the function g(x) by g(x) := δ/(x2+δ2), with Fourier transform g(k) givenby g(k) = √

π/2 e−|k|δ . Then∫A

θ(λ+ iδ, A) dλ =∫A

∫A

Im1

y − λ− iδdy dλ

=∫ ∞

−∞χA(λ)

∫ ∞

−∞χA(y)

δ

(λ− y)2 + δ2dy dλ

=∫ ∞

−∞χA(λ)

∫ ∞

−∞χA(y)g(λ− y) dy dλ

= √2π

∫ ∞

−∞χA(λ)(χA ∗ g)(λ) dλ.

Noting that χA(λ) is real, Parseval’s identity implies∫A

θ(λ+ iδ, A) dλ = √2π

∫ ∞

−∞χA(k) (χA ∗ g)(k) dk

= √2π

∫ ∞

−∞χA(k)χA(k)g(k) dk

= π

∫ ∞

−∞χA(k)χA(k)e

−|k|δ dk.

This leads to the expression

EA(δ) = E1A(δ) = |A| − 1

π

∫A

θ(λ+ iδ, A) dλ

= |A| −∫ ∞

−∞χA(k)χA(k)e

−|k|δ dk. (29)

Again using Parseval’s identity, we have

limδ→0

EA(δ) = |A| −∫ ∞

−∞χA(k)χA(k) dk

= |A| −∫ ∞

−∞χA(x)χA(x) dx = |A| − |A| = 0.

(This result may also be deduced from (27) on applying the Lebesgue dominatedconvergence theorem.)

We also findd

dδEA(δ) =

∫ ∞

−∞χA(k)χA(k)e

−|k|δ|k| dk � 0,

so that EA(δ) is a nondecreasing function of δ, converging to zero, and the finalpart of the theorem is verified. ✷

Remark. The estimate EA(δ) is optimal in the sense that equality is attainedin (27) by taking S = Ac and F(z) = z.

A special case is that in which A a finite interval (a, b).


COROLLARY 1. Let F(z) be an arbitrary Herglotz function, and let A be a finiteinterval (a, b). Let S be an arbitrary Borel subset of R. Then we have∣∣∣∣∫

A


ωδ(λ, S;F) dλ

∣∣∣∣� 2(b − a)

πtan−1 δ

b − a+ δ

πln

(1 + (b − a)2

δ2

).

Proof. By Theorem 1,∣∣∣∣∫ b

a

ω(λ, S;F) dλ −∫ b

a

ωδ(λ, S;F) dλ

∣∣∣∣� EA(δ) = 1

π

∫A

θ(λ+ iδ, Ac) dλ

= 1

π

∫ b

a

θ(λ+ iδ, (−∞, a] ∪ [b,∞)) dλ

= 1

π

∫ b

a

∫(−∞,a]∪[b,∞)

Im1

y − λ− iδdy dλ

= 1

π

∫ b

a

∫(−∞,a]∪[b,∞)

δ

(y − λ)2 + δ2dy dλ

= 1

π

∫ b

a

(π + tan−1 λ− b

δ− tan−1 λ− a

δ

)dλ

= (b − a)+ 1

π

∫ 0

a−btan−1 u

δdu− 1

π

∫ b−a

0tan−1 u

δdu

= (b − a)− δ

πln δ2 − 2(b − a)

πtan−1 b − a

δ+ δ

πln((b − a)2 + δ2)

= (b − a)− 2(b − a)

πtan−1 b − a

δ+ δ

πln

(1 + (b − a)2

δ2

).

For x > 0, we have tan−1 x + tan−1(x−1) = π/2. This yields the simpler expres-sion given in the statement of the corollary. ✷

4. Bounds on EA(δ)

From Theorem 1 and the following remark we see that the estimate EA(δ) forconvergence of value distribution as we approach the real axis is given by

EA(δ) = supS,F

∣∣∣∣∫A


ωδ(λ, S;F) dλ

∣∣∣∣,


where the supremum is over all Borel subsets S of R and all Herglotz functions F .From Equation (27), EA(δ) is given explicitly by

EA(δ) = 1

π

∫A

θ(λ+ iδ, Ac) dλ = 1

π

∫A

∫Ac

δ

(y − λ)2 + δ2dy dλ. (30)

Hence EA(δ) � 1π

∫Aπ dλ = |A|, and by symmetry between A and Ac it follows

that EA(δ) � |Ac|, i.e.

EA(δ) � min{|A|, |Ac|}.Hence, EA(δ) is finite if either |A| or |Ac| is finite.

In order to investigate further the dependence on A of EA(δ), we shall obtaina lower bound for EA(δ), which will imply in particular that EA(δ) is finite if andonly if either |A| or |Ac| is finite.

Consider first the special case in which A is a bounded set. Let I be a boundedclosed interval such that A ⊂ I , and suppose |A| = α, |I\A| = β. Then |I | =α + β, and for simplicity we shall take I to be the interval [0, α + β].

Following Equation (30), define a function FA(δ) by

FA(δ) = 1

π

∫A

θ(λ+ iδ, I\A) dλ = 1

π

∫A

∫I\A

δ

(y − λ)2 + δ2dy dλ. (31)

Here we can estimate the double integral on the right-hand side by making a changeof integration variables which preserves measures. A one-dimensional version ofthe required transformation is as follows.

Consider the continuous mapping f from I onto [0, α], defined by

x = f (λ) =∫ λ

0χA(t) dt (λ ∈ I = [0, α + β], x ∈ [0, α]),

where χA is the characteristic function of the set A.The mapping f has the following three properties, which we verify in order:

(i) Given any Borel subset S of I , we have |f (S)| � |S|; in particular, S mapssets of measure zero to sets of measure zero.

Proof. Given ε > 0, cover S by a family {In} of closed subintervals of I , havingtotal length less than |S| + ε. Then the family {f (In)} of subintervals of [0, α]covers the set f (S). Since |f (In)| = |A ∩ In| � |In|, it follows that |f (S)| �|S| + ε. Noting that ε > 0 was arbitrary, we have |f (S)| � |S|. ✷

(ii) |f (I\A)| = 0.Proof. According to standard results [7], almost all points of I\A are points of

density of I\A. Hence, by (i), it is sufficient to consider the set I\A consisting ofpoints of density of I\A.

Given any y ∈ I\A, we have lim|I|→0 |f (I)|/|I| = 0, where the limit is takenover closed intervals I containing y in their interior. Given any ε > 0, we can


find a Vitali covering of I\A, consisting of closed intervals I ⊂ I satisfying thecondition |f (I)|/|I| < ε. By the Vitali covering theorem [8], there exists a family{In} of disjoint intervals satisfying this condition, and having the property that|(I\A)\⋃

n In| < ε. We then have

|f (I\A)| �∣∣∣∣f(

(I\A)\⋃n

In

)∣∣∣∣ +∑n

|f (In)|

�∣∣∣∣(I\A)\⋃

n

In

∣∣∣∣ + ε∑n

|In| � ε + ε|I |.

Hence |f (I\A)| = |f (I\A)| = 0. ✷(iii) |f (A)| = |A|.Proof. This result follows immediately from (ii), since

|f (A)| = |f (A ∪ (I\A))| = |f (I )| = α = |A|. ✷More generally, one can show that, for Borel subsets X of I , |f (X)| = |X∩A|.

Thus the restriction to the set A of the mapping f from I to [0, α] is measure-preserving.

We can now extend this idea by considering the mapping 5 from I × I to[β, α + β] × [0, β], defined by (λ, y) → (x1, x2) (λ ∈ I, y ∈ I ), where

x1 = β +∫ λ

0χA(t) dt

x2 = β −∫ λ

y

χI\A(t) dt

, for y < λ;

x1 = β +∫ y

λ

χA(t) dt

x2 = β −∫ y

0χI\A(t) dt

, for y > λ. (32)

We are particularly interested in the restriction of the mapping 5 to the set of points(λ, y) in I × I for which λ = y and λ, y are, respectively, points of density of thesets A, I\A.

This restriction of 5 may be verified to be injective, and using properties ofthe one-dimensional map considered earlier it may be seen that 5 is measure-preserving in the sense that

|5((A× (I\A)) ∩ J)| = |(A× (I\A)) ∩ J|for any closed rectangle J not intersecting the diagonal λ = y. The set A× (I\A)and the rectangle [β, α + β] × [0, β] each have area αβ, and the effect of the


transformation 5 is to map the set A × (I\A) without distortion of area into therectangle. This measure preserving property of 5 is reflected in the fact that theJacobian of the transformation (32) is unity at points of density of A × (I\A),provided ∂xi/∂λ and ∂xi/∂y are taken as approximate derivatives.

Invariance of two-dimensional measure implies that integrals too are preserved.Next we show that, from (32), we have |x1 − x2| � |y − λ|.

In case that y < λ, we have

|x1 − x2| =∣∣∣∣∫ λ

0χA(t) dt +

∫ λ

y

χAc(t) dt

∣∣∣∣=

∣∣∣∣∫ λ

0χA(t) dt +

∫ λ

y

(1 − χA(t)) dt

∣∣∣∣=

∣∣∣∣(λ− y)+∫ y

0χA(t) dt

∣∣∣∣= (λ− y)+

∫ y

0χA(t) dt � λ− y = |λ− y|,

as required, and in case that λ < y, we have

|x1 − x2| =∣∣∣∣∫ y

λ

χA(t) dt +∫ y

0χAc(t) dt

∣∣∣∣=

∣∣∣∣∫ y

λ

(1 − χAc(t)) dt +∫ y

0χAc(t) dt

∣∣∣∣=

∣∣∣∣(y − λ)+∫ λ

0χAc(t) dt

∣∣∣∣= (y − λ)+

∫ λ

0χAc(t) dt � y − λ = |λ− y|,

as required.An application of the change of integration variables (λ, y) → (x1, x2) to the

integral on the right-hand side of Equation (31) thus leads to the inequality

FA(δ) � 1

π

∫ α+β

β

∫ β

0

δ

(x1 − x2)2 + δ2dx2 dx1. (33)

The inequality (33) expresses the result that the function FA(δ) defined by Equa-tion (31) is minimised, for given α = |A| and β = |I\A|, by taking A and I\Ato be separated in the sense that these sets are intervals given by A = (β, α + β],I\A = [0, β] or, alternatively, A = [0, α], I\A = (α, α + β].

The right-hand side of (33) may be expressed in terms of a single function F ,defined as

F(α) = 1

π

∫ ∞

α

∫ α

0

δ

(x1 − x2)2 + δ2dx2 dx1


= α

πtan−1

(δ

α

)+ δ

2πln

(1 + α2

δ2

). (34)

Using the invariance of the double integral under the transformation x1 → x1 +β, x2 → x2 + β, we have, with obvious notation,

F(α + β)− F(α) =∫ ∞

α+β

∫ α+β

0−

∫ ∞

α+β

∫ α+β

β

=∫ ∞

α+β

∫ β

0,

so that

F(α)+ F(β)− F(α + β) =∫ ∞

β

∫ β

0−

∫ ∞

α+β

∫ β

0=

∫ α+β

β

∫ β

0.

Hence we can rewrite the inequality (33) as

FA(δ) � F(α)+ F(β)− F(α + β) (|A| = α, |I\A| = β). (35)

This inequality,with FA(δ) defined as in Equation (31), holds quite generally if Ais contained in a bounded interval I , and does not depend on the assumption I =[0, α + β].

We can now use (35) to deduce corresponding bounds for EA(δ), where now A

is an arbitrary Borel subset of R. From Equation (30), we can write

EA(δ) = limN→∞

1

π

∫A∩IN

∫IN \A

δ

(y − λ)2 + δ2dy dλ,

where IN = [−N,N]. Now set αN = |A ∩ IN | and βN = |Ac ∩ IN |. We have

α = |A| = limN→∞ αN and β = |Ac| = lim

N→∞ βN.

Comparing with Equation (31), we now have, from (35),

EA(δ) � limN→∞

{F(αN)+ F(βN)− F(αN + βN)}, (36)

provided that this limit exists. There are two cases to be considered, summarisedin the following lemma.

LEMMA 1. (i) Suppose |A| = α < ∞. Then EA(δ) � F(α), where F is givenby (34).

(ii) EA(δ) is finite if and only if either |A| < ∞ or |Ac| < ∞.Proof. (i) Suppose |A| < ∞, implying that |Ac| = β = +∞. Using the explicit

expression in Equation (34) for the function F , and noting that αN → α, βN → ∞,αN+βN → ∞ in the limitN → ∞, we have limN→∞{F(βN)−F(αN+βN)} = 0.Hence, EA(δ) � limN→∞ F(αN) = F(α).

Note that the same argument shows that if |Ac| = β < ∞ then EA(δ) � F(β).(ii) We have already seen that EA(δ) is finite whenever either A or Ac has finite

measure. Suppose now that both A and Ac have infinite measure. In that case,


in (36) we have both αN → ∞ and βN → ∞. Using Equation (34) we maydeduce that, in the limit as N → ∞, we have the asymptotic expression

F(αN)+ F(βN)− F(αN + βN) ∼ δ

πln

(αNβN

αN + βN

).

In particular, the logarithm on the right-hand side diverges asN → ∞, and we haveEA(δ) = ∞ in this case. Hence, we have shown by contradiction that EA(δ) < ∞implies that either A or Ac has finite measure. ✷

5. Sparse Potentials

We consider the Dirichlet Schrödinger operator T = −(d2/dx2) + V , acting inL2(0,∞) with potential V (x). We say that V is a sparse potential if there existsa sequence of intervals {(ak, bk)} ≡ {Ik}, having length lk = (bk − ak), such thatlk → ∞ as k → ∞, and such that V (x) ≡ 0 for x ∈ Ik.

Let us suppose that we are in the limit-point case (cf. [2]), as the alternativeassumption of the limit-circle case is known to lead to purely discrete spectrumfor T (cf. [2]). In Section 1 we quoted the result, proved in [1], that, if A is anarbitrary measurable subset of the essential support of µa.c., then for arbitrary Borelsubsets S of R we have

limN→∞

{∣∣∣∣{λ ∈ A; v′(N, λ)v(N, λ)

∈ S

}∣∣∣∣ − 1

π

∫A

θ(mN+(λ), S) dλ

}= 0.

In the following theorem we obtain asymptotic expressions for each of these twointegrals in the case that V is a sparse potential.

THEOREM 2. Let V be a sparse potential, and {(ak, bk)} = {Ik}, k = 1, 2, 3, . . . ,any corresponding sequence of intervals on which V = 0, with length lk → ∞.Then if A and S are Borel subsets of R, with A bounded, it follows that there existsL0(ε) such that for all lk = bk − ak � L0(ε) we have∣∣∣∣ 1

π

∫A

θ(mak+ (λ), S) dλ− 1

π

∫A

θ(i√λ, S) dλ

∣∣∣∣ < ε (37)

and ∣∣∣∣∣∣∣∣{λ ∈ A; v′(bk, λ)v(bk, λ)

∈ S

}∣∣∣∣ − 1

π

∫A

θ(i√λ,−S) dλ

∣∣∣∣ < ε. (38)

Here −S is the set −S = {λ ∈ R;−λ ∈ S}. Here, for Im z � 0, with z = reiθ ,0 � θ � π , we define i

√z by i

√z = i

√rei

θ2 .


Proof. We begin with the proof of (37). Let ε > 0 be given, and assume for sim-plicity ε < |A|√2. From Theorem 1 it follows that there exists δ > 0, dependingonly on ε and the set A, such that∣∣∣∣ 1

π

∫A


π

∫A

θ(mak(λ+ iδ), S) dλ

∣∣∣∣ < ε

4(39)

and ∣∣∣∣ 1

π

∫A

θ(i√λ, S) dλ − 1

π

∫A

θ(i√λ+ iδ, S) dλ

∣∣∣∣ < ε

4. (40)

Now fix this value of δ. We define the subset Aδ of C+, consisting of all z ∈ C+ ofthe form z = λ+ iδ, for λ ∈ A. Thus Aδ is the translation of A by distance δ fromthe real z-axis.

Let uak(x, z), vak (x, z) be solutions of

−d2f (x, z)

dx2+ V (x)f (x, z) = zf (x, z)

subject to initial conditions

uak(ak, z) = 1, (uak )′(ak, z) = 0,

vak (ak, z) = 0, (vak )′(ak, z) = 1.

For each z ∈ Aδ the Weyl limit-point/limit-circle theory allows us to define a circleCbk ⊂ C+, consisting of all M such that U(x, z) := uak(x, z)+Mvak (x, z) satisfiesa real boundary condition at x = bk.

Next, let u(0)(x, z), v(0)(x, z) be solutions of

−d2f (x, z)

dx2= zf (x, z)

(i.e. we deal with the Schrödinger operator with zero potential here) subject toinitial conditions

u(0)(0, z) = 1, (u(0))′(0, z) = 0,

v(0)(0, z) = 0, (v(0))′(0, z) = 1.

Again, by the Weyl limit-point/limit-circle theory, we can define, for each z ∈ Aδ ,a circle C lk ⊂ C+, consisting of all M such that u(0)(x, z)+ Mv(0)(x, z) satisfies areal boundary condition at x = lk.

We show that Cbk = C lk .Let

V (x, z) := U(x + ak, z) = uak(x + ak, z)+ Mvak (x + ak, z).


Since

V (x + ak, z) = 0 for x ∈ [0, bk − ak] = [0, lk],V(x,z) satisfies the differential equation with zero potential in the interval [0, lk].Furthermore, V (x, z) satisfies a real boundary condition at x = lk. The functionsuak(x + ak, z) and vak (x + ak, z) also satisfy the differential equation with zeropotential in [0, lk], as well as the same initial conditions at x = 0 as u(0)(x, z) andv(0)(x, z), respectively. Hence we can identify uak(x + ak, z) and vak (x + ak, z)

with u(0)(x, z) and v(0)(x, z) for x ∈ [0, lk]. Thus

V (x, z) = u(0)(x, z)+ Mv(0)(x, z) (0 � x � lk).

Hence, we may identify M with M , and it follows that the two circles Cbk and C lk

are the same.We know that the m-function mak(z) for [ak,∞) of our given Schrödinger oper-

ator satisfies mak(z) ∈ Cbk . We also know that the m-function for the Schrödingeroperator with the zero potential, m(z) = i

√z, satisfies m(z) = i

√z ∈ C lk . Since

Cbk = C lk , we have, with rClk the radius of C lk ,

|mak(λ+ iδ)− i√λ+ iδ| � 2rClk

= 1

δ∫ lk

0 |v(0)(x, z)|2 dx= 1

δ∫ lk

0 | sin(x√λ+iδ)√

λ+iδ |2 dx.

(A proof of the expression for rClk can be found in [6].)But

∫ ∞0 |v(0)(x, z)|2 dx = ∞, and so rClk → 0 for lk → ∞. Hence, ∃L0(ε)

such that ∀lk = bk − ak � L0(ε)

|mak(λ+ iδ)− i√λ+ iδ| < σε

2√

2|A| , uniformly for λ ∈ A, (41)

where σ is given by

σ := infλ∈A Im(i

√λ+ iδ). (42)

Note that σ depends only on ε and A.Using (41) and (42), we have

γ (mak(λ+ iδ), i√λ+ iδ)

= |mak(λ+ iδ)− i√λ+ iδ|√

Im(mak(λ+ iδ))√

Im(i√λ+ iδ)

<σε

2√

2|A|√Im(mak(λ+ iδ))√

Im(i√λ+ iδ)

. (43)

Since ε < |A|√2, (41) implies that

| Immak(λ+ iδ)− Im(i√λ+ iδ)| < σ

2� 1

2 Im(i√λ+ iδ) for λ ∈ A,


which leads to Immak(λ+ iδ) > 12 Im(i

√λ+ iδ).

Hence, from (43) we have

γ (mak(λ+ iδ), i√λ+ iδ)

<σε

2|A| Im(i√λ+ iδ)

� ε

2|A| , uniformly for λ ∈ A.

We now use (17) to deduce a corresponding bound for angle subtended, namely

|θ(mak(λ+ iδ), S)− θ(i√λ+ iδ, S)| < πε

2|A| ,

and integration with respect to λ over A leads to∣∣∣∣ 1

π

∫A

θ(mak(λ+ iδ), S) dλ − 1

π

∫A


∣∣∣∣ < ε

2, (44)

for all lk = bk − ak � L0(ε).We finally have, using (39), (40) and (44),∣∣∣∣ 1

π

∫A


π

∫A

θ(i√λ, S) dλ

∣∣∣∣�

∣∣∣∣ 1

π

∫A


π

∫A

θ(mak(λ+ iδ), S) dλ

∣∣∣∣++

∣∣∣∣ 1

π

∫A

θ(mak(λ+ iδ, S) dλ − 1

π

∫A


∣∣∣∣++

∣∣∣∣ 1

π

∫A

θ(i√λ+ iδ, S) dλ− 1

π

∫A

θ(i√λ, S) dλ

∣∣∣∣<ε

4+ ε

2+ ε

4= ε,

for all lk = bk − ak � L0(ε), which proves (37).We now turn to the proof of (38). Let ε > 0 be given, as before, with ε < |A|√2.

With the same choice of δ as in (39) and (40), by Theorem 1 we can ensure that∣∣∣∣ 1

π

∫A

θ(i√λ,−S) dλ − 1

π

∫A

θ(i√λ+ iδ,−S) dλ

∣∣∣∣ < ε

4(45)

and ∣∣∣∣ 1

π

∫A

θ

(−v′(bk, λ+ iδ)

v(bk, λ+ iδ),−S

)−

∣∣∣∣{λ ∈ A; v′(bk, λ)v(bk, λ)

∈ S

}∣∣∣∣∣∣∣∣ < ε

4. (46)

Again fix this value of δ and let Aδ , as in the proof of (37), be the set consisting ofall z ∈ C+ of the form z = λ + iδ, for λ ∈ A. Since A is bounded, the closure Aδ

of Aδ is a compact subset of C+.Next we state the following lemma, which we shall reformulate as Lemma 3.


LEMMA 2. Let u(x, z), v(x, z) be solutions of −(d2f /dx2)+ Vf = zf , subjectto initial conditions

u(0, z) = 1, u′(0, z) = 0,

v(0, z) = 0, v′(0, z) = 1.

Let m(1) be any constant such that Imm(1) � 0. Then we have

limN→∞ γ

(−v′(N, z)v(N, z)

,−u′(N, z)+m(1)v′(N, z)u(N, z)+m(1)v(N, z)

)= 0,

where convergence is uniform in m(1) and uniform in z for z in any fixed compactsubset of C+.

Proof. See [1]. ✷Now let f be any solution of

−d2f

dx2+ Vf = zf

such that

Im

(−f ′(0, z)f (0, z)

)� 0.

For x > 0 we have

Im

(−f ′(x, z)f (x, z)

)= f f ′ − f ′f

2i|f |2 ,

so that Im(−(f ′(x, z)/f (x, z))) will be strictly positive for x > 0 if

d

dx

(f f ′ − f ′f

2i

)> 0.

It can easily be verified that

d

dx

(f f ′ − f ′f

2i

)= Im z|f |2 > 0 for f ≡ 0.

So −(f ′(x, z)/f (x, z)) has strictly positive imaginary part for all x > 0.We can write

−f ′(x, z)f (x, z)

= −u′(x, z)+m(1)v′(x, z)u(x, z)+m(1)v(x, z)

,

where m(1) is given by

−m(1) = −f ′(0, z)f (0, z)

.

Thus Imm(1) � 0, and we can reformulate Lemma 2 as follows.


LEMMA 3. Given any ε > 0 and any compact subset K of C+, ∃N0(ε,K, V )

such that

γ

(−v′(N, z)v(N, z)

,−f ′(N, z)f (N, z)

)< ε,

for all N � N0, z ∈ K and f satisfying

−d2f

dx2+ Vf = zf with Im

(−f ′(0, z)f (0, z)

)� 0.

With vak (x, z) and v(0)(x, z) defined as in the proof of (37), we have, as previ-ously,

vak (x + ak, z) = v(0)(x, z) for x ∈ [0, lk].Now set

f (x, z) = v(x + ak, z) for x ∈ [0, lk],so that

−f ′(0, z)f (0, z)

= −v′(ak, z)v(ak, z)

has positive imaginary part, by the same argument as used earlier.We can now use Lemma 3 in the case of zero potential, with N = lk, v(x, z)

replaced by v(0)(x, z) and the above choice of f (x, z), to show that we can chooseL0(ε) sufficiently large that, for lk � L0(ε), we have

γ

(−(vak)′(bk, z)

vak (bk, z),−v′(bk, z)

v(bk, z)

)<

ε

4|A| ,

for all z ∈ Aδ .As in the proof of (37), we first convert this γ estimate into an estimate of angle

subtended, namely∣∣∣∣θ(−(vak )′(bk, λ+ iδ)

vak(bk, λ+ iδ),−S

)− θ


v(bk, λ+ iδ),−S

)∣∣∣∣ < πε

4|A| ,

leading to∣∣∣∣ 1

π

∫A

θ

(−(vak )′(bk, λ+ iδ)


)dλ−

− 1

π

∫A

θ


v(bk, λ + iδ,−S

)dλ

∣∣∣∣ < ε

4, (47)

for lk � L0(ε).Note here that L0(ε) is independent of the potential V .


With

v(0)(x, z) = sin(x√z)√

z,

where Re(i√z) < 0, we have

−(v(0))′(x, z)v(0)(x, z)

= i√z

(−1 − e2ix√z

−1 + e2ix√z

)→ i

√z as x → ∞.

Since

(vak )′(bk, z)vak (bk, z)

= (v(0))′(lk, z)v(0)(lk, z)

,

we can find L0(ε), independent of V , such that for lk � L0(ε) we have∣∣∣∣−(vak)′(bk, λ+ iδ)

vak (bk, λ+ iδ)− i

√λ+ iδ

∣∣∣∣ < σε

4√

2|A| , uniformly for λ ∈ A, (48)

where σ is defined by (42). Hence, using (48), we have

γ

(−va

′k (bk, λ+ iδ)

vak(bk, λ+ iδ), i

√λ+ iδ

)<

ε

4|A| , uniformly for λ ∈ A,

which leads to∣∣∣∣θ(−(vak )′(bk, λ+ iδ)


)− θ(i

√λ+ iδ,−S)

∣∣∣∣ < πε

4|A| ,

and thus to∣∣∣∣ 1

π

∫A

θ



)dλ−

− 1

π

∫A


∣∣∣∣ < ε

4, (49)

for lk � L0(ε).Using (45), (46), (47) and (49), we arrive at∣∣∣∣∣∣∣∣{λ ∈ A; v

′(bk, λ)v(bk, λ)

∈ S

}∣∣∣∣ − 1

π

∫A

θ(i√λ,−S) dλ

∣∣∣∣�

∣∣∣∣∣∣∣∣{λ ∈ A; v′(bk, λ)v(bk, λ)

∈ S

}∣∣∣∣ − 1

π

∫A

θ


v(bk, λ+ iδ),−S

)∣∣∣∣++

∣∣∣∣ 1

π

∫A

θ


v(bk, λ+ iδ),−S

)dλ−


− 1

π

∫A

θ



)dλ

∣∣∣∣ ++

∣∣∣∣ 1

π

∫A

θ

(−(vak )′(bk, λ + iδ)

vak (bk, λ+ iδ),−S

)dλ−

− 1

π

∫A


∣∣∣∣ +

+∣∣∣∣ 1

π

∫A

θ(i√λ+ iδ,−S) dλ− 1

π

∫A

θ(i√λ,−S) dλ

∣∣∣∣<ε

4+ ε

4+ ε

4+ ε

4= ε, (50)

for lk � L0(ε) := max{L0(ε), L0(ε)}. Thus (38) is proved and Theorem 2 fol-lows. ✷

Since we could redefine the sequence of intervals {Ik}, replacing each interval(ak, bk) in the sequence by a pair of intervals (ak, ck), (ck, bk), with ck = (ak +bk)/2, Equations (37) and (38) remain valid with ak, respectively bk , replaced by ckon the left-hand side. If A is a subset of R− and S is a subset of R+ for which theright-hand side of (38) is nonzero, this implies that v′(ck, λ)/v(ck, λ) and m

ck+ (λ)will have different asymptotic value distribution, for λ ∈ A. On the other hand, ifin addition A is a subset of an essential support of the spectral measure µa.c., thenaccording to (9), the asymptotic value distributions should be the same. It followsthat there can be no absolutely continuous measure for λ < 0.

Equations (37) and (38), together with (9), may also be used to prove, for var-ious classes of sparse potentials, that the spectral measure for λ > 0 is purelysingular. As a simple example of this argument, consider the potential V (x) =∑∞

n=1 δ(x − xn), with (xn+1 − xn) → +∞ as n → ∞. (The theory presented inthis paper can easily be extended to include such distributional potentials.) We canthen define a sequence of intervals {Ik}, with Ik = (xk, xk+1), and let A ⊂ R+ bea subset of an essential support of µa.c.. Noting that θ(i

√λ, S) = θ(i

√λ,−S) for

λ > 0, Equations (37) and (38), together with (9), imply that v′(xk, λ)/v(xk, λ) hasthe same asymptotic value distribution for λ ∈ A, in the limit k → ∞, whether xk istaken just to the right, or just to the left of the singularity of the potential. However,at the δ singularity x = xk, the function v′(xk, λ)/v(xk, λ) has discontinuity 1.Hence, the two asymptotic distributions cannot agree, and we may deduce that inthat case there is no absolutely continuous measure for λ > 0. More generally,one has the qualitative understanding that absolutely continuous spectrum is onlyallowed if the potential, in the regions where it is nonzero, fails to disturb theasymptotics of v′(x, λ)/v(x, λ), where x is an endpoint of one of the intervalsIk. For a recent treatment of sparse potentials, see [10]. For proof of absence ofabsolutely continuous spectrum see, e.g., [11–13].


6. Further Extensions and Developments

The ideas of value distribution for real valued functions, and the associated valuedistribution for Herglotz functions, are particularly relevant in the analysis of as-ymptotic behaviour and spectral properties for the Schrödinger equation, wherethese ideas are closely linked to the geometrical structure of hyperbolic space.Applications to various classes of potential are given in [1]. Results such as thoseof Theorem 2 of the present paper may be extended and applied to a wider class ofso-called sparse potentials. For example, we may describe a potential V as sparsein this wider sense if, given any ε,N > 0, there exists an interval I having length|I| = N , such that

∫I(V (x))

2 dx < ε. (Note that a general L2 potential is sparse inthis sense.) Results for potentials of this kind will be described elsewhere.

Another fruitful direction of application of geometrical analysis to spectral prop-erties of Schrödinger operators is the analysis of the Weyl m-function and valuedistribution for convergent sequences of potentials. As an example of the kind ofresult which can be obtained, consider a sequence {Vn} of potential functions suchthat Vn(x) converges uniformly to V (x) for x � 0, in the limit n → ∞. Let mn(z)

be the m-function corresponding to the potential Vn, and m(z) the m-function cor-responding to V . Then one may show, in the limit n → ∞, that the associated valuedistribution for mn converges to that for m. Current research, partly motivated byresults of Deift and Killip in [14] for L2 potentials, is focussed on developing com-parable results, for appropriate classes of potentials V , where uniform convergenceis replaced by convergence in L2-norm.

Some of the results of the present paper, and of [1], can be further generalisedto cover a description of value distribution in which the underlying measure isnot Lebesgue measure but some other Herglotz-type measure. For results in thisdirection see [15], and further developments to be published elsewhere.

References

1. Breimesser, S. V. and Pearson, D. B.: Asymptotic value distribution for solutions of theSchrödinger equation, Math. Phys. Anal. Geom. 3 (2000), 385–403.

2. Coddington, E. A. and Levinson, N.: Theory of Ordinary Differential Equations, McGraw-Hill,New York, 1955.

3. Eastham, M. S. P. and Kalf, H.: Schrödinger-type Operators with Continuous Spectra, Pitman,Boston, 1982.

4. Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space, Pitman,London, 1981.

5. Pommerenke, C.: Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992.6. Pearson, D. B.: Value distribution and spectral theory, Proc. London Math. Soc. 68(3) (1994),

127–144.7. Oxtoby, J. C.: Measure and Category, Springer, New York, 1971.8. Saks, S.: Theory of the Integral, Hafner Publ., New York, 1933.9. Pearson, D. B.: Quantum Scattering and Spectral Theory, Academic Press, London, 1988.

10. Simon, B. and Stolz, G.: Operators with singular continuous spectrum, V. Sparse potentials,Proc. Amer. Math. Soc. 124(7) (1996), 2073–2080.


11. McGillivray, I., Stollmann, P. and Stolz, G.: Absence of absolutely continuous spectra formultidimensional Schrödinger operators with high barriers, Bull. London Math. Soc. 27 (1995),162–168.

12. Simon, B. and Spencer, T.: Trace class perturbations and the absence of absolutely continuousspectrum, Comm. Math. Phys. 125 (1989), 113–126.

13. Stolz, G.: Spectal theory for slowly oscillating potentials, II. Schrödinger operators, Math.Nachr. 183 (1997), 275–294.

14. Deift, P. and Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödingeroperators with square summable potentials, Comm. Math. Phys. 203 (1999), 341–347.

15. Christodoulides, Y.: Spectral theory of Herglotz functions and their compositions and theSchrödinger equation, PhD thesis (2002), University of Hull.


59

Improved Epstein–Glaser Renormalization inCoordinate Space I. Euclidean Framework

JOSÉ M. GRACIA-BONDÍAForschungszentrum BiBoS, Fakultät der Physik Universität Bielefeld, D-33615 Bielefeld, Germany,Departamento de Física Teórica, Universidad de Zaragoza, E-50009 Zaragoza, Spain, andDepartamento de Física Teórica I, Universidad Complutense, E-28040 Madrid, Spain

(Received: 4 March 2002)

Abstract. In a series of papers, we investigate the reformulation of Epstein–Glaser renormalizationin coordinate space, both in analytic and (Hopf) algebraic terms. This first article deals with analyticalaspects. Some of the (historically good) reasons for the divorces of the Epstein–Glaser method, bothfrom mainstream quantum field theory and the mathematical literature on distributions, are madeplain; and overcome.

Mathematics Subject Classifications (2000): 81T15, 46F99.

Key words: Epstein–Glaser renormalization, scaling.

1. Introduction

This is the first of a series of papers, the companions [1, 2] often being denoted,respectively, II and III.

We find it convenient to summarize here the aims of these papers, in reverseorder. Ever since Kreimer perceived a Hopf algebra lurking behind the forest for-mula [3], the question of encoding the systematics of renormalization in such astructure (and the practical advantages therein) has been in the forefront. Connesand Kreimer were able to show, using the ϕ3

6 model as an example, that renormal-ization of quantum field theories in momentum space is encoded in a commutativeHopf algebra of Feynman graphs H , and the Riemann–Hilbert problem with val-ues in the group of loops on the dual of H [4, 5]. The latter makes sense onlyin the context of renormalization by dimensional regularization [6, 7], physicists’method of choice. Now, whereas it is plausible that the Hopf algebra approach torenormalization is consistent with all main renormalization methods, there is muchto be learned by a systematic verification of this conjecture. Paper III focuses oncombinatorial-geometrical aspects of this approach to perturbative renormalizationin QFT within the framework of the Epstein–Glaser (EG) procedure [8].

One can argue that all that experiments have established is (striking) agreementwith (renormalized) momentum space integrals [9]. Be that as it may, renormaliza-tion on real space is more intuitive, in that momentum space formulations “rather

60 JOSE M. GRACIA-BONDIA

obscure the fact that UV divergences arise from purely short-distance phenom-ena” [10]. For the questions of whether and how configuration space-based meth-ods exhibit the Hopf algebraic structure, the EG method was a natural candidate.It enjoys privileged rapports with external field theory [11–13], possesses a starkreputation for rigour, and does not share some limitations of dimensional regular-ization – allowing for renormalization in curved backgrounds [14], for instance.

In spite of its attractive features, EG renormalization still remains outside themainstream of QFT. The (rather rigorous) QFT text by Itzykson and Zuber hasonly the following to say about it: “. . . the most orthodox procedure of Epstein andGlaser relies directly on the axioms of local field theory in configuration space. It isfree of mathematically undefined quantities, but hides the multiplicative structureof renormalization” [15, p. 374]. Raymond Stora, today the chief propagandistof the method, had commented: “In spite of its elegancy and accuracy this the-ory suffers from one defect, namely it does not yield explicit formulae of actualcomputational value” [16]. Indeed.

Over the years, some of the awkwardness of the original formalism was dis-pelled in the work by Stora. The ‘splitting of distributions’ was reformulated in [17]as a typical problem of extension (through the boundaries of open sets) in distrib-ution theory. Moreover, in [18] it was made clear that an (easier) Euclidean analogof the EG construction does exist. Beyond being interesting on its own right (forinstance for the renormalization group approach to criticality), it allows performingEG renormalization in practice by a (sort of) ‘Wick rotation’ trick – the subject ofpaper II of this series.

When tackling the compatibility question of EG renormalization and theConnes–Kreimer algebra, two main surviving difficulties are brought to light. Thefirst is that, while the Hopf algebra elucidation of Bogoliubov’s recursive procedureis defined graph-by-graph, in the EG approach it is buried under operator aspects ofthe time-ordered products and the S-matrix, not directly relevant for that question.This problem was recently addressed by Pinter [19, 20] and also in [21]; the lastpaper, however, contains a flaw, examined in III.

The second difficulty, uncovered in the course of the same investigation, has todo with prior, analytical aspects of Epstein and Glaser’s basic method of subtrac-tion. For, it was curious to observe, the extension method by Epstein and Glaserhas remained divorced as well from the literature on distributions, centered mainlyon analytical continuation and ‘finite part’ techniques. One scours in vain for anyfactual link between EG subtraction and the household names of mathematicaldistribution theory. And so the vision of casting all of quantum field theory in thelight of distribution analysis [22, 23] has remained unfulfilled.

In the present paper we are concerned with the second of the mentioned difficul-ties. This means in practice that we deal with primitively UV divergent diagrams.(Nonprimitive diagrams are dealt with in III.) By means of a seemingly minute de-parture from the letter, if not the spirit, of the EG original prescription we succeedto deliver its missing link to the standard literature on extension of distributions.

IMPROVED EPSTEIN–GLASER RENORMALIZATION IN COORDINATE SPACE I 61

Then we proceed to show the dominant place our improved subtraction method oc-cupies with regard to dimensional regularization in configuration space; differentialrenormalization [24]; ‘natural’ renormalization [25]; and BPHZ renormalization.

The benefits of the improved prescription do not stop there: it goes on to re-markably simplify the task of constructing covariant renormalizations in II, andthe Hopf-geometrical constructions in III.

An important sideline of this paper is the use of the theory of Cesàro summa-bility of distributions [26, 27] in dealing with the infrared difficulties; this helps toclarify the logical dependence of the BPHZ procedure on the causal one, alreadypointed out in [29]. Improved BPHZ methods for massless fields ensue as well.

The main theoretical development is found in Section 2. Afterwards, we proceedby way of alternating discussions and examples. In order to deliver the argumentwithout extraneous complications, we work out diagrams belonging to scalar theo-ries. Most examples are drawn from the massless ϕ4

4 model: masslessness is morechallenging and instructive, because of the attendant infrared problems, and moreinteresting for the renormalization group calculations performed in III. Eventuallywe bring in examples in massive theories as well.

2. Renormalization in Configuration Space

2.1. THE NEW PRESCRIPTION

All derivatives in this paper, unless explicitly stated otherwise, are in the sense ofdistributions. We tacitly use the translation invariance of Feynman propagators andamplitudes; in particular, the origin stands for the main diagonal.

Let d denote the dimension of the coordinate space. Typically, d will be 4n. Anunrenormalized Feynman amplitude f (�), or simply f , associated to a graph �, issmooth away from the diagonals. We say that � is primitively divergent when f (�)is not locally integrable, but is integrable away from zero. Denote by Fprim(R

d) ↪→L1

loc(Rd \ {0}) this class of amplitudes.

By definition, a tempered distribution f ∈ S′(Rd) is an extension or renormal-ization of f if

f [φ] := 〈f , φ〉 =∫

Rd

f (x)φ(x) ddx

holds whenever φ belongs to S(Rd \ {0}).Let

f (x) = O(|x|−a) as x → 0, (1)

with a an integer, and let k = a − d � 0. Then, f /∈ L1loc(R

d). But f can beregarded as a well-defined functional on the space Sk+1(R

d) of Schwartz functionsthat vanish at the origin at order k + 1. Thus the simplest way to get an extensionof f would appear to be standard Taylor series surgery: to throw away the k-jet


of φ at the origin, in order to define f by transposition. Denote this jet by jk0φ andthe corresponding Taylor remainder by Rk0φ. We have by that definition

〈f , φ〉 = 〈f,Rk0φ〉. (2)

Using Lagrange’s integral formula for the remainder:

Rk0φ(x) = (k + 1)∑

|β|=k+1

xβ

β!∫ 1

0dt (1 − t)k∂βφ(tx),

where we have embraced the usual multiindex notation, and exchanging integra-tions, one appears to obtain an explicit integral formula for f :

f (x) = (−)k+1(k + 1)∑

|β|=k+1

∂β[xβ

β!∫ 1

0dt(1 − t)ktk+d+1

f

(x

t

)]. (3)

Lest the reader be worried with the precise meaning of (1), we recall that in QFTone usually considers a generalized homogeneity degree, the scaling degree [30].The scaling degree σ of a scalar distribution f at the origin of Rd is defined to be

σf = inf{s : lim

λ→0λsf (λx) = 0

}for f ∈ S′(Rd),

where the limit is taken in the sense of distributions. Essentially, this means thatf (x) = O(|x|−σf ) as x → 0 in the Cesàro average sense [31]. Then [σf ] andrespectively [σf ] − d – called the singular order – occupy the place of a in (1) andof k.

The trouble with (3) is that the remainder is not a test function, so, unless theinfrared behaviour of f is very good, we end up in (2) with an undefined integral.In fact, in the massless theory f is an homogeneous function with an algebraic sin-gularity, the infrared behaviour is pretty bad, and −d is also the critical exponent.A way to avoid the infrared problem is to weight the Taylor subtraction. Epsteinand Glaser [8] introduced weight functions w with the properties w(0) = 1,w(α)(0) = 0 for 0 < |α| � k, and projector maps φ �→ Wwφ on S(Rd) givenby

Wwφ(x) := φ(x)− w(x)jk0φ(x). (4)

The previous ordinary Taylor surgery case corresponds to w ≡ 1, and the identity

Ww(wφ) = wW1φ

tells us that Ww indeed is a projector, since Ww(wxγ ) = 0 for |γ | � k.

Look again at (4). There is a considerable amount of overkilling there. Thepoint is that, in the homogeneous case, a worse singularity at the origin entails abetter behaviour at infinity. So we can, and should, weight only the last term of the


Taylor expansion. This leads to the definition employed in this paper, at variancewith Epstein and Glaser’s:

Twφ(x) := φ(x) − jk−10 (φ)(x)− w(x)

∑|α|=k

xα

α! φ(α)(0). (5)

Just w(0) = 1 is now required in principle from the weight function.An amazing amount of mathematical mileage stems from this simple physical

observation. To begin with, Tw is also a projector. To obtain an integral formula forit, start from

Twφ = (1 − w)Rk−10 φ + wRk0φ,

showing that it Tw interpolates between Rk0 , guaranteeing a good UV behaviour,and Rk−1

0 , well behaved enough in the infrared. By transposition, using (3), wederive

Twf (x) = (−)kk∑|α|=k

∂α[xα

α!∫ 1

0dt(1 − t)k−1

tk+df

(x

t

)(1 − w

(x

t

))]+

+ (−)k+1(k + 1)∑

|β|=k+1

∂β[xβ

β!∫ 1

0dt(1 − t)ktk+d+1

f

(x

t

)w

(x

t

)]. (6)

This is the central formula of this paper.

2.2. ON THE AUXILIARY FUNCTION

It is important to realize what is (and is not) required of the weight function w,apart from a good behaviour at the origin: in view of the smoothness and goodproperties of f away from the origin, we have a lot of leeway, and, especially, wdoes not have to be a test function, nor to possess compact support. Basically, whatis needed is that w decay at infinity in the weak sense that it sport momenta ofsufficiently high order.

We formalize this assertion for greater clarity. First, one says that the distribu-tion f is of order |x|l (with l not a negative integer) at infinity, in the Cesàro sense,if there exists a natural number N , and a primitive fN of f of order N , such that fNis locally integrable for |x| large and the relation fN(x) = O(|x|N+l ) as |x| ↑ ∞holds in the ordinary sense. Now, for any real constant γ , the space Kγ is formedby those smooth functions φ such that ∂αφ(x) = O(|x|γ−|α|) as |x| ↑ ∞, foreach |α|. A topology for Kγ is generated by the obvious family of seminorms, andthe space K is defined as the inductive limit of the spaces Kγ as γ ↑ ∞. Considernow the dual space K ′ of distributions. The following are equivalent [26–28]:

• f ∈ K ′.• f satisfies f (x) = o(|x|−∞) in the Cesàro sense as |x| ↑ ∞.


• There exist constants µα such that

f (λx) ∼∑α�0

µαδ(α)(x)

λ|α|+1

in the sense of distributions, as λ ↑ ∞.• All the moments 〈f (x), xα〉 exist in the sense of Cesàro summability of inte-

grals (they coincide with the aforementioned constants µα).

Any element of K ′ which is regular and takes the value 1 at zero qualifies as aweight ‘function’. For instance, one can take for w an exponential function eiqx ,with q �= 0. This vanishes at ∞ to all orders, in the Cesàro sense, and so it is aperfectly good infrared problem-buster auxiliary function. The fact that eiqx ∈ K ′,for q �= 0, means that, outside the origin in momentum space, the Fourier transformof elements φ ∈ K can be computed by a standard Cesàro evaluation

φ(q) = 〈exp(iqx), φ(x)〉.Of course, for this auxiliary function the original Equation (4) no longer applies,since it has no vanishing derivatives at the origin. But (4) can be replaced by themore general

Wwφ(x) := φ(x)− w(x)∑

0�|α|�k

xα

α!(φ

w

)(α)(0). (7)

This was seen, at the heuristic level, by Prange [29]; see the discussion on theBPHZ formalism in Subsection 5.3, where the ‘Cesàro philosophy’ comes into itsown.

These observations are all the more pertinent because the contrary prejudiceis still widespread. For instance, the worthy thesis [20], despite coming on thefootsteps of [29], yet unfortunately exhibits it; on its page 30: “. . . [the exponen-tial] function is not allowed in the W -operation because it does not have compactsupport”. Of course it is allowed: then the Fourier transformed subtraction Ww

of Epstein and Glaser becomes exactly the standard BPHZ subtraction, aroundmomentum p = q. What Tw becomes will be revealed later.

2.3. PROPERTIES OF THE T -PROJECTOR

Consider now the functional variation of the renormalized amplitudes with respectto w. One has⟨

δ

δwTwf,ψ

⟩:= d

dλTw+λψf

∣∣∣∣λ=0

for Tw, and similarly for Ww, by definition of functional derivative. It is practicalto write now

δα := (−)|α| δ(α)

α! ,


for this combination is going to appear with alarming frequency. From (4) wewould obtain

δ

δwWwf [·] = −

∑|α|�k

f [xα ·] δα, (8)

whereas (5) yieldsδ

δwTwf [·] = −

∑|α|=k

f [xα ·]δα,

independently of w in both cases. Malgrange’s theorem says that different renor-malizations of a primitively divergent graph differ by terms proportional to thedelta function and all its derivatives δ(α), up to |α| = k. Thus there is no canon-ical way to construct the renormalized amplitudes, the inherent ambiguity beingrepresented by the undetermined coefficients of the δ’s, describing how the chosenextension acts on the finite codimension space of test functions not vanishing in aneighborhood of 0.

There is, however, a more ‘natural’ way – in which the ambiguity is reduced toterms in the higher-order derivatives of δ, exclusively. This is guaranteed by ourchoice of Tw.

In practice, one works with appropriate 1-parameter (or few-parameter) familiesof auxiliary functions, big enough to be flexible, small enough to be manageable.Recall than in QFT, with c = h = 1, the physical dimension of length is inversemass. Let then the variable µ have the dimension of mass. We consider the changein Twf when the variable w changes from w ≡ w(µx) to w((µ + δµ)x), whichintroduces the Jacobian δw/δµ = (∂w(µx))/∂µ, yielding

∂

∂µTw(µx)f = −

∑|α|=k

〈f, xν∂νw(µx)xα〉δα. (9)

Here we have assumed that f has no previous dependence on µ.Enter now the (rotation-invariant) choice wµ(x) := H(µ−1 − |x|), where H is

the Heaviside step function: it not only recommends itself for its simplicity, but itturns out to play a central theoretical role. The parameter µ corresponds in our con-text to ’t Hooft’s energy scale in dimensional regularization – see Subsection 5.1;the limits µ ↓ 0 and µ ↑ ∞ correspond to the case w = 1 and respectively to the‘principal value’ of f ; in general they will not exist.

Write Tµf for the corresponding renormalizations. With the help of (9), oneobtains

µ∂

∂µTµf =

∑|α|=k

f [δ(µ−1 − |x|)|x|xα]δα. (10)

For f homogeneous (of order −d − k as it happens), the expression is actuallyindependent of µ, the coefficients of the δα being

cα =∫

|x|=1f xα =

∫|x|=A

f |x|xα, (11)


with |α| = k and any A > 0. Note that similar extra terms, with |α| < k, comingout of the formulae (8) would indeed be µ-dependent.

Compute the Tµ in the massless (homogeneous) case, whereupon one can pull fout of the integral sign. We get

Tµf (x) = (−)k{∑

|α|=k∂α

[xαf (x)

α! (1 − (1 − µ|x|)k)]

+

+ (k + 1)∑

|β|=k+1

∂β[xβf (x)

β!∫ µ|x|

1dt(1 − t)kt

]}. (12)

Formula (12) is simpler than it looks: because of our previous remark on (10), allthe µ-polynomial terms in the previous expression for Tµf must cancel. Let usthen denote, for k � 1,

Hk :=k∑l=1

(−)l+1

l

(k

l

)=

(k

1

)− 1

2

(k

2

)+ · · · − (−)k 1

k. (13)

At least for µ|x| � 1, the expression for Tµf becomes

Tµf (x) = (−)k(k + 1)∑

|β|=k+1

∂β[xβf (x)

β! (logµ|x| +Hk)].

By performing the derivative with respect to logµ directly on this formula,one obtains in the bargain interesting formulae for distribution theory. Namely, forany f homogeneous of degree −d − k:

(−)k(k + 1)∑

|β|=k+1

∂β[xβf (x)

β!]

=∑|α|=k

(∫|x|=1

f xα)δα(x). (14)

Thus, the final result is

Tµf (x) = (−)k(k + 1)∑

|β|=k+1

∂β[xβf (x)

β! logµ|x|]

+Hk∑|α|=k

cαδα(x) (15)

with the cα given by (11). The resulting expression is actually valid for all x: awayfrom the origin it reduces to f (x), as it should. It is in the spirit of differentialrenormalization, since f is renormalized as the distributional derivative of a regularobject that coincides with f away from the singularity.

Let us put Equation (15) to work at once. When performing multiplicativerenormalization in the causal theory [2], the relevant property of a renormalizedamplitude turns out to be its dilatation or scaling behaviour. This is not surprisingin view of the form of our integral Equation (6). Now, a further consequence of


the choice of operator Tw is that it modifies the original homogeneity in a minimalway. Had we stuck toWw (using (14)), the relatively complicated form

Wµf (λ·) = λ−k−d(Wµf + log λ

∑|α|=k

cαδα +∑|α|<k

aα(λk−|α| − 1)δα),

for some aα , would ensue; whereas for Tµ from (15) one obtains

[f ]R,µ(λx) := Tµf (λx) = λ−k−d(

[f ]R,µ(x)+ log λ∑|α|=k

cαδα(x)

); (16)

or

E[f ]R,µ(x) := (−k − d)[f ]R,µ(x)+∑|α|=k

cαδα(x),

where E denotes the Euler operator∑

|β|=1 xβ∂β . From now on, the notations

[f ]R,µ and Tµf will be interchangeably used; when the dependence on µ neednot be emphasized, we can write [f ]R instead.

We invite the reader to prove (16) directly, by reworking the argument used forthe case k = 0 in [32, pp. 307–308].

Let us record here the obvious fact that when we employ two different prescrip-tions compatible with our renormalization scheme, the difference in the results isgiven by

[f ]R1 = [f ]R2 +∑|β|=k

Cβδβ, (17)

for some constants Cβ . ‘Different prescriptions’ could mean that we use differentweight functions, or different choices of renormalization conditions, or simply inthe value of the parameter µ. In this respect, we note that the Fourier transformsof our real space renormalized amplitudes (see the end of Section 5) do not obeystandard renormalization conditions on momentum space, so in particular contextsthey might be in need of modification for this purpose.

2.4. SUPPLEMENTARY REMARKS

Following [33], with a difference by a factor of 2 in the definition, we convenientlyformalize the first order operator from the (larger) space S′

1(Rd) of continuous

linear functionals on Schwartz functions that vanish at the origin, appearing in theforegoing:

S :=∑|β|=1

∂βxβ.

It ‘regularizes’ S′1, mapping it onto S′. Clearly, on tempered distributions S =

d+E. Therefore S kills any homogeneous distribution of order −d, like δ – but not


the homogeneous functionals! From (15), it is clear that (the analog in ξ -space of) Scaptures the Wodzicki residue density in the theory of pseudodifferential operators– see for instance the discussion in Chapter 7 of [32]. Analogously, define

Sk+1 := (k + 1)!∑

|β|=k+1

∂βxβ

β! .

This sends onto S′ the space of functionals S′k+1(R

d), dual of the space of Schwartzfunctions which vanish at the origin up to order k + 1. On well-behaved distribu-tions, it is equivalent to (S+ k) · · · (S+ 1)S: we think of Sk+1 as an ordered powerof S1, with the coordinate multiplications remaining to the right of the differentialoperators. For massless models, our formulae come close to simply iterating S inthis way, as done in [33].

For theoretical purposes, it should be kept in mind that the T -operator remainsin the general framework of the Epstein–Glaser theory: after all, one could alwaysfind weight functions w(µ) such that Ww(µ) is identical to Tµ. In particular, thesingular order of Tµf is the same as that of f [14]. However, we see already that afiner classification needs to be introduced.

DEFINITION 1. A distribution f = f1 is called associate homogeneous of or-der 1 and degree a when there exists a homogeneous distribution f0 of degree asuch that

f1(λx) = λa(f1(x)+ g(λ)f0(x)),

for some function g(λ). It is readily seen that only the logarithm function can footthe bill for g. Then, a distribution fn is called associate homogeneous of order nand degree a when there exists an associate homogeneous distribution fn−1 of ordern− 1 and degree a such that

fn(λx) = λa(fn(x)+ log λfn−1(x)).

Clearly the renormalization of primitively divergent graphs in massless theories,using Tµ, gives rise to associate homogeneous distributions of order 1. To passassociate homogeneous distributions of order 1 thru the same machine (12), inorder to obtain a renormalized closed expression, is routine. Assuming no priordependence of f1 on µ, one would have to use now

µ∂

∂µTµf1 =

∑|α|=k

(∫f1x

α − logµ∫f0x

α

)δα

instead of (10), to simplify the output of (6). We omit the straightforward details. Ofcourse, if g = [f ]R;µ for f primitive, then Tµg = g; but diagrams with a renormal-ized subdivergence provide less trivial examples. The complete renormalization ofmassless theories gives rise exclusively to associate homogeneous distributions [2].


As remarked in [34], on the finite dimensional vector space spanned by anhomogeneous distribution and its associates up to a given order, the Euler operatortakes the Jordan normal form. The Sk+1 operators are nilpotent on that space, whena = −d − k. For example,

S

[1

x4

]R

= S

(1

x4

),

and therefore S2[1/x4]R = 0. In consonance with this, Equation (15) remains validfor f primitively divergent of the associated homogeneous type.

The main theoretical development of the T -operator closes here. We have foundtwo instances in the QFT literature of the improved causal method advocated byus (5): the ‘special R operation’ introduced in the outstanding (apparently un-published) manuscript [35]; and, unwittingly, ‘natural’ renormalization [25] – seeSubsection 5.2. The scaling properties are discussed in [35]; but no explicit for-mulae for the renormalized amplitudes are given there. Also, after this paper waswritten, we realized that the scaling properties of Epstein–Glaser renormalizationare discussed in [36]. We turn to matters of illustration and comparison.

3. Some Examples

We compute the simplest primitive diagrams relevant for the four-point functionof φ4

4 theory – for these logarithmically divergent graphs, there is of course nodifference between Tw and Ww. The following notation will be used in the sequel:

2d,m :=∫

|x|=1x2mi = 2�(m+ 1

2 )π(d−1)/2

�(m+ 12d)

, (18)

where i labels any component. In particular, 2d,0 =: 2d , the area of the sphere indimension d. The quotients of the 2d,m are rational:

2d,m

2d= (2m− 1)!!(2m+ d − 2)(2m+ d − 4) · · · d .

The ‘propagator’, given by the formula

DF(x) = |x|2−d

(d − 2)2d,

when d �= 2, and by DF(x) = (log |x|)/22 when d = 2, is simply the Greenfunction for the Laplace equation:

7DF(x) = −δ(x).Consider the ‘fish’ diagram in ϕ4

4 theory, giving the first correction to the four-pointfunction. The corresponding amplitude is proportional to x−4. On using (15) withk = 0,[

1

x4

]R

= ∂β[xβ log(µ|x|)

x4

]= S1

log(µ|x|)x4

.


Next we look at µ(∂/∂µ)(TµD2F ). By direct computation, on the one hand,

µ∂

∂µ

[1

x4

]R

= ∂β[xβ

x4

]= −1

27

(1

x2

)= 24δ(x),

and on the other, according to (10):

µ∂

∂µ

[1

x4

]R

= c0δ(x), with residue c0 =⟨

1

x4, δ(µ−1 − |x|)|x|

⟩R4

= 24,

which serves as a check. In this case, as µ varies from 0 to ∞, all possible renor-malizations of D2

F are obtained. One can as well directly check here the equation:

Tµf (λx) = λ−4[Tµf (x)+24 log λδ(x)], (19)

for f (x) = 1/x4. Equation (19) contains the single most important informationabout the fish graph and is essential for the treatment of diagrams in which itappears as one of the subdivergences [2].

Next among the primitive diagrams relevant for the vertex correction comes thetetrahedron (also called the ‘open envelope’) diagram. In spite of appearances, itis a three-loop graph, as one of the circuits depends on the others; it is the lowest-order diagram in φ4

4 theory with the full structure of the four-point function. Theunrenormalized amplitude f ∈ Fprim(R

12) is of the form

f (x, y, z) = 1

x2y2z2(x − y)2(y − z)2(x − z)2 ,

which is logarithmically divergent overall. A funny thing about this diagram isthat the amplitude for it looks exactly the same in momentum space – see, forinstance, [37]. Denote by s the collective variable (x, y, z). Then

[f ]R = S(x,y,z)

[log(µ|s|)

x2y2z2(x − y)2(y − z)2(x − z)2].

Again, the most important information from the diagram concerns its dilatationproperties. Proceeding as above, we obtain

µ∂

∂µ[f ]R =: Iδ(x),

where, for any A > 0,

I =∫s2=A

|s| ds

x2y2z2(x − y)2(y − z)2(x − z)2 .

This integral is computable with moderate effort. First, one rescales variables: y =|x|u and z = |x|v, to obtain

I =∫

S3d

(x

|x|) ∫

d4u d4v

u2v2(x/|x| − u)2(u− v)2(x/|x| − v)2 .


The calculation is then carried out by means of ultraspherical polynomial [38, 39]techniques. We recall that these polynomials are defined from

(1 − 2xr + r2)−n =∞∑k=0

Cnk (x)rn,

for r < 1. There follows an expansion for powers of the propagator

1

(x − y)2n = 1

|y|n∞∑k=0

Cnk (xy)

( |x||y|

)k,

if, for instance, |x| < |y|. Using their orthogonality relation∫Sl

d

(x

|x|)Cnk

(xu

|x||u|)Cnl

(xv

|x||v|)

= δklCnk

(uv

|u||v|)n2d

k + n,to perform the angular integrals (in our case n = 1, l = 3), we obtain I as the sumof six radial integrals, corresponding to regions like |u| < |v| < 1, and so forth.Each one is equivalent to 2π6ζ(3). This yields finally the residue 12π6ζ(3) – thegeometrical factor 24 is always present. In consequence, now

Tµf (λx) = λ−12[Tµf (x)+ 12π6ζ(3) log λδ(x)].This is the first diagram which has a nontrivial topology, from the knot theoryviewpoint, and thus the appearance of a ζ -value is expected [37].

Consider now the two-loop ‘setting sun’ diagram that contributes to the two-point function in ϕ4

4 theory; it will prove instructive. One has to renormalize 1/x6,and the singular order is 2. Off (15) we read that[

1

x6

]R

= 1

2S3

[log(µ|x|)x6

]+ 3π2

87δ(x). (20)

Clearly, our formulae come rather close to simply iterating the operator S, as donein [33]. The last term obviously does not make a difference for the dilatationproperties; but we shall soon strengthen the case for not dropping it. One has[

1

x6

]R

(λx) = λ−6

([1

x6

]R

+ 24

8log λ7δ(x)

).

The reader may check that using Ww instead of Tw would bring to (20) theextra term π2µ2δ(x), with an unwelcome µ-power dependence. As we know, thiscomplicates the dilatation properties for the diagram. The terms polynomially de-pendent on µ are like the ‘junk DNA’ of the Epstein–Glaser formalism, as theycarry no useful information on the residues of QFT [2].

More generally, for quadratic divergences (such as also appear in the first (two-vertex) contribution to the two-point function of the ϕ6

3 and ϕ36 theories), one con-

structs the extension

[|x|−d−2]R = 1

2S3

(logµ|x|xd+2

)+ 32d

4d7δ(x) (21)


and

[|x|−d−2]R(λx) = λ−d−2

([|x|−d−2]R + 2d

2dlog λ7δ(x)

).

4. The Comparison with the Mathematical Literature

4.1. ON THE REAL LINE

Now we must muster support for the choice of Tw and Tµ. For the basics of distrib-ution theory, we recommend [40]. For concrete computations, a good place to startis the treatment by Hörmander in Section 3.2 of [41], of the extension problem forthe distributions

f (x) = x−l+ , x

−l− , |x|−l , |x|−lsgn(x), x−l

on the real line. Of course, these are not independent: x− is just the reflection of x+with respect to the origin, then |x|−l = x−l

+ + x−l− , and so on; note that x−1 is just

the ordinary Cauchy principal value of 1/x.On our side, for instance,

xf (x) = H(x) for f (x) = x−1+ ; xf (x) = sgn(x) for f (x) = |x|−1;

and so on. Then our formulae (15), for l odd, give

[x−l+ ]R = (−)l−1

(l − 1)!dl

dxl(H(x) log(µ|x|))+Hl−1δ

l−1,

or, say,

[|x|−l]R = (−)l−1

(l − 1)!dl

dxl(sgn(x) log(µ|x|))+ 2Hl−1δ

l−1,

and, for l even, simply

[|x|−l]R = (−)l−1

(l − 1)!dl

dxllog |x|. (22)

Hörmander invokes the natural method of analytic continuation of xz+, with zcomplex, plus residue subtraction at the simple poles at the negative integers. Ourformulae coincide with Hörmander’s – see, for example, his (3.2.5) – providedthat (a) we take µ = 1; and (b) Hk defined in (13) equals (as anticipated in thenotation) the sum of the first k terms of the harmonic series! This turns out to bethe case, although the proof, that the curious reader can find in [42, Ch. 6], is notquite straightforward. Thus we understand that in (15) and similar formulae Hk justmeans

∑kj=1 1/j .

Encouraged by this indication of being on the right track, we take a closerlook at the analytic continuation method. The point is that the function z �→

IMPROVED EPSTEIN–GLASER RENORMALIZATION IN COORDINATE SPACE I 73∫ ∞

0 xzφ(x) dx for �z > −1 is analytic, its differential being dz∫ ∞

0 xz log xφ(x) dx.Let us now consider the analytic continuation definition for xz+, where for simplic-ity we first take −2 < �z < −1. One gathers that

〈xz+, φ〉 =∫ ∞

0xzR0

0φ(x) dx.

We recall the proof of this:

〈xz+, φ〉 :=⟨

1

z + 1

d

dxxz+1

+ , φ

⟩= − 1

z+ 1limε↓0

∫ ∞

ε

xz+1φ′(x) dx.

A simple integration by parts, taking v = xz+1 and u = φ(x) − φ(0), completesthe argument.

Iterating the procedure, one obtains

〈xz+, φ〉 =∫ ∞

0xzRl−1

0 φ(x) dx

for −l − 1 < �z < −l, with l a positive integer. At z = −l, however, this formulafails because of the attendant infrared problem. Let us then compute the first twoterms of the Laurent development of xz: in view of

〈xz+, φ〉 =∫ µ−1

0xzRl−1

0 φ(x) dx +∫ ∞

µ−1Rl−2

0 φ(x) dx + φ(l−1)(0)µ−(z+l)

(l − 1)! (z + l) ,

the pole part is isolated. Therefore

limz→−l

[xz+ − (−)l−1δ(l−1)(x)

(l − 1)!(z + l)]

= Tµ(xl+)− δl−1(x) logµ.

Hörmander goes on to consider Hadamard’s finite part: that is, for x+, onestudies∫ ∞

ε

xzφ(x) dx,

where φ is always a test function, for any z ∈ C, and discards the multiples ofpowers ε−θ , for nonvanishing θ with �θ � 0, and the multiples of log ε. He provesthat this finite part coincides with the result of the analytic continuation method.

We do not need to review his proof, as we can show directly the identity of ourresults with finite part, by the following trick:

∫ ∞

ε

φ(x)

xldx =

l−1∑j=0

∫ µ−1

ε

φ(j)(0)

j ! xj−l dx +l−2∑j=0

∫ ∞

µ−1

φ(j)(0)

j ! xj−l dx +

+∫ µ−1

ε

x−lRl−10 φ(x) dx +

∫ ∞

µ−1x−lRl−2

0 φ(x) dx.


Then, as ε ↓ 0, the two last terms give rise to the Tµ(1/xl) renormalization and thesurviving finite terms cancel, except for the expected contribution −((φ(l−1)(0))/((l − 1)!)) logµ, coming from the first sum.

Denote the finite part of x−l+ by Pf(H(x))/xl , where Pf stands for pseudofunc-tion (or for partie finite, according to taste). In summary, we have proved:

PROPOSITION 1. On the real line, the T -operator leads to a one-parametergeneralization of the finite part and analytic continuation extensions, to wit,

[x−l+ ]R := Tµ(x

−l+ ) = Pf

H(x)

xl+ δl−1(x) logµ.

This generalization is in the nature of things. Actually, the finite part and ana-lytic continuation methods are not nearly as uniquely defined as some treatmentsmake them appear. For instance, at the negative integers the definition of the finitepart of xz changes if we substitute Aε for ε; and, analogously, one can slip in adimensionful scale in analytical prolongation formulae. The added flexibility ofthe choice of µ is convenient.

We parenthetically observe that the nonhomogeneity of Tµ, and then of Pf, isdirectly related to the presence of logarithmic terms in the asymptotic expansionfor the heat kernels of elliptic pseudodifferential operators [26].

Finally, we remark that the Laurent development for xz+|z=−l continues:

φ(l−1)(0)

ε(l − 1)! + Pf(x−l+ )+ ε

2Pf(x−l

+ log x+)+ ε2

3! Pf(x−l+ log2 x+)+ · · · (23)

with ε := z + l and the obvious definition for

Pf(x−l+ logm x+) = [x−l

+ logm x+]R;µ=1.

4.2. DIMENSIONAL REDUCTION

The phrase ‘dimensional reduction’ is used in the sense of ordinary calculus, it doesnot refer here to the method of renormalization of the same name. The reader mayhave wondered why we spend so much time on elementary distributions on R. Thereason, as it turns out, is that an understanding of the one-dimensional case is allthat is needed for the renormalization of |x|−d−k, for any k and in any dimension d;thus covering the basic needs of Euclidean field theory. For instance, one can define[x−4]R on R4 from knowledge of x−1

+ on R.Denote r := |x| and let f (r) be an amplitude on Rd , depending only on the

radial coordinate, in need of renormalization. We are ready now to simplify (15)by a method that generalizes Proposition 1 to any number of dimensions.

Given an arbitrary test function φ, consider its projection onto the radial-sum-values function φ �→ Pφ given by

Pφ(r) :=∫

|y|=1φ(ry).


We compute the derivatives of Pφ at the origin: (Pφ)(2m+1)(0) = 0 and

(Pφ)(2m)(0) = 2d,m7mφ(0).

To prove this, whenever all the β’s, and thus n, are even, use

(Pφ)(n)(0) =∑|β|=n

n! ∂βφβ!

∣∣∣∣x=0

∫|y|=1

yβ11 . . . y

βnn

=∑|β|=n

n! ∂βφβ!

∣∣∣∣x=0

2�(β1+12 ) · · ·�(βn+1

2 )

�(n+d2 ),

in consonance with (18); the integral vanishes otherwise. Note that Pφ can beconsidered as an even function defined on the whole real line. Then, whenever theintegrals make sense,

〈f (r), φ(x)〉Rd = 〈f (r)rd−1, Pφ(r)〉R+ ,

which in particular means that extension rules for H(r)f (r) on R give extensionrules for f (r)rd−1 on Rd . This we call dimensional reduction.

Before proceeding, let us put the examined real line extensions in perspective,by investigating how satisfactory our results are from a general standpoint, andwhether alternative renormalizations with better properties might exist. Note first,from (15):

PfH(x)

x= d

dx(H(x) log |x|).

For z not a negative integer, the property

x xz+ = xz+1+ (24)

obtains; and excluding z = 0 as well, we have

d

dxxz+ = z xz−1

+ . (25)

One can examine how the negative integer power candidates fare in respect ofthese two criteria: of course, except for x−l , which keeps all the good properties,homogeneity is irretrievably lost.

Actually, it is x−l+ that we need. One could define a renormalization [x−l+ ]diff

of x−l+ simply by

[x−l+ ]diff := (−)l−1 1

(l − 1)!dl

dxl(H(x) log |x|),

so [x−1+ ]diff = Pf(H(x)/x); and automatically the second (25) of the requirements

d

dx[x−l

+ ]diff = −l[x−l−1+ ]diff


would be fulfilled. This would be ‘differential renormalization’ in a nutshell. Itdiffers from the other extensions studied so far: from our previous results,

[x−l+ ]diff = [x−l

+ ]R + (−)l(Hl−1 + logµ)δl−1(x).

On the other hand, it is seen that

d

dxTµ(x

−l+ ) = −lTµ(x−l−1

+ )+ δl(x),

so that Tµ does not fulfill that second requirement; but in exchange, it does fulfillthe first one (24):

x+Tµ(x−l+ ) = Tµ(x

−l+1+ ).

There is no extension of xa+ for which both requirements simultaneously hold.It looks as if we are faced with a choice between [·]diff and Tµ(·) – which is

essentially Pf(·) – each one with its attractive feature. But the situation is in truthnot symmetrical: in higher-dimensional spaces the analogue of the first requirementcan be generalized to the renormalization of |x|−l ; whereas the analog of the secondthen cannot be made to work – have a sneak preview at (26).

Estrada and Kanwal define then, for k � 0 [43, 44],⟨Pf

(1

rd+k

), φ(x)

⟩Rd

:=⟨Pf

(1

rk+1

), Pφ(r)

⟩R+

;⟨[

1

rd+k

]diff

, φ(x)

⟩Rd

:=⟨[

1

rk+1

]diff

, Pφ(r)

⟩R+.

In view of (22), the case k odd is very easy, and then all the definitions coincide:

Pf

(1

rd+k

)= Tµ

(1

rd+k

)=

[1

rd+k

]diff

= rz|z=−d−k,

the function rz having a removable singularity at −d − k. However, in most in-stances in QFT k happens to be even, so we concentrate on this case. We note thatthe long paper [45] deals directly with Hadamard’s finite part on Rd .

We are not in need of new definitions. By going through the motions of changingto radial plus polar coordinates and back, one checks that, assuming a sphericallysymmetric weight function w, the evaluation 〈Twf (r), φ(x)〉 is equal to⟨

f (r), φ(x) − φ(0)− 7φ(0)

2!d r2 − · · · − w(r)2d,m7mφ(0)

(2m)!2d r2m

⟩;

the right-hand side being invariant under Tw. This was perhaps clear from thebeginning, from symmetry considerations. It means in particular that the differentputative definitions of Tµ on Rd obtained from Tµ on the real line all coincide withthe original definition, that is:


PROPOSITION 2. The Tµ operators are mutually consistent under dimensionalreduction.

Moreover,

r2qTµ(r−d−2m) = Tµ(r

−d−2m+2q)

follows, by using the easy identity

r27mδ(x) = 2m(2m+ d − 2)7m−1δ(x).

Therefore, it is now clear that

Tµ(r−d−2m) = Pf(r−d−2m)+ 2d,m7

mδ(x)

2d(2m)! logµ.

It remains to compute the derivatives. A powerful technique, based on ‘trun-cated regularization’ and calculation of the derivatives across surface jumps, wasdeveloped and clearly explained in [43]. It is rather obvious that for k − d odd the‘naïve’ derivation formulae (see right below) will apply. Whereas for k − d = 2meven, they obtain extra delta function terms; in particular for the powers of theLaplacian

7n[

1

rd+2m

]diff

= (d + 2m+ 2n− 2) · · · (d + 2m+ 2)(d + 2m)(2m+ 2) · · · ×× (2m+ 2n)

[1

rd+2m+2n

]diff

+ 2d,m

(2m)!n∑l=1

7nδ(x)

2m+ 2l − 1. (26)

The first term is what we termed the ‘naïve’ formula.Estrada and Kanwal do not explicitly give the powers of 7 for finite part. But

from it is a simple task to compute

7n[

1

rd+2m

]R

= (d + 2m+ 2n− 2) · · · (d + 2m+ 2)(d + 2m)(2m+ 2) · · · ×× (2m+ 2n)

[1

rd+2m+2n

]R

− 2d,m

(2m)!n∑l=1

(4(m+ l)+ d − 2)7nδ(x)

2(m+ l)(2m+ 2l + d − 2).

No one seems to have computed explicitly the distributional derivatives of thePf(x−l

+ logm x+) and the correspondingly defined Pf(r−l logm r), although theymight be quite helpful for Euclidean QFT on configuration space.

We next enterprise to tackle a comparison with methods of renormalization inreal space in the physical literature. Of those there are not many: it needs to besaid that the flame-keepers of the Epstein–Glaser method [46] actually work in


momentum space (using dispersion relation techniques). Euclidean configurationspace dimensional regularization, on the other hand, starting from [39], evolvedinto a powerful calculational tool in the eighties. With the advent of ‘differentialrenormalization’ [24] in the nineties, regularization-free coordinate space tech-niques came into their own: they are the natural ‘market competitors’ for the ideaspresented here.

We deal first with dimensional regularization.

5. Comparison with the QFT Literature

5.1. DIMENSIONAL REGULARIZATION AND ‘MINIMAL SUBTRACTION’

Dimensional regularization on real space, for primitively divergent diagrams, canbe identified with analytic continuation. To get the basic idea, it is perhaps conve-nient to perform first a couple of blind calculations. Start from the identity

µε |x|−d+ε = µε

εSx(|x|−d+ε).

Then, expanding in ε, on use of (15), it follows that

µε |x|−d+ε = 2dδ(x)

ε+ Sx log(µ|x|)

|x|d + O(ε).

The first term is a typical infinite (as 1/ε) counterterm of the dimensionally regular-ized theory. The order of the delta function derivative, 0 in this case, tells us that weare dealing with a logarithmic divergence. The coefficient 2n of the counterterm,or QFT residue, coincides with our scaling coefficient of Section 2. The secondterm is precisely [1/xd ]R , our renormalized expression.

Let us go to quadratic divergences. A brute-force computation establishes forthem the differential identity

µε |x|−d−2+ε = µε

2ε(1 − 32ε + 1

2ε2)S3(|x|ε−d−2). (27)

On the other hand, from (15),

S3(|x|−d−2) = 2d

d7δ(x). (28)

Performing in (27) the expansion with respect to ε, this yields

µε |x|−d−2+ε = 2d

2dε7δ(x)+ 1

2S3

(logµ|x|xd+2

)+ 32d

4d7δ(x)+ O(ε).

That is,

µε |x|−d−2+ε = 2d

2d

7δ(x)

ε+ [|x|d−2]R + O(ε).


A pattern has emerged: as before, there is a unique counterterm in 1/ε; the residuecoincides with our scaling coefficient; the order of the delta function derivativereminds us of the order of the divergence we are dealing with; and the ‘con-stant’ regular term is precisely [1/|x|d+2]R constructed in (21) according to ourrenormalization scheme.

The correspondence between the two schemes, at the present level, is absoluteand straightforward. It is then a foregone conclusion that we shall have µ-independ-ent residues, always coincident with the scaling factors, for the simple poles of1/|x|d+2m, and that the first finite term shall coincide with Tµ, provided we identifyour scale with ’t Hooft’s universal one. This is an immediate consequence of theLaurent development (23), transported to Rd by dimensional reduction. In symbols

µε |x|ε−d−2m = 2d,m

(2m)!ε7mδ(x)+ [|x|−d−2]R + O(ε). (29)

This substantiates the claim that Tµ effects a kind of minimal subtraction. Letus point out, in the same vein, that already in [26] the analytic continuation ofRiemann’s zeta function was evaluated as the outcome of a quantum field theory-flavoured renormalization process.

A word of warning is perhaps in order here. Performing the Fourier transformof these identities, we do not quite obtain the usual formulae for dimensionalregularization in momentum space. The nonresemblance is superficial, though, andrelated to choices of ‘renormalization prescriptions’. The beautiful correspondenceis ‘spoiled’ (modified) as well for diagrams with subdivergences, because in di-mensional regularization contributions will come to O(ε0) from the higher termsof the ε-expansion, when multiplied by the unavoidable singular factors; but, again,the difference is not deep: we show in III how one organizes the Laurent expan-sions with respect to d so as to make the correspondence with the T -subtractiontransparent.

Much was made in [47], and rightly so, of the importance of the perturbativeresidues in the dimensional regularization scheme. Residues for primitive diagramsare the single most informative item in QFT. The coefficients of higher-order polesare determined by the residues – consult the discussion in [48]. Now, the appealof working exclusively with well-defined quantities, as we do, would be muchdiminished if that information were to disappear in our approach. But we knowit is not lost: it is stored in the scaling properties.

5.2. DIFFERENTIAL RENORMALIZATION AND ‘NATURAL RENORMALIZATION’IN QFT

Differential renormalization, in its original form, turns around the following exten-sion of 1/x4 (in R4):

[1/x4]R,FJL := −1

47

logµ2x2

x2. (30)


At present, two main schools of differential renormalization seem still in vogue:the original and more popular ‘(constrained) differential renormalization’ of theSpanish school – see for instance [49] – and the ‘Russian school’ – inauguratedin [33]. This second method, as already reported, reduces to systematic use ofthe operators Sk+1, i.e., to our formulae (15) without the delta terms. Whereasthe first school has its forte in concrete 1-loop calculations for realistic theories,assuming compatibility of differentiation with renormalization, the second initiallystressed the development of global renormalization formulae for diagrams withsubdivergences, and the compatibility of Bogoliubov’s rules with renormalization.

Hereafter, we refer mainly to the original version. It proceeded from its men-tioned starting point to the computation of more complicated diagrams by reduc-tions to two-vertex diagrams. This involves a bewildering series of tricks, witnessmore of the ingenuity of the inventors than of the soundness of the method. V.g.,the tetrahedron diagram (considered already) is rather inelegantly renormalized bythe substitution 1/x2 �→ x2[1/x4]R,FJL. They get away with it, in that particularcase, because their expression is still not infrared divergent. But in nonprimitivediagrams infrared infinities may arise in relation with the need to integrate theproduct of propagators over the coordinates of the internal vertices in the diagram,and, in general, under the procedures of differential renormalization it is impossibleto avoid incurring infrared problems [50].

Even for primitively divergent diagrams, differential renormalization is not freeof trouble. In his extremely interesting paper, Schnetz [25] delivers a critique ofdifferential renormalization. In elementary fashion, notice that

xµ log(µ2x2)

x4= −1

2∂µ

[1 + log(µ2x2)

x2

],

and so

[1/x4]R = −1

47

1 + log(µ2x2)

x2.

This is to say,

[1/x4]R − [1/x4]R,FJL = π2δ(x). (31)

We contend that ‘our’ [1/x4]R and not [1/x4]R,FJL is the right definition. Of course,one is in principle free to add certain delta terms to each individual renormalizationand proclaim that to be the ‘right’ definition. However, 1/x4 on R4 is dimensionallyreduced to x−1

+ on R+ and because, as already pointed out, differential renormaliza-tion of this distribution is consistent with [x−1

+ ]R for µ = 1, the [·]R,FJL definitionis inconsistent with any of the natural alternatives we established in the previoussubsection. (It would clearly induce back an extra δ term in the definition of [x−1

+ ]Ron the real line, fully unwelcome in the context.)

In other words, if we want to make use both of sensible rules of renormaliza-tion for the radial integral (namely, including differential renormalization at this


level) and of Freedman, Johnson and Latorre’s formulae, we have to relinquish thestandard rules of calculus. This Schnetz noticed.

Schnetz proposes instead a ‘natural renormalization’ procedure on R4, boilingdown to the rule

7n+1 log(µ2x2)

x2= −

[4n+1n!(n+ 1)!

x2n+4

]R

+(

8π2Hn + 4π2

n+ 1

)7nδ(x) (32)

whose first instance is precisely the previous Equation (31). This he found byheuristically defining ‘natural renormalization’ as the one that relates renormaliza-tion scales at different dimensions without changing the definition of ordinary inte-grals or generating r-dependence in the renormalization of r-independent integrals;and by elaborate computations to get rid of the angular integrals.

His calculation is any rate correct, and the results can be read off (for d = 4,m = 0) our (26), taking into account (30) and (31). We have proved that ouroperator Tw in the context just amounts to ‘natural renormalization’.

Shortly after the inception of the differential renormalization, it naturally oc-curred to some people that a definite relation should exist between it and dimen-sional regularization. However, because of the shortcomings of the former, theylanded on formulae both messy and incorrect [51]. The reader is invited to comparethem with our (29).

The more refined version of differential renormalization in [33], coincides withour formulae for logarithmic divergences and eludes the main thrust of Schnetz’scritique; however, we have seen that in general it does not yield the Laurent devel-opment of the dimensionally regularized theory either. On the other hand, it mustbe said that the emphasis in [24, 25] in bringing in the Laplacian instead of theless intuitive albeit more fundamental Sk operators has welcome aspects, not onlybecause of the enhanced feeling of understanding, but also in that it makes thetransition to momentum space a trivial affair, as soon as the Fourier transform ofthe (evidently tempered) distribution x−2 log(µ2x2) is known.

The trinity of basic definitions in differential renormalization is then replacedby the identities

[1

x4

]R

= −1

27

logµ|x|x2

+ π2δ(x);

[1

x6

]R

= − 1

1672 logµ|x|

x2+ 5

8π27δ(x);

[logµ|x|x4

]R

= −1

47

log2µ|x| + logµ′|x|x2

+ π2

2δ(x);

the δ’s being absent in standard differential renormalization. In the next Section 6we shall see another demonstration of their importance.


The kinship of the EG method with differential renormalization à la Smirnovand Zavialov was recognized by Prange [29]; he was stumped for nonlogarithmicdivergences, though. See [52] in the same vein.

5.3. THE CONNECTION WITH BPHZ RENORMALIZATION

We still have left some chips to cash. We elaborate next the statement that BPHZsubtraction has no independent status from Epstein–Glaser, and that the validityof that renormalization method is just a corollary of the latter. This involves just atwo-line proof.

The Fourier transforms of the causally renormalized amplitudes exist at least inthe sense of tempered distributions. They are in fact rather regular. Taking Fouriertransforms is tantamount to replacing the test function by an exponential, which,according to the Cesàro theory of [26, 27], can preclude smoothness of the mo-mentum space amplitude only at the origin. The appearance of an (integrable)singularity at p = 0 is physically expected in a theory of massless particles.

Let us fix our conventions. We define Fourier and inverse Fourier transforms by

F [φ](p) := φ(p) :=∫

ddx

(2π)d/2e−ipxφ(x),

and

F−1[φ](p) := φ(p) :=∫

ddx

(2π)d/2eipxφ(x),

respectively. It follows that

(xµφ)ˇ(p) = (−i)µ∂µφ(p),where µ denotes a multiindex; so that, in particular,

(xµ)ˇ(p) = (−i)µ(2π)d/2δ(µ)(p).Also,

∂µφ(0) = (−i)µ(2π)−d/2〈pµ, φ〉. (33)

From this and the following consequence of (2):

〈F [f ], F−1[φ]〉 = 〈F [f ], F−1[Rk0φ]〉,there follows at once

F [f ](p) = Rk0F [f ](p).This is nothing but the BPHZ subtraction rule in momentum space.

We hasten to add:


• An expression such as F [f ] is not a priori meaningless: it is a well de-fined functional on the linear subspace of Schwartz functions φ whose firstmomenta

∫pαφ(p) ddp up to order k + 1 happen to vanish. (This is the

counterpart FSk+1, according to (33), of the distributions on real space actingon test functions vanishing up to order k + 1 at the origin.)

• Moreover, explicit expressions for these functionals on the external variablesare given precisely by the unrenormalized momentum space amplitudes!This circumstance constitutes the (deceptive) advantage of the BPHZ formal-ism for renormalization. We say deceptive because – as persuasively arguedin [35] – the BPHZ method makes no effective use of the recursive propertiesof renormalization (paper III) and then, when using it, prodigious amounts ofenergy must go into proving convergence of, and/or computing, the (ratherhorrendous) resulting integrals, into showing that the Minkowskian counter-parts define bona fide distributions, etc. Much more natural to remain on thenutritious ground of distribution theory on real space, throughout. But this hasnever been done.

• Also, the ∂µF [f ](0) for |µ| � k exist for massive theories.

For zero-mass models, the basic BPHZ scheme runs into trouble; this is duenaturally to the failure of ∂µf (0) to exist for |µ| = k, on account of the infraredproblem. Now, one can perform subtraction at some external momentum q �= 0,providing a mass scale. This is just the Fourier-mirrored version of standard EGrenormalization, with weight function e−iqx ; one only has to remember to use (7)instead of (4).

It is patent, though, that this last subtraction is quite awkward in practice, andwill introduce in the Minkowskian context a noncovariance which must be com-pensated by further subtractions. This prompted Lowenstein and Zimmermannto introduce their ‘soft mass insertions’ [53]. Which amounts to an epicycle toomany.

In the light of the approach advocated in this paper, there exist several simplerand more physical strategies.

• One strategy is to recruit our basic formula (5) in momentum space

F [f ](p)− jk−10 F [f ](p)−

∑|µ|=k

∂µF [fw](p)pµµ! .

Still with w(x) = e−iqx , this leads at once to

F [f ](p)− jk−10 F [f ](p)−

∑|µ|=k

∂µF [f ](q)pµµ! .

Note that the difference between two of these recipes is polynomial in pµ,with |µ| = k only, as it should. This can be more easily corrected for Lorentzcovariance, should the need arise [1].

• A second method is to exploit homogeneity in adapting our recipes for directuse in momentum space, in the spirit of [54] and [55].


• A third one is to perform Fourier analysis on our previous results. One has∫d4x

(2π)2e−ipx log(µ|x|)

x2= − 1

p2log

(C|p|2µ

),

where C := eγ ! 1.781072 . . . with γ the Euler–Mascheroni constant. Then,from (32), for instance for the ‘fish’ diagram in the ϕ4

4 model:[1/x4]R(p) = 1

4 [1 − log(C2p2/4µ2)],and more generally:

[1/x2k+4]R(p) = (−)k+1p2k

4k+1k!(k + 1)![

2 log|p|2µ

−B(k + 1)−B(k + 2)

],

where B(x) := d/dx(log �(x)) has been invoked, and we recall thatB(n) = −γ +Hn−1.

For the setting sun diagram in the ϕ44 model, in particular:

[1/x6]R = p2

16

(log

|p|γ2µ

− 5

4

). (34)

6. Some Examples in Massive Theories

The aim of this short section is to dispel any idea that the usefulness of EG-typerenormalization, and in particular of the T -subtraction, is restricted to masslessmodels. The overall conclusion, though, is that the massless theory keeps a nor-mative character. Our purposes being merely illustrative, we liberally borrow fromSchnetz [25], Prange [29], and Haagensen and Latorre [56].

The first example is nothing short of spectacular. Suppose we add to our orig-inal Lagrangian for ϕ4

4 a mass term 12m

2ϕ2 and treat it as a perturbation, for thecalculation of the new propagator. Then we would have for DF(x):

1

x2−

∫dx′ m2

(x − x′)2x′2 +∫

dx′ dx′′ m4

(x − x′)2(x′ − x′′)2x′′2 − · · ·

This ‘nonrenormalizable’ interaction is tractable with our method. We work inmomentum space, so we just have to consider the renormalization of 1/p2k fork > 1. This is read directly off (34), by inverting the roles of p and x, with theproviso that µ gets replaced by 1/µ, in order to keep the correct dimensions. Thenthe result is

DF(x) = 1

x2+ m2

2

∞∑n=0

m2nx2n

4nn!(n+ 1)!(

logµ|x|

2−B(n+ 1)−B(n+ 2)

).

On naturally identifying the scale µ = m, one obtains on the nose the exactexpansion of the exact result

DF(x) = m

|x|K1(m|x|).


Here K1 is the modified Bessel function of order 1. Had we kept the original EGsubtraction with a H(µ− |p|) weight, we would earn a surfeit of terms with extrapowers of µ, landing in a serious mess.

It is also interesting to see how well or badly fare the other ‘competitors’.Differential renormalization gives a expression of similar type but with differentcoefficients:

1

x2+ m2

2

∞∑n=0

m2nx2n

4nn!(n+ 1)!(

logµ|x|

2+ 2γ

).

To obtain the correct result, it is necessary to substitute a different mass scaleµn for each integral and to adjust ad hoc an infinity of such parameters. Dimen-sional regularization (plus ‘minimal’ subtraction of a pole term for each summandbut the first) fares slightly better, as it ‘only’ misses the B(n + 2) terms [25].The distribution-theoretical rationale for the success of the ‘illegal’ expansion per-formed is explained in [25].

Let us now look at the fish diagram in the massive theory. It is possible to use (3)instead of (6). Make the change of variables:

t = |x|s

; dt = −|x| ds

s2.

For the renormalized amplitude, one gets

−S[m2

x4

∫ ∞

|x|ds sK1(ms)

2

].

Now,∫ds sK1(ms)

2 = s2

2

(K2

0 (s)+ 2K0(s)K1(s)

s−K2

1 (s)

)

can be easily checked from

K ′0(s) = K1(s), K ′

1(s) = −K0(s)− K1(s)

s.

The final result is then

7

[m2

2

(K2

0 (m|x|)−K21 (m|x|)+ K0(m|x|)K1(m|x|)

m|x|)].

Had we used (6), the upper limit of the integral would become 1/µ, and the resultwould be modified by

m2

4µ2x2(K2

1 (m/µ)−K2(m/µ)K0(m/µ)).

At the ‘high energy’ limit, as µ ↑ ∞ and |x| ↓ 0, this interpolates between theprevious result and the renormalization in the massless case.


However, this method becomes cumbersome already for renormalizing D3F . It

is convenient to modify the strategy, and to use in this context differential renor-malization, corrected in such a way that the known renormalized mass zero limitis kept. This idea succeeds because of the good properties of our subtraction withrespect to the mass expansion. For instance, away from zero [56],

(mK1(m|x|)

|x|)3

= m2

16(7− 9m2)(7−m2)(K0(m|x|)K2

1 (m|x|)+K30 (m|x|)).

Note the three-particle ‘threshold’. To this Haagensen and Latorre add a term ofthe form

π 2

4log

2µ

γm7δ(x),

to which, for reasons sufficiently explained, we should add a term of the form(5π2/8)7δ(x). A term proportional to δ (thus a mass correction) is also present.As they indicate, it is better fixed by a renormalization prescription.

7. Conclusion

We have delivered the missing link of the EG subtraction method to the standardliterature on extension of distributions. The improved subtraction method sits at thecrossroads in regard to dimensional regularization in configuration space; differ-ential renormalization; ‘natural’ renormalization; and BPHZ renormalization. Thediscussions in the previous sections go a long way to justify the conjecture (madeby Connes, and independently by Estrada) that Hadamard’s finite part theory isin principle enough to deal with quantum field theory divergences. To accomplishthat feat, however, it must go under the guise of the T -projector; this gives thenecessary flexibility to deal with complicated diagrams with subdivergences [2].

Our method leads to emphasize the scaling properties of EG renormalization.It turns out that the very same properties play the essential role in the proof of ul-traviolet renormalizability of quantum field theory on curved backgrounds, finallydelivered by Hollands and Wald in an outstanding series of papers [57, 58] – seealso [59].

Acknowledgements

I am indebted to Ph. Blanchard for much encouragement and discussions, to S. Laz-zarini for a computation pertaining to Subsection 5.1, to C. Brouder for a couple ofgood suggestions, and to D. Kreimer and J. C. Várilly for comments on an earlierversion of the manuscript. Support from VI-UCR is acknowledged.


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10. Collins, J. C.: Renormalization, Cambridge Univ. Press, Cambridge, 1984.11. Dosch, H. G. and Müller, V. F.: Fortschr. Phys. 23 (1975), 661.12. Bellissard, J.: Comm. Math. Phys. 41 (1975), 235.13. Gracia-Bondía, J. M.: Phys. Lett. B 482 (2000), 315.14. Brunetti, R. and Fredenhagen, K.: Comm. Math. Phys. 208 (2000), 623.15. Itzykson, C. and Zuber, J.-B.: Quantum Field Theory, McGraw-Hill, New York, 1980.16. Stora, R.: Lagrangian field theory, In: C. DeWitt-Morette and C. Itzykson (eds), Proc. Les

Houches School, Gordon and Breach, New York, 1973.17. Popineau, G. and Stora, R.: A pedagogical remark on the main theorem of perturbative

renormalization theory, Unpublished preprint, CPT & LAPP-TH (1982).18. Stora, R.: A note on elliptic perturbative renormalization on a compact manifold, Unpublished

undated preprint, LAPP-TH.19. Pinter, G.: Ann. Phys. 8 10 (2001), 333.20. Pinter, G.: Epstein–Glaser renormalization: finite renormalizations, the S-matrix of D4 theory

and the action principle, Doktorarbeit, DESY, 2000.21. Gracia-Bondía, J. M. and Lazzarini, S.: Connes–Kreimer–Epstein–Glaser renormalization,

hep-th/0006106.22. Güttinger, W.: Phys. Rev. 89 (1953), 1004.23. Bogoliubov, N. N. and Parasiuk, O. S.: Acta Math. 97 (1957), 227.24. Freedman, D. Z., Johnson, K. and Latorre, J. I.: Nuclear Phys. B 371 (1992), 353.25. Schnetz, O.: J. Math. Phys. 38 (1997), 738.26. Estrada, R., Gracia-Bondía, J. M. and Várilly, J. C.: Comm. Math. Phys. 191 (1998), 219.27. Estrada, R.: Proc. Roy. Soc. London A 454 (1998), 2425.28. Estrada, R. and Kanwal, R. P.: A Distributional Approach to Asymptotics, Theory and

Applications (2nd edn), Birkhäuser, Boston, 2002.29. Prange, D.: J. Phys. A 32 (1999), 2225.30. Steinmann, O.: Perturbation Expansions in Axiomatic Field Theory, Lecture Notes in Phys. 11,

Springer, Berlin, 1971.31. Estrada, R.: Internat. J. Math. and Math. Sci. 21 (1998), 625.32. Gracia-Bondía, J. M., Várilly, J. C. and Figueroa, H.: Elements of Noncommutative Geometry,

Birkhäuser, Boston, 2001.33. Smirnov, V. A. and Zavialov, O. I.: Theoret. and Math. Phys. 96 (1993), 974.34. Gelfand, I. M. and Shilov, G. E.: Generalized Functions I, Academic Press, New York, 1964.35. Kuznetsov, A. N., Tkachov, F. V. and Vlasov, V. V.: Techniques of distributions in perturbative

quantum field theory I, hep-th/9612037, Moscow, 1996.36. Grigore, D. R.: Ann. Phys. (Leipzig) 10 (2001), 473.37. Kreimer, D.: Knots and Feynman Diagrams, Cambridge Univ. Press, Cambridge, 2000.38. Andrews, G. E., Askey, R. and Roy, R.: Special Functions, Cambridge Univ. Press, Cambridge,

1999.


39. Chetyrkin, K. G., Kataev, A. L. and Tkachov, F. V.: Nuclear Phys. B 174 (1980), 345.40. Blanchard, Ph. and Brüning, E.: Generalized Functions, Hilbert Spaces and Variational

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Reading, MA, 1989.43. Estrada, R. and Kanwal, R. P.: Proc. Roy. Soc. London A 401 (1985), 281.44. Estrada, R. and Kanwal, R. P.: J. Math. Anal. Appl. 141 (1989), 195.45. Blanchet, L. and Faye, G.: gr-qc/0004008, Meudon, 2000.46. Scharf, G.: Finite Quantum Electrodynamics: the Causal Approach, Springer, Berlin, 1995.47. Kreimer, D.: Talks at Abdus Salam ICTP, Trieste, March 27, 2001 and Mathematical Sciences

Research Institute, Berkeley, April 25, 2001.48. Gross, D. J.: Applications of the renormalization group to high-energy physics, In: R. Balian

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89

Trace Functionals for a Classof Pseudo-Differential Operators in R

n

FABIO NICOLADipartimento di Matematica, via Carlo Alberto 10, 10123 Turin, Italy. e-mail: [email protected]

(Received: 25 March 2002; in final form: 11 September 2002)

Abstract. In this paper we define trace functionals on the algebra of pseudo-differential operatorswith cone-shaped exits to infinity. Furthermore, we improve the Weyl formula on the asymptoticdistribution of eigenvalues and make use of it in order to establish inclusion relations between theinterpolation normed ideals of compact operators in L2(Rn) and the above operator classes.

Mathematics Subject Classifications (2000): 47G30, 58J42.

Key words: Dixmier trace, noncommutative residue, pseudo-differential operators, trace functionals,Weyl formula.

1. Introduction

In this paper we study pseudo-differential operators in Rn with symbols satisfying

estimates of product type. The basic ideas of this calculus go back to Shubin [29],Parenti [22], Feygin [10], Grushin [13], Cordes [5, 6], and Schrohe [23], and itsproperties follow from the general Weyl calculus of Hörmander, see [15], ChapterXVIII. In fact, the corresponding symbol classes Sµ,ρ are just the classes S(m, g)with weight function m(x, ξ) = 〈x〉ρ〈ξ 〉µ and slowly varying metric

gx,ξ = |dx|21 + |x|2 + |dξ |2

1 + |ξ |2 . (1.1)

These operator classes play an important role in scattering theory, as the resol-vent of the Laplacian can be viewed as a holomorphic family taking values inthe space L−2,0 (Lµ,ρ = Op(Sµ,ρ)). Actually, here as in other applications, it ismostly the subalgebra Lµ,ρcl(ξ,x) of operators which are classical both in x and ξ (seeDefinition 2.1, below) which arises, cf. the book by Schulze [27].

We observe that the corresponding calculus can be easily transferred to noncom-pact manifolds with cone-shaped exits to infinity, i.e. defined in terms of changesof variables which are classical symbols in x; however, in the following, we keepthe R

n frame for simplicity.The first problem under investigation here is the existence of trace function-

als (i.e. functionals which vanish on commutators) on the algebra⋃

µ,ρ∈ZLµ,ρ

cl(ξ,x)

90 FABIO NICOLA

of all operators of integer order. This problem was studied in [31, 32] for clas-sical operators on compact manifolds and extended to several operator algebrasby Guillemin [12], Fedosov et al. [8, 9], Melrose [19], Melrose and Nistor [20],Schrohe [24–26], and Boggiatto and Nicola [2].

Following the ideas of [20], we start from the usual trace functional

Tr(aw) =�

a(x, ξ) dx dξ, (1.2)

defined on trace class operators in L2(Rn) and extend it using holomorphic fam-ilies; here aw denotes the pseudo-differential operator with Weyl symbol a. Pre-cisely, if in (1.2) we replace a by any holomorphic family a(τ, z) ∈ S

µ+z,ρ+τcl(ξ,x)

such that a(0, 0) = a, we see that Tr(a(τ, z)w) is defined and holomorphic for�z < −µ− n, �τ < −ρ − n, and extends to a meromorphic function of τ , z withat most simple poles on the surfaces z = −µ− n+ j , τ = −ρ − n+ k, j, k ∈ N.In a neighborhood of (0, 0) ∈ C × C we shall have

Tr(a(τ, z)w) = 1

τzTrψ,e(a

w)− 1

zTrψ(a

w)− 1

τTre(a

w)+∑h,i�0

ch,iτhzi,

defining in this way the functionals Trψ,e, Trψ , Tre. We shall determine an ex-plicit expression for each of these functionals and see that Trψ,e is a trace on thealgebra

⋃µ,ρ∈Z

Lµ,ρ

cl(ξ,x), whereas the restrictions Trψ and Tre of Trψ and Tre to⋃µ∈Z

⋂ρ∈Z

Lµ,ρ

cl(ξ,x) and⋃

ρ∈Z

⋂µ∈Z

Lµ,ρ

cl(ξ,x), respectively, define traces on these al-gebras. The uniqueness of the traces involved is investigated for quotient algebras.

In Section 4 we study inclusion relations between the classes Lµ,ρcl(ξ,x), whereµ < 0, ρ < 0, and the interpolation normed ideals L(p,∞)(L2(Rn)), 1 � p < ∞,(cf. [4, 11]). In fact, the spaces Lµ,ρcl(ξ,x), under our assumption on µ and ρ, are allcontained in the ideal K(L2(Rn)) of compact operators in L2(Rn), obtaining, inparticular, trace class operators when µ < −n, ρ < −n. On the other hand, thespaces L(p,∞)(L2(Rn)), defined as the sets of compact operators with an eigenvaluesequence µk(|T |) = O(k−1/p), also are contained in K(L2(Rn)) (more preciselythey define a filtration of K(L2(Rn))) and in turn contain the ideal B1(L

2(Rn)) oftrace class operators.

In order to establish such inclusion relations we investigate the asymptotic ba-haviour of the spectrum of elliptic self-adjoint operators of positive order. We makeuse of the results of Hörmander [14] on the spectrum of operators with positiveWeyl symbol p ∈ S(p, g), to improve the Weyl formula of [18], obtaining betterestimates of the remainder. We point out that in the case µ = ρ in the Weylformula for the counting function N(λ) a factor log λ appears (cf. [21], and [1],Section 2.7 for the case of ordinary differential operators). This phenomenon leadsus to introduce new normed ideals, denoted by L

(p,∞)

log (L2(Rn)), 1 � p < ∞, seeDefinition 4.2.

The limit cases of the normed ideals L(1,∞)(L2(Rn)) and L(1,∞)

log (L2(Rn)) areparticularly important as they correspond to domains of two Dixmier traces (cf. [7]).

TRACE FUNCTIONALS 91

Dixmier traces, in a broad sense, are defined taking a class of compact operatorsfor which the usual trace diverges at a given (suitable) rate. Then, with any suchoperator it is associated, via a normalizing sequence, a bounded sequence, and2-dilatation invariant states in l∞(N) provide (nonnormal) traces. Informally, ifα = (αN) is a convenient sequence, we define

Trα,ωT = limω

1

αN

N∑k=0

µk(T ) = Dixmier trace for T � 0, µk(T ) ↘,

see Section 4 for the precise definition.We emphasize that, in a strict sense, by Dixmier trace is generally meant a trace

which ‘sums’ logarithmic divergences, i.e. whose domain is the idealL(1,∞)(L2(Rn)). However, we shall consider the above, more general construc-tion, since we shall also need to consider the trace associated with the sequenceαN = (logN)2 and therefore with domain L(1,∞)

log (L2(Rn)). Indeed, we shall provethe following theorem.

THEOREM 1.1. We have

Trψ,e(aw) = 2n2 Trα,ω(a

w) for a ∈ S−n,−ncl(ξ,x) , (1.3)

Trψ(aw) = nTrα,ω(a

w) for a ∈ S−n,ρcl(ξ,x) with ρ ∈ Z, ρ < −n, (1.4)

Tre(aw) = nTrα,ω(a

w) for a ∈ Sµ,−ncl(ξ,x) with µ ∈ Z, µ < −n, (1.5)

independently of ω, where αN = (logN)2 in (1.3), whereas αN = logN in (1.4)and (1.5).

It is well known that the analogous result for classical operators on compactmanifolds is due to Connes [3].

After completing the present paper, we were acquainted of the recent contribu-tion of Lauter and Moroianu [17], which obtain results corresponding to those putforward in Section 3 within the framework of the double-edge pseudo-differentialcalculus on manifolds with fibered boundaries.

2. Basic Calculus and Holomorphic Families

We briefly recall the definitions of some symbol classes and the basic properties ofthe corresponding operators; we follow Schulze [27] in notations and terminology.

The symbol classes Sµ,ρ := Sµ,ρ(Rn × Rn), µ, ρ ∈ R are defined by the

inequalities

|∂αx ∂βξ a(x, ξ)| � Cα,β(1 + |ξ |)µ−|β|(1 + |x|)ρ−|α|,

for all x, ξ ∈ R. We denote by Lµ,ρ the space of the corresponding pseudo-differential operators. If a is a symbol in Sµ,ρ , we shall choose the notation aw

for the pseudo-differential operator with Weyl symbol a.

92 FABIO NICOLA

In order to consider classical symbols, we need to introduce some functionspaces.

We denote by S(µ)ξ (S[µ]ξ ) the space of all functions a ∈ C∞(Rn × (Rn \ {0}))

(resp. in C∞(Rn × Rn)) homogeneous of degree µ with respect to ξ (resp. homo-

geneous for |ξ | � 1); in the same way we define the spaces S(ρ)x and S[ρ]x . Then we

set

Sµ,[ρ] := Sµ,ρ ∩ S[ρ]x and S[µ],ρ := Sµ,ρ ∩ S[µ]

ξ ,

and we define Sµ,[ρ]cl(ξ) ⊂ Sµ,[ρ] as the space of all a(x, ξ) ∈ Sµ,[ρ] such that there are

elements aj ∈ S[µ−j ]ξ ∩ S[ρ]

x , j ∈ N, with

a(x, ξ) −N∑j=0

aj (x, ξ) ∈ Sµ−(N+1),ρ

for every N ∈ N. Analogously we obtain the spaces S[µ],ρcl(x) .

DEFINITION 2.1. A symbol a ∈ Sµ,ρ(Rn × Rn) is called classical in x and ξ if

it has the following properties:

(i) there are symbols aj ∈ S[µ−j ],ρcl(x) , j ∈ N, such that

a(x, ξ) −N∑j=0

aj (x, ξ) ∈ Sµ−(N+1),ρ,

for every N ∈ N.(ii) there are symbols bk ∈ S

µ,[ρ−k]cl(ξ) , k ∈ N, such that

a(x, ξ) −N∑k=0

bk(x, ξ) ∈ Sµ,ρ−(N+1),

for every N ∈ N.

We shall denote by Sµ,ρcl(ξ,x) the space of classical symbols in x, ξ and by Lµ,ρcl(ξ,x) thespace of the corresponding pseudo-differential operators.

For instance, the symbol a(x, ξ) = (1 + |x|2)ρ/2(1 + |ξ |2)µ/2 is in Sµ,ρcl(ξ,x).As with the standard calculus, one sees that every classical symbol a determines

its asymptotic expansions in homogeneous terms in a unique way, i.e. there areunique mappings

σµ−jψ : Sµ,ρcl(ξ,x) → S

(µ−j)ξ , (2.1)

σ ρ−ke : Sµ,ρcl(ξ,x) → S(ρ−k)

x , (2.2)


for all j, k ∈ N. Furthermore, one can also consider the homogeneous componentof degree ρ − k (with respect to x) of σµ−j

ψ (a) (which in turn coincides with thehomogeneous component of degree µ − j of σρ−k

e (a)); in this way one obtainsunique mappings

σµ−j,ρ−kψ,e : Sµ,ρcl(ξ,x) → S

(µ−j)ξ ∩ S(ρ−k)

x (2.3)

for all k, j ∈ N.

DEFINITION 2.2. A symbol a(x, ξ) ∈ Sµ,ρ(Rn × Rn) is called elliptic if for

suitable constants C,R > 0 it satisfies the inequality

|a(x, ξ)| � C(1 + |x|)ρ(1 + |ξ |)µ, (2.4)

for all (x, ξ) ∈ Rn × R

n, with |x| + |ξ | � R.

For a classical symbol a ∈ Sµ,ρ

cl(ξ,x) the ellipticity condition (2.4) is equivalent torequiring that the principal symbols σµψ (a), σ

ρe (a), σ

µ,ρψ,e (a) do not vanish on their

definition domains.It is also useful to define a scale of weighted Sobolev spaces as follows:

DEFINITION 2.3. Let s, r ∈ R; we set Hs,δ(Rn) = {〈x〉−δu;u ∈ Hs(Rn)}.

PROPOSITION 2.4. Let A ∈ Lµ,ρ(Rn). Then A induces a continuous mapHs,δ(Rn) → Hs−µ,δ−ρ(Rn).

For the proof of Proposition 2.4 see, for example, [27].In the following, we will be interested in the following operator algebras, con-

structed beginning from Lµ,ρ

cl(ξ,x).

DEFINITION 2.5. Let

I := Op(S(R2n)) and A :=⋃ρ∈Z

⋃µ∈Z

Lµ,ρ

cl(ξ,x)/I.

We define the two-sided ideals of A

Iψ =⋃µ∈Z

⋂ρ∈Z

Lµ,ρ

cl(ξ,x)/I, Ie =⋃ρ∈Z

⋂µ∈Z

Lµ,ρ

cl(ξ,x)/I

and the quotient algebras

Aψ = A/Ie, Ae = A/Iψ, Aψ,e = A/(Iψ + Ie).

In the next section, we use holomorphic families of classical operators, accord-ing to the following definition (cf. [20]):

94 FABIO NICOLA

DEFINITION 2.6. Let .1, .2 ⊂ C be open subsets of the complex plane andlet h1: .1 → C, h2: .2 → C be holomorphic functions. We call holomorphic(symbol) families of order (h1(z), h2(τ )) functions of the particular product type

.1 ×.2 � (τ, z) �→ a(τ, z)

= [x]h2(τ )[ξ ]h1(z)b(τ, z, x, ξ) ∈ S�h1(z),�h2(τ )

cl(ξ,x) ,

where [·] denotes an arbitrary strictly positive C∞ function on Rn with [y] = |y|

for |y| � 1 and b(τ, z, x, ξ) ∈ S0,0cl(ξ,x) is holomorphic as a function of (τ, z) for

every fixed x, ξ .We denote by FSh1(z),h2(τ )

cl(ξ,x) the space of these holomorphic families.

Remark 2.7. Given a(x, ξ) ∈ Sµ,ρ

cl(ξ,x), there always exists a holomorphic fam-

ily a(τ, z) ∈ Sµ+z,ρ+τcl(ξ,x) with a(0, 0) = a; it suffices to choose a(τ, z, x, ξ) =

[x]τ [ξ ]za(x, ξ) from the previous definition.

3. Trace Functionals

The main result of this section is the explicit construction of trace functionals foreach of the algebras in Definition 2.5. These traces come from residues of the traceof holomorphic operator families, according to ideas of Wodzicki [31].

Firstly, we recall that on the ideal of regularizing operators every trace is amultiple of the functional

Tr(aw) = (2π)−n�

a(x, ξ) dx dξ, (3.1)

i.e. the usual operator trace. The integral (3.1) extends by continuity to a ∈ Sµ,ρ

provided µ < −n, ρ < −n. In order to extend it further, we need to regularizethe resultant divergent integral; we do this using holomorphic families (cf. Defini-tion 2.6).

LEMMA 3.1. If a(τ, z) ∈ FSµ+z,ρ+τcl(ξ,x) is a holomorphic symbol family, then the

function t (τ, z) := Tr(a(τ, z)w) is defined and holomorphic for �z < −µ − n,�τ < −ρ− n, and extends to a meromorphic function of τ , z with, at most, simplepoles on the surfaces

z = −µ− n+ j, τ = −ρ − n+ k, j, k ∈ N.

Proof. We can write

a(τ, z, x, ξ) = [x]τ [ξ ]za′(τ, z, x, ξ),

where a′(τ, z, x, ξ) ∈ Sµ,ρ

cl(ξ,x) is holomorphic with respect to τ , z. So we have

t (τ, z) = (2π)−n�

[x]τ [ξ ]za′(τ, z, x, ξ) dx dξ.


Now we write

t (τ, z) = t1(τ, z)+ t2(τ, z)+ t3(τ, z)+ t4(τ, z),

where t1, t2, t3, t4 are the integrals respectively on

A1 = {|x| � ε, |ξ | � 1}, A2 = {|x| � ε, |ξ | � 1},A3 = {|x| � ε, |ξ | � 1}, A4 = {|x| � ε, |ξ | � 1}.

To prove Lemma 3.1, it would suffice to set ε = 1, but in view of future develop-ments, it is useful to work with an arbitrary ε � 1.

Clearly t1(τ, z) is an entire function.As t2 is concerned, we note that for |ξ | � 1 and every p ∈ N, p � 1, we have

a(x, ξ) =p−1∑j=0

σµ−jψ (a(τ, z))(x, ξ/|ξ |)|ξ |z+µ−j + rp(τ, z, x, ξ),

with a remainder rp ∈ Sz+µ−p,τ+ρcl(ξ,x) . Substituting this expression for a(x, ξ) in the

integral

t2(τ, z) =∫

|x|�ε

∫|ξ |�1

a(x, ξ) dx dξ

and introducing polar coordinates for the integration in the variables ξ , we obtain

t2(τ, z) = −(2π)−np−1∑j=0

1

z+ µ+ n− j×

×∫

|x|�ε

∫Sn−1

σµ−jψ (a(τ, z)) dθ dx + Rp,ε(τ, z), (3.2)

where Rp,ε(τ, z) is holomorphic for �z < −µ− n+ p and all τ ∈ C.Interchanging the roles of the variables x, ξ we obtain

t3(τ, z) = −(2π)−nq−1∑k=0

ετ+ρ+n−k

τ + ρ + n− k×

×∫

|ξ |�1

∫Sn−1

σ ρ−ke (a(τ, z)) dθ dξ + R′

q,ε(τ, z), (3.3)

where R′q,ε(τ, z) is holomorphic for �τ < −ρ − n+ q and all z ∈ C.

Finally, repeating the same argument twice, we get

t4(τ, z) = (2π)−np−1∑j=0

q−1∑k=0

1

z+ µ+ n− j

ετ+ρ+n−k

τ + ρ + n− k×

×∫

Sn−1

∫Sn−1

σµ−j,ρ−kψ,e (a(τ, z)) dθ dθ ′ +

+p−1∑j=0

1

z+ µ+ n− jR′′q,j,ε(τ, z)+ R′′′

p,ε(τ, z), (3.4)

96 FABIO NICOLA

where R′′q,j,ε(τ, z) is holomorphic for �τ < −ρ − n + q and all z ∈ C, and

R′′′p,ε(τ, z) is holomorphic for �z < −µ+ n+ p and all τ ∈ C.

So, we have verified that t (τ, z) extends to a meromorphic function on �τ <−ρ−n+q, �z < −µ+n+p. As p and q are arbitrary, this concludes the proof. ✷

Remark 3.2. Note that if a(τ, z) ∈ FSµ+z,ρ+τcl(ξ,x) is a holomorphic family with

a(0, 0) = 0, then Tr(a(τ, z)w) is holomorphic near (0, 0).

Now, let a ∈ Sµ,ρ

cl(ξ,x), µ, ρ ∈ Z be a classical symbol and let a(τ, z) be aholomorphic family with a(0, 0) = a (cf. Remark 2.7). Consider the functionalsdefined by

τzTr(a(τ, z)w)

= Trψ,e(aw)− τ Trψ(a

w)− z Tre(aw)+ τ 2V + τzV ′ + z2V ′′, (3.5)

where V , V ′, V ′′ are holomorphic near (0, 0). In view of Remark 3.2, they do notdepend on the choice of the holomorphic family a(τ, z).

PROPOSITION 3.3. The functionals Trψ,e, Trψ , Tre defined in (3.5) have thefollowing explicit expressions:

Trψ,e(aw) = (2π)−n

∫Sn−1

∫Sn−1

σ−n,−nψ,e (a) dθ dθ ′, (3.6)

Trψ(aw) = (2π)−n lim

ε→+∞

(∫|x|�ε

∫Sn−1

σ−nψ (a) dθ dx − (2π)n log ε Trψ,e(a

w)−

−ρ+n∑i=1

εi

i

∫Sn−1

∫Sn−1

σ−n,i−nψ,e (a) dθ dθ ′

), (3.7)

Tre(aw) = (2π)−n lim

ε→+∞

(∫|ξ |�ε

∫Sn−1

σ−ne (a) dθ dξ − (2π)n log ε Trψ,e(a

w)−

−µ+n∑i=1

εi

i

∫Sn−1

∫Sn−1

σi−n,−nψ,e (a) dθ dθ ′

). (3.8)

Proof. We refer to the proof of Lemma 3.1, where we now take p > µ + n,q > ρ + n.

Expression (3.6) follows at once as limit lim(τ,z)→(0,0) τzTr(a(τ, z)w) using theexpressions (3.2), (3.3), (3.4).

To prove (3.7), we observe that we can obtain Trψ as

Trψ(aw) = − lim

τ→0τ−1 lim

z→0(τzTr(a(τ, z)w)− Trψ,e(a

w)). (3.9)


When we perform the most internal limit, the expressions τzt1(τ, z) and τzt3(τ, z)vanish as well as τzR′′′

p,ε(τ, z) and the terms of the sums in (3.2) and (3.4) forj �= µ − n. What remains obviously is independent of ε but, on the other hand,as ε → +∞ the expression τR′′

q,µ+n,ε(τ, 0) and the terms of the sum in (3.4) withk > ρ + n tend to zero uniformly for small τ . Then we have

limz→0

(τzTr(a(τ, z)w)− Trψ,e(aw))

= (2π)−nτ limε→0

(−∫

|x|�ε

∫Sn−1

σµ−jψ (a(τ, 0)) dθ dx+

+ ετ

τ

∫Sn−1

∫Sn−1

σ−n,−nψ,e (a(τ, 0)) dθ dθ ′ − (2π)n

τTrψ,e(a

w)+

+ρ+n−1∑k=0

ετ+ρ+n−k

τ + ρ + n− k

∫Sn−1

∫Sn−1

σ−n,ρ−kψ,e (a(τ, 0)) dθ dθ ′

),

from which, by (3.9), (3.7) follows. At the same way one proves (3.8). ✷Remark 3.4. Let us note that the restrictions Trψ and Tre of Trψ and Tre to⋃

µ∈ZLµ,−n−1cl(ξ,x) and

⋃ρ∈Z

L−n−1,ρcl(ξ,x) are given by

Trψ(aw) =

∫Rnx

∫Sn−1

σ−nψ (a) dθ dx, a ∈

⋃µ∈Z

Sµ,−n−1cl(ξ,x) , (3.10)

Tre(aw) =

∫Rnξ

∫Sn−1

σ−ne (a) dθ dξ, a ∈

⋃ρ∈Z

S−n−1,ρcl(ξ,x) , (3.11)

and Trψ and Tre turn out to be just the finite parts of the integrals in (3.10) and (3.11)when a ∈ ⋃µ∈Z,ρ∈Z

Sµ,ρ

cl(ξ,x). Furthermore, the functional Trψ and Tre vanish on Ieand Iψ , respectively, so that they are well defined on Aψ and Ae as continuousextensions of Trψ and Tre.

THEOREM 3.5. The functional Trψ,e defines a trace on the algebra A whichvanishes on Iψ and Ie and therefore it induces traces on Aψ , Ae and Aψ,e. On Iψand Ie , trace functionals are given, respectively, by Trψ and Tre defined in (3.10)and (3.11). For all these algebras, the above functionals are the unique traces upto multiplication by a constant.

Proof. In all cases, the statement easily follows from the same arguments of theproof of Theorem 1.4 of [9]. See also [2] for a version in R

n, and the other paperson Wodzicki’s residue listed in the references. To avoid an overweight of the paper,we prefer then to omit any detail. ✷

Remark 3.6. Theorem 3.5 tells us that for each of the algebras in Definition 2.5,vanishing of the corresponding trace characterizes commutators.

98 FABIO NICOLA

4. Dixmier Traces

We begin by reviewing the construction of nonnormal Dixmier traces. We con-sider traces whose natural domains are contained in the ideal K(H) of compactoperators on the Hilbert space H .

For T ∈ K(H), let µn(T ), n ∈ N, be the sequence of the eigenvalues of |T |,counted with their multiplicity and labelled in decreasing order and let σN(T ) =∑N

n=0 µn(T ), N ∈ N. For a fixed sequence α of positive numbers αN such that

(i) αN → +∞;(ii) α0 > α1 − α0 and αN+1 − αN � αN+2 − αN+1 for N ∈ N;

(iii) α−1N α2N → 1,

we define the ideal

Iα(H) := {T ∈ K(H) : α−1N σN(T ) ∈ l∞(N)}.

Then, consider a linear form ω on Cb(1,∞), the space of the continuous bound-ed function on [1,∞], with ω � 0, ω(1) = 1 and ω(f ) = 0 if limx→+∞ f (x) = 0.Given a bounded sequence a = (an)n�1, we construct the function

fa =∑n�1

anχ[n−1,n) ∈ L∞(R+)

and define the ω-limit limω an = ω(Mfa) where, for g ∈ L∞(R+),

Mg(t) := 1

log t

∫ t

1

g(s)

sds

is the Cesàro mean of g. In the case of convergent sequences, the ω-limit coincideswith the usual limit.

DEFINITION 4.1. Let α = (αN) be a sequence as above and T ∈ Iα(H), T � 0.We define the Dixmier trace of T as Trα,ω(T ) = limω α

−1N σN(T ).

Dixmier’s trace extends to a linear map on Iα(H).The case of the sequence αN = logN will be of particular relevance in the

following; we shall use the notation Trω for the Dixmier trace associated withthat sequence (cf. [4]), whereas we shall denote by L(1,∞)(H) its domain, cf. thefollowing more general definition.

DEFINITION 4.2. For 1 < p < ∞ we define the subspace L(p,∞)(H) ⊂ K(H)

as the set of all compact operators T with σN(T ) = O(N1−1/p). Similarly, wedefine L(1,∞)(H) ⊂ K(H) by the condition σN(T ) = O(logN).

For 1 < p < ∞, we define the subspace L(p,∞)

log (H) ⊂ K(H) as the set of all

compact operators T with σN(T ) = O(N1−1/p(logN)−1/p); L(1,∞)

log (H) ⊂ K(H)

will be defined by the condition σN(T ) = O((logN)2).


All these spaces are normed ideals contained in K(H) and containing the idealB1(H) of trace class operators.

Remark 4.3. Let us observe that for p = 1 the ideal L(1,∞)log (H) is the natural

domain of the Dixmier trace associated with the sequence αN = (logN)2. In shortwe shall denote it by Tr′ω.

All spaces Lµ,ρcl(ξ,x) with µ < 0, ρ < 0 are contained in K(L2(Rn)). In order toestablish relations between these spaces and the ideals in Definition 2.5, we haveto study the asymptotic bahaviour of the spectrum of such operators. We beginby observing that Theorem 3.4 of Hörmander [14], when applied to our symbolclasses, gives the following result for operators of positive order (in our case, withthe metric gx,ξ given in (1.1), we have

gσx,ξ = 〈ξ 〉2|dx|2 + 〈x〉2|dξ |2 and h2(x, ξ) := sup gx,ξ /gσx,ξ = 〈x〉−2〈ξ 〉−2).

PROPOSITION 4.4. Let a ∈ Sµ,ρ

cl(ξ,x), µ > 0, ρ > 0, be a positive elliptic symbol.Then the corresponding operator aw is self-adjoint in L2(Rn); it is bounded frombelow and has a discrete spectrum {λj }j∈N diverging to +∞.

Under the hypotheses of Proposition 4.4, it makes sense to consider the functionN(λ) := ∑

j :λj�λ 1 which ‘counts’ the number of eigenvalues not greater than λ.We are going to give an asymptotic estimation for this function.

In the proof of the following theorem, we shall use the notation f (y) ≺ g(y) forfunctions f, g: Y → R when there exist a constant C > 0 such that f (y) � Cg(y)

for all y ∈ Y ; the constant C may depend on the parameters, indices, etc., possiblyappearing in the expression of f and g, but not on y ∈ Y .

THEOREM 4.5 (Weyl Formula). Let a ∈ Sµ,ρ

cl(ξ,x), µ > 0, ρ > 0, be a strictlypositive elliptic symbol and denote by N(λ) the counting function associated withthe operator aw. Then for every

0 < δ1 <2

3ρ, 0 < δ2 <

2

3µ,

we have

N(λ) =

Cµλ

nµ log λ+ O(λ

nµ ), for µ = ρ,

C ′µλ

nµ + O(λ

nµ

−δ1), for µ < ρ,

C ′′ρλ

nρ + O(λ

nρ−δ2), for µ > ρ,

(4.1)

where

Cµ = (2π)−n

nµ

∫Sn−1

∫Sn−1

σµ,µ

ψ,e (a)− nµ dθ dθ ′, (4.2)

C ′µ = (2π)−n

n

∫Rnx

∫Sn−1

σµψ (a)

− nµ dθ dx, (4.3)

C ′′ρ = (2π)−n

n

∫Rnξ

∫Sn−1

σ ρe (a)− nρ dθ dξ. (4.4)

100 FABIO NICOLA

Proof. It follows from Theorem 4.1 of Hörmander [14] that

|N(λ)−W(λ)| ≺ W(λ+ λ1−δ)−W(λ− λ1−δ), (4.5)

for 0 < δ < 2/(3 max{µ, ρ}) and

W(λ) = (2π)−n∫a(x,ξ)�λ

dx dξ. (4.6)

Hence we have to estimate the Weyl term W(λ). We first consider the case µ = ρ.It is easy to convince ourselves, by the ellipticity of a, that for every fixed

x ∈ Rn and large λ, the subset {a(x, ξ) � λ} ⊂ R

nξ is star-shape with respect to

the origin (in fact for fixed x, u ∈ Rn, |u| = 1, a(x, tu) is increasing as a function

of t for large t), so that, if we introduce polar coordinates (r, θ) in the integral withrespect to the variable ξ in (4.6), we shall have to integrate on a set of the type

{(x, r, θ) ∈ Rn × R+ × S

n−1 : r � f (λ, x, θ)},for a suitable nonnegative function f (λ, x, θ). Now again by (2.4), which we sup-pose is satisfied for every (x, ξ) ∈ R

n × Rn in view of the strict positivity of a, we

have

max{C ′λ1µ (1 + |x|)−1 − 1, 0}

� f (λ, x, θ) � max{Cλ 1µ (1 + |x|)−1 − 1, 0}, (4.7)

for suitable constants C,C ′ > 0, so that, in particular, f (λ, x, θ) vanishes for

|x| � Cλ1µ − 1. Then we have

(2π)nW(λ) = 1

n

∫|x|�Cλ 1

µ −1

∫Sn−1

f (λ, x, θ)n dθ dx. (4.8)

Now we write a = σµψ (a) + a′, where a′ ∈ C∞(Rn × (Rn \ {0})) satisfies the

estimate

|a′(x, ξ)| ≺ (1 + |x|)µ(1 + |ξ |)µ−1. (4.9)

Let a0(x, θ) = σµ

ψ (a)(x, ξ(1, θ)). By definition of f (λ, x, θ), by (4.9), and thehomogeneity of σµψ (a), we deduce

|λ−1a0(x, θ)f (λ, x, θ)µ − 1| ≺ λ−1(1 + |x|)µ(1 + f (λ, x, θ))µ−1. (4.10)

Then, in (4.8), we write f as (f µ)1µ and replace f µ by

f µ = λa−10 (1 + (λ−1a0f

µ − 1));we obtain

(2π)nW(λ)

= 1

nλnµ

∫|x|�Cλ 1

µ −1

∫Sn−1

a0(x, θ)− nµ (1 + R(λ, x, θ))

nµ dθ dx, (4.11)


where, in view of (4.10) and (4.7), on the integration domain the function R =λ−1a0f

µ − 1 satisfies the estimate

|R(λ, x, θ)| ≺ λ−1(1 + |x|)µ(1 + f (λ, x, θ))µ−1 ≺ λ− 1µ (1 + |x|). (4.12)

From (4.11), it follows that

(2π)nW(λ) = 1

nλnµ

∫|x|�Cλ 1

µ −1

∫Sn−1

a0(x, θ)− nµ dθ dx + g(λ), (4.13)

with

|g(λ)| ≺ λnµ−1

∫|x|�Cλ 1

µ −1(1 + |x|)−n+1 dx, (4.14)

as one sees from the fact that (1 + t)α ≺ t for 0 � t � T < +∞ and byusing (4.12). Introducing polar coordinates in (4.14), we get g(λ) = O(λ

nµ ), and

therefore it remains only to estimate the integral in (4.13). To do this, we write

a0(x, θ) = σµ,µ

ψ,e (a)(x, ξ(1, θ)) + a′′(x, θ);arguing as above, we easily deduce

(2π)nW(λ)

= 1

nλnµ

∫|x|�Cλ 1

µ −1

∫Sn−1

σµ,µψ,e (a)

− nµ dθ dx + g(λ)+ g′(λ), (4.15)

with g′(λ) = O(λnµ ). Switching to polar coordinates in the integral with respect to

the variables x in (4.15) and using the homogeneity of σµ,µψ,e (a), we obtain

W(λ) = Cµλnµ log λ+ O(λ

nµ ), (4.16)

where Cµ is given in (4.2). Replacing λ with λ± λ1−δ in (4.16), we get

W(λ ± λ1−δ) = Cµλnµ log λ+ O(λ

nµ ),

from which, by (4.5), the first formula in (4.1) follows.In the same way, a simpler version of the above argument proves the other two

formulas in (4.1). In the case µ < ρ, for instance, one obtains

W(λ) = C ′µλ

nµ + O(λ

n−1µ ),

where C ′µ is given in (4.3) and therefore

W(λ ± λ1−δ) = C ′µλ

nµ + O(λ

nµ−δ

),

where 0 < δ < 2/3ρ.This concludes the proof. ✷

102 FABIO NICOLA

Of course, (4.1) could be rewritten in the form N(λ) = W(λ)+R(λ), where W(λ)is given by (4.6) and

R(λ) = O(λn/µ) for µ = ρ,

R(λ) = O(λn/µ−δ1) for µ < ρ,

R(λ) = O(λn/ρ−δ2) for µ > ρ.

However, the computation of the volume W(λ) and therefore the more explicitformula (4.1), will be essential in the following.

We shall need the following simple lemma.

LEMMA 4.6. For 1 � p < ∞, let gp be the inverse function of fp: (1,∞) →R+, fp(x) = xp log x. Then

(a) if (an) and (bn) are positive sequences with an ∼ bn, we have gp(an) ∼gp(bn);

(b) for every positive sequence (kn) diverging to +∞, we have gp(kn) ∼(pkn/ log kn)1/p.

Proof. (a) The statement immediately follows observing that, for 0 < x < x′,we have

0 <log gp(x)− log gp(x′)

x − x′ � 1

px,

as one verifies by Lagrange’s formula.(b) Note that fp((pkn/ log kn)1/p) ∼ kn and then use (a). ✷

THEOREM 4.7. Let µ < 0, ρ < 0, with µ � −n or ρ � −n, so that Lµ,ρcl(ξ,x) ⊂K(L2(Rn)) but Lµ,ρcl(ξ,x) �⊂ B1(L

2(Rn)). Then the following inclusions hold:

Lµ,ρ

cl(ξ,x) ⊂

L(−n/µ,∞)

log (L2(Rn)), if µ = ρ,

L(−n/µ,∞)(L2(Rn)), if µ > ρ,

L(−n/ρ,∞)(L2(Rn)), if µ < ρ.

(4.17)

Furthermore, we have

Trψ,e(aw) = 2n2 Tr′ω(a

w), for a ∈ S−n,−ncl(ξ,x) , (4.18)

Trψ(aw) = nTrω(a

w), for a ∈ S−n,ρcl(ξ,x) with ρ ∈ Z, ρ < −n, (4.19)

Tre(aw) = nTrω(a

w), for a ∈ Sµ,−ncl(ξ,x) with µ ∈ Z, µ < −n, (4.20)

independently of ω.


Proof. We verify the first inclusion in (4.17). The other cases can be proved inthe same way.

Consider first the case of an elliptic operator Lµ,µcl(ξ,x) � A > 0, −n � µ < 0 withreal Weyl symbol a, and therefore defining a isomorphism L2(Rn) → H−µ,−µ(Rn)

(because IndA = 0). Then its inverse A−1: H−µ,−µ(Rn) → L2(Rn) satisfies thehypotheses of Theorem 4.1 (possibly after the addition of a multiple of the identityoperator), so that for its counting function we have the formula

NA−1(λ) ∼ Cµλ− nµ log λ, (4.21)

with

Cµ = −(2π)−n

nµ

∫Sn−1

∫Sn−1

σµ,µ

ψ,e (a)− nµ dθ dθ ′.

Standard arguments (cf. [30], Proposition 13.1) show that (4.21) is equivalent tothe following formula for the eigenvalues λk of A−1:

λ− nµ

k log λk ∼ C−1µ k,

which, by Lemma 4.6, implies

λk ∼ g− nµ(C−1

µ k) ∼ (−nk/(µCµ log k))−µ/n.

For the eigenvalues of A that are λ−1k , we obtain the formula

λ−1k ∼ (−nk/(µCµ log k))µ/n. (4.22)

From (4.22), it follows that

N∑k=1

λ−1k ∼

(− n

µC−1µ

)µn∫ N

1

(log x

x

)−µn

dx

∼ n

n+µ(− n

µC−1µ

)µn N1+µ

n (logN)−µn , for − n < µ < 0,

12 C−n(logN)2, for µ = −n.

(4.23)

Hence, A ∈ L(−n/µ,∞)

log (L2(Rn)). As L(−n/µ,∞)

log (L2(Rn)) is an ideal of B(H), thefirst inclusion in (4.17) follows, since one can write P ∈ S

µ,µ

cl(ξ,x) as P = (PA−1)A,where PA−1 is bounded in L2(Rn).

We now come to the relations (4.18), (4.19), (4.20) between the traces Trψ,e,Trψ , Tre and the Dixmier traces. We limit ourselves to proving (4.18).

It follows from (4.23) that (4.18) holds for an elliptic operator A > 0 withthe real Weyl symbol. By linearity, it suffices to prove that such operators spanS

−n,−ncl(ξ,x) up to trace class operators (on which both Trψ,e and Tr′ω vanish). Now,

104 FABIO NICOLA

every operator can be written as sum of two self-adjoint operators and, if P = pw

is self-adjoint but not elliptic, we write p = (p + Cq) − Cq where

q(x, ξ) = (1 + |x|)µ(1 + |ξ |)µ and C = − infp/q + 1.

So P is seen as the difference of two elliptic operators with a positive Weyl symbol.Therefore, by Lemma 3.2 of Hörmander [14], we can limit ourselves to considerP � 0 elliptic. Then, as by Fredholm theory V = KerP is a finite-dimensionalsubspace of S(Rn), the orthogonal projection PV on V is regularizing. Now P =(P + PV ) − PV , and P + PV is elliptic, strictly positive, with real Weyl symbol.This concludes the proof. ✷

Acknowledgements

I am thankful to Professor L. Rodino for very helpful suggestions about the subjectof this paper.

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20. Melrose, R. and Nistor, V.: Homology of pseudodifferential operators I, Manifolds withboundary, Preprint, MIT 1996.

21. Nilsson, N.: Asymptotic estimates for spectral function connected with hypoelliptic differentialoperators, Ark. Mat. 5 (1965), 527–540.

22. Parenti, C.: Operatori pseudo-differentiali in Rn e applicazioni, Ann. Mat. Pura Appl. 93

(1972), 359–389.23. Schrohe, E.: Spaces of weighted symbols and weighted Sobolev spaces on manifolds,

In: Lecture Notes in Math. 1256, Springer, New York, 1987, pp. 360–377.24. Schrohe, E.: Traces on the cone algebra with asymptotics, Actes des Journées de Saint Jean

de Monts, Journées Equations aux Dérivées Partielles 1996, Ecole Polytechnique, Palaiseau,1996.

25. Schrohe, E.: Noncommutative residues and manifold with conical singularities, J. Funct. Anal.150 (1997), 146–174.

26. Schrohe, E.: Wodzicki’s noncommutative residue and traces for operator algebras on mani-folds with conical singularities, In: L. Rodino (ed.), Microlocal Analysis and Spectral Theory,Kluwer Acad. Publ., Dordrecht, 1997, pp. 227–250.

27. Schulze, B. W.: Boundary Value Problems and Singular Pseudo-differential Operators, Wiley,Chichester, 1998.

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319.30. Shubin, M. A.: Pseudodifferential Operators and Spectral Theory, Springer, Berlin, 1987.31. Wodzicki, M.: Spectral asymmetry and noncommutative residue, Thesis, Stekhlov Inst. Math.,

Moscow, 1984.32. Wodzicki, M.: Noncommutative residue, Chapter I. Fundamentals, In: Manin, Yu. I. (ed.),

K-theory, Arithmetic and Geometry, Lecture Notes in Math. 1289, New York, 1987, pp. 320–399.


107

Integrable Equations of the Formqt = L1(x, t, q, qx, qxx)qxxx + L2(x, t, q, qx, qxx)

AHMET SATIRKoza Sokak 136/1, GOP 06670 Ankara, Turkey. e-mail: [email protected], [email protected]

(Received: 18 February 2002; in final form: 20 August 2002)

Abstract. Integrable equations of the form qt = L1(x, t, q, qx, qxx)qxxx+L2(x, t, q, qx, qxx) areconsidered using linearization. A new type of integrable equations which are the generalization ofthe integrable equations of Fokas and Ibragimov and Shabat are given.

Mathematics Subject Classifications (2000): 58F07, 35N10.

Key words: linearization, integrable equations.

1. Introduction

There are several methods [1] (existence of infinitely many conserved quantities,infinite number of symmetries, the Painlevé test, bi-Hamiltonian formulation) toexamine the integrability of nonlinear partial differential equations, although intwo dimensions most of these methods imply each other [2]. Another approach tointegrability is linearization technique [3]. In [4], the preliminary classification ofqt = P(q, qx, qxx, qxxx) was given using linearization. Later, the classification ofqt = P(x, t, q, qx , qxx) was considered in [5] such that if the linearized equationof a given differential equation supports an eigenvalue equation, then the givendifferential equation is integrable. The advantage of using this technique is that thegiven nonlinear equation can be of any form including explicit dependence on theindependent variables.

In this work, integrable equations of the form

qt = L1(x, t, q, qx , qxx)qxxx + L2(x, t, q, qx , qxx)

are considered. The result of the calculations gives the integrable equations in theform

qt = r1qxxx + r1,xqxx + r1ρ1

2q3x +

(3r1ρ

2− r2

2

)qx (1)

with the condition that

r1,t = (18r1,xxxr21 − 18r1,xxr1,xr1 + 8r3

1,x − 9r1,xr1r2 + 27r21 r2,x)/(18r1), (2)

108 AHMET SATIR

where r1 = r1(x, t), r2 = r2(x, t), ρ1 is a constant and ρ is a function of q. Resultis also compared with Fokas’ generalized symmetry approach [6].

2. Linearization

We can describe the linearization method for the type of equations

qt = P(x, t, q, qx , qxx, . . .) (3)

in the following way. First, we linearize the given differential equation. In otherwords, we replace q (and its derivatives) in (3) by q + ε� and differentiate bothsides of the resulting expression with respect to ε and take the limit ε → 0.

�t = DP(�), (4)

where DP is the Fréchet derivative [7]. The equation above can also be written as

�t =N∑i=0

Pi�i =N∑i=0

∂P

∂qi�i, (5)

where N is the order of differential equation,

qo = q, q1 = qx, q2 = qxx,

�0 = �, �1 = �x, �2 = �xx

and so on. In the symmetry approach (4) is the main equation, � is the symmetryof the differential equation and it is a function of x, t, qi .

The compatible eigenvalue equation is

H� = 0, (6)

where H depends on qi and x, t . If its order (highest derivative in H ) is N , then(6) may be written as

�N =N−1∑i=0

Ai�i, (7)

whereA0, A1, . . . , AN−1 are functions of qi , x, t and k. Compatibility of (7) and (5)

�N,t −�t,N = 0 (8)

using (4) and (7) will give

N−1∑i=0

�iWi = 0. (9)

Letting

Wi = 0 (10)

INTEGRABLE EQUATIONS 109

we obtain a system of partial differential equations among Pi , Ai and their partialderivatives. Expanding Ai’s in terms of the parameter k will give

Ai =N∑j=0

Aij kj , (11)

where Aij ’s are functions of qi , x, t . Now substituting (11) into (10), we obtainan overdetermined system of differential equations. Letting each coefficient of ki

to vanish results in, firstly, a set of algebraic equations among Aij and Pi and,secondly, a system of partial differential equations. The solution of this systemwill determine the eigenvalue equation (6) which can be integrated to give

�� = k�, (12)

where � is the recursion operator. Letting

� =∑

kn�n, (13)

then one gets

��n = �n−1. (14)

Hence, � is transforming a symmetry into another one [7].

3. Integrable Equations of the Formqt = L1(x, t, q, qx, qxx)qxxx + L2(x, t, q, qx, qxx)

We consider differential equations of the following form:

qt = L1(x, t, q, qx , qxx)qxxx + L2(x, t, q, qx , qxx). (15)

The linearization of the equation above can be given as

�t = γ�xxx + δ�xx + α�x + β�. (16)

Here α, β, γ , δ are functions of x, t , q, qx , qxx . We consider an eigenvalue equationhaving the same order as (15)

�xxx = A�xx + B�x + C�. (17)

Using the expansion of A, B and C as

A = A0 + A1k + A2k2,

B = B0 + B1k + B2k2,

C = C0 + C1k + C2k2,

(18)

where Ai ,Bi and Ci are functions of x, t, q, qx , qxx, qxxx . The compatibility of (16)and (17) will give the algebraic equations

A1 = 0, A2 = 0, B1 = E1/δ2/3, B2 = E2/δ

2/3, B3 = 0, (19)

C1 = −(E1(δx + 3A0δ + 2γ ))/(3δ5/3), C2 = 0, C3 = 0 (20)

110 AHMET SATIR

and evolution equations

γt = −(27A0,xxδxδ3E1 − 54A0,xxδ

3E1γ + 45A0,xδ2xδ

2E1 ++ 108A0,xδxA0δ

3E1 + 36A0,xδxδ2E1γ − 216A0,xA0δ

3E1γ ++ 162A0,xB0δ

4E1 − 252A0,xδ2E1γ

2 ++ 108αxxδ

3E1 + 72αxδ2E1γ + 81B0,xxδ

4E1 ++ 189B0,xδxδ

3E1 + 162B0,xA0δ4E1 + 270B0,xδ

3E1γ ++ 108βxδ

3E1 + 162C0,xδ4E1 − 27δxxxA0δ

3E1 −− 54δxxxδ

2E1γ + 90δxxδxA0δ2E1 + 90δxxδxδE1γ +

+ 27δxxγxδ2E1 + 54δxxA

20δ

3E1 + 36δxxA0δ2E1γ +

+ 108δxxB0δ3E1 − 45δ3

xA0δE1 − 40δ3xE1γ −

− 45δ2xγxδE1 + 45δxγxxδ

2E1 − 18δxγxA0δ2E1 +

+ 162δxA0B0δ3E1 + 180δxA0δE1γ

2 + 108δxB0δ2E1γ +

+ 162δxC0δ3E1 + 200δxE1γ

3 − 54E1,t δ2γ −

− 9γxxxδ3E1 + 54γxxA0δ

3E1 + 36γxxδ2E1γ −

− 18γ 2x δ

2E1 − 108γxA20δ

3E1 − 396γxA0δ2E1γ −

− 36γxαδ2E1 + 108γxB0δ

3E1 − 300γxδE1γ2)/(36δ2E1), (21)

δt = −(27A0,xδxδ2E1 − 54A0,xδ

2E1γ + 54αxδ2E1 +

+ 81B0,xδ3E1 − 18δxxxδ

2E1 + 36δxxδxδE1 ++ 27δxxA0δ

2E1 − 20δ3xE1 − 9δ2

xA0δE1 −− 9δxγxδE1 + 18δxA0δE1γ − 18δxαδE1 ++ 54δxB0δ

2E1 + 60δxE1γ2 − 27E1,t δ

2 ++ 27γxxδ

2E1 − 54γxA0δ2E1 − 90γxδE1γ )/(18δE1), (22)

A0,t = A0,xxxδ + 3A0,xxδx ++ 3A0,xxA0δ + A0,xxγ + 3A2

0,xδ + 3A0,xδxx ++ 7A0,xδxA0 + 3A0,xγx + 3A0,xA

20δ + 2A0,xA0γ +

+ A0,xα + 3A0,xB0δ + 3αxx + αxA0 + 3B0,xxδ + 6B0,xδx ++ 3B0,xA0δ + 2B0,xγ + 3βx + 3C0,xδ ++ δxxxA0 + 2δxxA

20 + 3δxxB0 + δxA

30 +

+ 3δxA0B0 + 3δxC0 + γxxx + 2γxxA0 + γxA20 + 2γxB0, (23)

B0,t = 3A0,xxB0δ + 3A0,xB0,xδ + 6A0,xδxB0 ++ 3A0,xA0B0δ + 2A0,xB0γ + 3A0,xC0δ + αxxx −− αxxA0 + 2αxB0 + B0,xxxδ + 3B0,xxδx +

INTEGRABLE EQUATIONS 111

+ B0,xxγ + 3B0,xδxx + B0,xδxA0 + 3B0,xγx ++ B0,xα + 3B0,xB0δ + 3βxx − 2βxA0 ++ 3C0,xxδ + 6C0,xδx + 2C0,xγ + δxxxB0 ++ 2δxxA0B0 + 3δxxC0 ++ δxA

20B0 + δxA0C0 + 2δxB

20 +

+ 3γxxB0 + γxA0B0 + 3γxC0, (24)

C0,t = 3A0,xxC0δ + 3A0,xC0,xδ ++ 6A0,xδxC0 + 3A0,xA0C0δ + 2A0,xC0γ ++ 3αxC0 + 3B0,xC0δ + βxxx −− βxxA0 − βxB0 + C0,xxxδ ++ 3C0,xxδx + C0,xxγ + 3C0,xδxx ++ C0,xδxA0 + 3C0,xγx + C0,xα ++ δxxxC0 + 2δxxA0C0 + δxA

20C0 +

+ 2δxB0C0 + 3γxxC0 + γxA0C0. (25)

The solution to the system (21)–(25) will give the linearized form of the inte-grable equations as follows

�t = r1�xxx + r1,x�xx +[

3r1/31 ρ

2q2x + r2

2

]�x + 3

2R1/31 ρqqx� (26)

with compatible eigenvalue equations

�xxx =[qxx

qx− 2r1,x

3r1

]�xx +

[− ρ1q

2x − ρ

r2/31

+ qxxr1,x

3qxr1+

+ r1,xx

3r1+ r2

1,x

9r21

+ E1

3r1k + E2

3r1k2

]�x +

+[

− qxxρ

r2/31 qx

+ 3qxρq

2r2/31

+ r1,xρ

3r5/31

]�, (27)

where r1 = r1(x, t), r2 = r2(x, t), ρ1 is a constant and ρ is a function of q with thecondition

ρqqq + 4ρ1ρq = 0. (28)

Because of the nonautonomity of the differential equations, there is another condi-tion on r1 and r2 in the form

r1,t = (18r1,xxxr21 − 18r1,xxr1,xr1 + 8r3

1,x − 9r1,xr1r2 + 27r21 r2,x)/(18r1). (29)

The integrable equation is in the form

qt = r1qxxx + r1,xqxx + r1ρ1

2q3x +

(3r1ρ

2− r2

2

)qx. (30)

112 AHMET SATIR

In the limit r1 = r1(x, t) → r1 = constant, r2 = r2(x, t) → r2 = constant,Equation (29) will disappear, we obtain the equations classified by Fokas [6] andIbragimov and Shabat [8].

4. Conclusion

In this work, integrable equations of the form

qt = L1(x, t, q, qx , qxx)qxxx + L2(x, t, q, qx , qxx)

are considered using linearization. A new type (1) of integrable equations which arethe generalization of the integrable equations of Fokas and Ibragimov and Shabatare given. The classification of most general type of qt = f (x, t, q, qx , qxx, qxxx)

partial differential equations has not been completed yet due to present limitationsin our computing system; we hope to have an extension of the system in the nearfuture.

The classification of partial differential equations with explicit dependence onspacetime is useful in the study of attractors for the solutions of partial differ-ential equations [9] and the nonlocality of partial differential equations may giverise to important contributions in theories such as superconductivity in Pippard’smodification of the London theory [10].

Acknowledgement

I would like to thank Anne Boutet de Monvel and the referee for constructivecomments.

References

1. Fokas, A. S.: Symmetries and integrability, Stud. Appl. Math. 77 (1987), 253–299.2. Wadati, M., Sanuki, H. and Konno, K.: Relationship among inverse method, Bäcklund

transformation and infinite number of conservation laws, Progr. Theoret. Phys. 53 (1975),419–436.

3. Satır, A.: PhD Thesis, Middle East Technical University, Ankara, Turkey, 1994.4. Satır, A.: Preliminary classification of qt = f (q, qx, qxx, qxxx), J. Math. Phys. 37 (1996),

3050–3061.5. Satır, A.: Classification of qt = f (x, t, q, qx, qxx), Stud. Appl. Math. 102 (1999), 205–219.6. Fokas, A. S.: A symmetry approach to exactly solvable evolution equations, J. Math. Phys. 21

(1980), 1318–1325.7. Olver, P. J.: Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin, 1993.8. Ibragimov, N. K. and Shabat, A. B.: Evolutionary equations with nontrivial Lie–Bäcklund

group, Functional Anal. Appl. 14 (1980), 19–28.9. Hale, J. K.: Attractors and dynamics in partial differential equations, CDSNS95-225, Georgia

Institute of Technology.10. Tinkham, M.: Superconductivity, Gordon and Breach, New York, 1965.


113

Rate of Convergence in Homogenizationof Parabolic PDEs

LUIS J. ROMAN1, XINSHENG ZHANG2 and WEIAN ZHENG3,�1Department of Mathematics, University of California, Irvine, CA 92697, U.S.A.e-mail: [email protected] of Statistics, East China Normal University, Shanghai, China3Department of Statistics, East China Normal University, Shanghai, Chinaand Department of Mathematics, University of California, Irvine, CA 92697, U.S.A.e-mail: [email protected]

(Received: 16 October 2001; in final form: 10 February 2003)

Abstract. We consider the solutions to ∂/∂tu(n) = a(n)(x)�u(n) where {a(n)(x)}n=1,2,... arerandom fields satisfying a ‘well-mixing’ condition (which is different to the usual ‘strong mixing’condition). In this paper we estimate the rate of convergence of u(n) to the solution of a heat equa-tion. Since our equation is of simple form, we get quite strong result which covers the previoushomogenization results obtained on this equation.

Mathematics Subject Classification (2000): 60J60.

Key words: stochastic homogenization.

There have been many excellent results on homogenization problems (see [3] andreferences therein). Most of them assumed that the coefficients in the related equa-tions are periodic. However, G. C. Papanicolaou and R. S. Varadhan [6] showed in1978 a remarkable example in which no periodicity was assumed. In order to betterunderstand the situation, we introduce a ‘well-mixing’ condition which does notrequire any periodicity of the coefficients. Apart from the periodic case, which hasbeen widely discussed in the existing literature, most of the results in the homog-enization of random operators, show the convergence of the solutions towards thesolution of the homogenized equation, without estimating the rate of convergence.The first successful attempt to give such an estimate is in the work of Yurinski [7],which has been improved upon recently in [2] for the case where the coefficientsof the operator are assumed to be stationary random fields and satisfying strong oruniform mixing conditions. Their work is concerning an elliptic operator. In thispaper we treat a parabolic operator and give specific examples in which the rate ofconvergence is given by a power of the microscopic length scale. In comparisonwith M. Kleptsyna and A. Piatniski’s recent work [4], we have no hypothesis onperiodicity and the differentiability of the coefficients.

� Research partially supported by NSF Grant DMS-0203823.

114 LUIS J. ROMAN ET AL.

Let (�,F , P ) be a probability space and let Zd be the set of all d-dimensionalintegers. For each z ∈ Zd and positive integer m, denote

Bm,z = {x ∈ Rd, 2−mzi < xi � 2−m(zi + 1)},which is a block in Rd with side length 2−m.

DEFINITION 1. Let b(x) be a given function. We say that a sequence of randomfields {b(n)(x, ω)}n is well-mixing with mean b and rate less than q(m, n) → 0(n → ∞) if for each fixed m,

supz,n

{[q(m, n)]−1E

∣∣∣∣∫Bm,z

(b(n)(x, ω) − b(x)) dx

∣∣∣∣}

is bounded.

Remark 1. Our definition of ‘well-mixing’ is different to the well-known (uni-form) strong mixing condition even when d = 1. Given a family of randomvariables a(x, ω), (0 � x < ∞), let us denote Ft = σ {a(t, ω)} (which is nota σ -filtration here),

F�t = σ {a(x, ω), x � t} and F�t = σ {a(x, ω), x � t}.Ft satisfies a uniformly strong mixing condition with coefficient φ(β) = cβ−α ifthere are constants c � 0 and α > 0 such that

|E[ξη] − E[ξ ]E[η]| � φ(β)E1/2[ξ 2]E1/2[η2], ∀ξ ∈ F�t , ∀η ∈ F�t+β.

In Example 3 we will show that the uniform strong mixing condition implies well-mixing condition if the variances of {a(x, ω)}x are bounded.

EXAMPLE 1. Given α > 0, let 1 − α � ξz � 1 + α (z ∈ Zd) be a family ofindependent and identically distributed random variables defined on �. Define

b(n)(x, ω) :=∑z∈Zd

χBn,z(x)ξz(ω). (1)

Then ∀n > m,

E

∣∣∣∣∫Bm,z

(b(n)(x, ω) − Eξ0) dx

∣∣∣∣�

√E

∣∣∣∣∫Bm,z

(b(n)(x, ω) − Eξ0) dx

∣∣∣∣2

=√√√√E

∣∣∣∣ ∑Bn,z′ ⊂Bm,z

2−nd (ξz′ − Eξz′)

∣∣∣∣2

RATE OF CONVERGENCE IN HOMOGENIZATION OF PARABOLIC PDES 115

=√ ∑

Bn,z′ ⊂Bm,z

2−2ndE|(ξz′ − Eξz′)|2

= supz′

√2−(m+n)dVar(ξz′)

= 2−(m+n)d/2√

Var(ξ0). (2)

Therefore b(n) is well-mixing with mean Eξ0 and rate less than 2−(m+n)d/2√Var(ξ0).

EXAMPLE 2. Let us consider the chess board model. Take two constants b1 < b2.Define on Rd that b(n)(x) = b1 (if x ∈ Bn,z ⊂ Rd and the sum of the coordinatesof z (

∑di=1 zi) is an even number) and b(n)(x) = b2 (if x ∈ Bn,z and

∑di=1 zi is an

odd number). Thus b(n) assumes alternatively the values b1 and b2. It is easy to seethat b(n) is well-mixing with mean 1

2

∑di=1 zi and rate less than 2−n.

EXAMPLE 3. Let us consider a more general situation. Given a family of randomvariables {a(t, ω)}t�0 and let Ft be the associated σ -algebra. Suppose that

(1) Ft satisfies a uniformly strong mixing condition with coefficient φ(β) = cβ−α ;(2) E[a(t, ω)] is periodic in t with period γ ;(3) E|a(t, ω)|2 � 1 (∀t).

It is easy to see from (3) that |E[a(x, ω)a(y, ω)] − E[a(x, ω)]E[a(y, ω)]| isbounded. So we have from (1) and (3) that there is a constant C such that

|E[a(x, ω)a(y, ω)] − E[a(x, ω)]E[a(y, ω)]| � C(1 + |x − y|)−α.

Denote a(n)(t, ω) = a(nt, ω). We have for Bm,z ⊂ [0,∞),

Var

(∫Bm,z

a(n)(x, ω) dx

)

= E

∣∣∣∣∫Bm,z

a(n)(x, ω) dx −∫Bm,z

E[a(n)(x, ω)] dx

∣∣∣∣2

= E

[(∫Bm,z

a(n)(x, ω) dx −∫Bm,z

E[a(n)(x, ω)] dx

)×

×(∫

Bm,z

a(n)(y, ω) dy −∫Bm,z

E[a(n)(y, ω)] dy

)]

= E

[∫Bm,z

a(n)(x, ω) dx∫Bm,z

a(n)(y, ω) dy −

−∫Bm,z

E[a(n)(x, ω)] dx∫Bm,z

E[a(n)(y, ω)] dy

]


= E

[∫Bm,z

a(nx, ω) dx∫Bm,z

a(ny, ω) dy −

−∫Bm,z

E[a(nx, ω)] dx∫Bm,z

E[a(ny, ω)] dy

]

�∫Bm,z

∫Bm,z

C(1 + |n(x − y)|)−α dx dy

�∫Bm,z

[∫ y+2−m

y−2−m

C(1 + |n(x − y)|)−α dx

]dy

� 2−m+1∫ 2−m

0C(1 + nx)−α dx.

When |α − 1| > 0,∫ 2−m

0C(1 + nx)−α dx = C|1 − α|−1n−1|(1 + n2−m)1−α − 1| = O(n−α);

and when α = 1,∫ 2−m

0C(1 + nx)−α dx = Cn−1| log(1 + n2−m)| = O(n−1 log n).

On the other hand,

limn→∞ E

[∫Bm,z

a(n)(x, ω) dx

]

= limn→∞ E

[∫Bm,z

a(nx, ω) dx

]

= limn→∞

∫Bm,z

E[a(nx, ω)] dx

= 2−mγ −1∫ γ

0E[a(x, ω)] dx,

where we use the periodicity hypothesis (2) in the last equality. Denote

q(m, n) ={

2−m+1∫ 2−m

0C(1 + nx)−α dx

}−1/2

,

then

q(m, n)E

∣∣∣∣∫bm,z

(a(n)(x, ω) − γ −1

∫ γ

0E[a(x, ω)]

)dx

∣∣∣∣� q(m, n)Var1/2

(∫bm,z

a(n)(x, ω) dx

)� 1

Therefore a(n) is well-mixing with mean γ −1∫ γ

0 E[a(x, ω)] dx and rate q(m, n).


Let us now consider the two following PDEs:

∂

∂tu(n)(t, x, ω) = a(n)(x, ω)�u(n)(t, x, ω), u(n)(0, x) = u0(x) (3)

and

∂

∂tu(t, x) = a0�u(t, x), u(0, x) = u0(x), (4)

where u0(x) ∈ C20(R

d) and {1/a(n)}n is well-mixing with mean 1/a0 and rateq(m, n). The existence of the solution to the above equation is well-known asexplained in (5).

Equation (4) is just the heat equation. The following result for (3) is well known:

LEMMA 1. Given u0 bounded with compact support, there is a constant 0 < c1

such that when |x − y| < 1 and t � 1,

|u(n)(x, t) − u(n)(y, t)| � |x − y|c1

t1/2ξ(x),

where ξ ∈ L1(Rd) ∩ L2(R

d) and has exponential decay.Proof. Equation (3) can be considered as a special form of an equation of the

following type:

∂

∂tv = 1

q(x)

∑ij

∂

∂xi

[q(x)Aij (x)

∂

∂xjv

], (5)

where q(x) and Aij (x) satisfy an uniform ellipticity condition. Denote by K(x, t, y)

the kernel associated to the above equation, then the solution of (3) is given byv(x, t) = ∫

K(x, t, y)u0(y) dy and K is bounded by Aronson’s inequality [1]

K(x, t, y) � C

td/2exp

{−|x − y|2

Ct

}, (6)

where the constant C only depends on the ellipticity constraints.Moreover, from the stationarity of Equation (3), it is sufficient to prove the

lemma when u0 ∈ C30(R

d) and a(n)(x) are smooth (and therefore u(n)(x) is alsosmooth). For fixed n and t , if we denote v(s, x) = u(n)(t − s, x), then

∂

∂sv + a(n)�v = 0, ∀0 < s < t.

Thus, according to Krylov ([5], p. 133), when |x − y| �√t ,

|u(n)(x, t) − u(n)(y, t)| � κ|x − y|α

tα/2sup

{(z,s); |z−x|�√t,s<t}

{|u(n)(z, s)|} (7)

with constants α and κ independent of the smoothness of a(n)(·).


Given each pair (x, y) such that |x − y| � 1, we denote N = [|x − y|/√t] + 1,(i.e., the smallest integer larger than |x − y|/√t). Then

|u(n)(x, t) − u(n)(y, t)|

�N−1∑i=0

∣∣∣∣u(n)

(x + (i + 1)(y − x)

N, t

)− u(n)

(x + i(y − x)

N, t

)∣∣∣∣� κN

(|x − y|/N)α

tα/2sup

{(z,s); |z−x|�|y−x|,s<t}{|u(n)(z, s)|}

� κN1−α |x − y|αtα/2

sup{(z,s); |z−x|�|y−x|,s<t}

{|u(n)(z, s)|}

� κN1−α |x − y|αtα/2

ξ1(x),

where

ξ1(x) = sup{(z,s); |z−x|�1,s<1}

{∣∣∣∣∫

u0(y)K(x, s, y) dy

∣∣∣∣}.

Thus

|u(n)(x, t) − u(n)(y, t)| � κ

{ |x − y|αtα/2

+ 2|x − y|√

t

}ξ1(x)

and we get the lemma. ✷On the other hand, �u is also a solution of the heat equation. Therefore

�u(x, t) =∫

�u0(y)(4πa0t)−d/2 exp

{−|x − y|2

4a0t

}dy

is bounded and Lipschitz continuous. From (3) and (4), we get

1

a0

∂

∂tu(t, x) − 1

a(n)

∂

∂tu(n)(t, x, ω) = �[u(t, x) − u(n)(t, x, ω)]

or1

a(n)

∂

∂t[u(t, x) − u(n)(t, x, ω)]

= �[u(t, x) − u(n)(t, x, ω)] +(

1

a(n)− 1

a0

)∂

∂tu(t, x).

Multiplying both sides by (u − u(n)), we get

[u(t, x) − u(n)(t, x, ω)] 1

a(n)

∂

∂t[u(t, x) − u(n)(t, x, ω)]

= [u(t, x) − u(n)(t, x, ω)]�[u(t, x) − u(n)(t, x, ω)]++ [u(t, x) − u(n)(t, x, ω)]

(1

a(n)− 1

a0

)∂

∂tu(t, x).


Integrating on both sides the above equation with respect to dx dt , we obtain∫1

2a(n)[u(t, x) − u(n)(t, x, ω)]2 dx

=∫ t

0

∫[u(s, x) − u(n)(s, x, ω)]�[u(s, x) − u(n)(s, x, ω)] dx ds +

+∫ t

0

∫[u(s, x) − u(n)(s, x, ω)]

(1

a(n)− 1

a0

)∂

∂su(s, x) dx ds

= −∫ t

0

∫|∇[u(s, x) − u(n)(s, x, ω)]|2 dx ds +

+∫ t

0

∫[u(s, x) − u(n)(s, x, ω)]

(1

a(n)− 1

a0

)a0�u(s, x) dx ds,

where we used the integral by parts formula in the last equation, which can be donebecause of the exponential decay of [u(s, x)−u(n)(s, x, ω)] and the L2-boundednessof ∇[u(s, x) − u(n)(s, x, ω)]. Therefore∫

1

2a(n)[u(t, x) − u(n)(t, x, ω)]2 dx

�∫ t

0

∫[u(s, x) − u(n)(s, x, ω)]

(1

a(n)(x)− 1

a0

)a0�u(s, x) dx ds. (8)

Let us estimate the right side of the last inequality. By Lemma 1, when|x − y| � r and x, y ∈ B ⊂ Rd ,

|[u(s, x) − u(n)(s, x, ω)]a0�u(s, x) − [u(s, y) − u(n)(s, y, ω)]a0�u(s, y)|� |[u(s, x) − u(n)(s, x, ω)]a0�u(s, x) − [u(s, y) − u(n)(s, y, ω)]a0�u(s, x)| +

+ |[u(s, y) − u(n)(s, y, ω)]a0�u(s, x) − [u(s, y) − u(n)(s, y, ω)]a0�u(s, y)|� |[u(s, x) − u(s, y)]a0�u(s, x)| +

+ |[u(n)(s, x, ω) − u(n)(s, y, ω)]a0�u(s, x)| ++ |[u(s, y) − u(n)(s, y, ω)]a0[�u(s, x) − �u(s, y)]|

� Crc1

s1/2supx∈B

{η(x)}, (9)

where η(x) has exponential decay. We revise η so that we also have

�u(s, x) � η(x). (10)

Decompose now Rd = ⋃z Bm,z and select one point ym,z from each Bm,z. By

using (9),

E

{∫1

2a(n)[u(t, x) − u(n)(t, x, ω)]2 dx

}

�∑z

∫ t

0E

∣∣∣∣∫Bm,z

[u(s, ym,z) − u(n)(s, ym,z, ω)]×


×(

1

a(n)(x)− 1

a0

)a0�u(s, ym,z) dx

∣∣∣∣ ds +

+ C∑z

t1/22−(c1+d)mdc1 supx∈Bm,z

{η(x)}. (11)

From here, C will denote an arbitrary constant. As a direct consequence, we obtainthe following theorem:

THEOREM 1. Suppose that {1/a(n)} is well-mixing with mean 1/a0 and rate lessthan q(m, n), then there is a constant c2 such that when t � 1,

E

∣∣∣∣∫

[u(t, x) − u(n)(t, x, ω)]2 dx

∣∣∣∣ � c2t1/2[2(m−1)dq(m, n) + 2−c1m]

Proof. From (11), we get easily

E

∣∣∣∣∫

1

2a(n)[u(t, x) − u(n)(t, x, ω)]2 dx

∣∣∣∣�

∑z

∫ t

0

∣∣∣∣E{[u(s, ym,z) − u(n)(s, ym,z, ω)]a0�u(s, ym,z)×

×∫Bm,z

(1

a(n)(x)− 1

a0

)dx

∣∣∣∣ ds +

+ C∑z


{η(x)}

=∑z

∑2−mz∈B1,z

∫ t

0

∣∣∣∣E{[u(s, ym,z) − u(n)(s, ym,z, ω)]a0�u(s, ym,z)×

×∫Bm,z

(1

a(n)(x)− 1

a0

)dx

∣∣∣∣ ds +

+ C∑z

∑2−mz∈B1,z


{η(x)}

� C∑z

2(m−1)d sup(z; 2−mz∈B(1,z))

∫ t

0E

{[u(s, ym,z) − u(n)(s, ym,z, ω)]×

× a0�u(s, ym,z)

∣∣∣∣∫Bm,z

(1

a(n)(x)− 1

a0

)dx

∣∣∣∣ ds

}+

+ C∑z

t1/22−c1mdc1 supx∈B1,z

{η(x)}

� C∑z

2(m−1)d∫ t

0sup

2−mz∈B(1,z){η(ym,z)} ×

× sup2−mz∈B(1,z)

{E

∣∣∣∣∫Bm,z

(1

a(n)(x)− 1

a0

)dx

∣∣∣∣}

ds +


+ C∑z


{η(x)}

� C∑z

2(m−1)d t sup2−mz∈B(1,z)

{η(ym,z)} supz

{E

∣∣∣∣∫Bm,z

(1

a(n)(x)− 1

a0

)dx

∣∣∣∣}

+

+ C∑z


{η(x)}

� C2(m−1)d t supz

E

∣∣∣∣∫Bm,z

(1

a(n)(x)− 1

a0

)dx

∣∣∣∣ + Ct1/22−c1mdc1 , (12)

where we used (9), (10), and the fact that η is a function with exponential decay.Thus

E

∣∣∣∣∫

1

2a(n)|u(t, x) − u(n)(t, x, ω)|2 dx

∣∣∣∣� C[t2(m−1)dq(m, n) + t1/22−c1mdc1 ]. (13)

Hence, the theorem follows. ✷As an immediately consequence we have the following corollary:

COROLLARY 1. In Example 1,

E

∣∣∣∣∫

|u(t, x) − u(n)(t, x, ω)|2 dx

∣∣∣∣ � c3t1/22−c1[nd/(2c1+d)], (14)

where c3 is a constant.Proof. Select m = [nd/(2c1 + d)] as the integer part of nd/(2c1 + d), then

(2c1 + d)m � nd. Thus md − nd � −2c1m and

2mdq(m, n) = 2[m−n]d/2 � 2−c1m.

Theorem 1 yields

E

∣∣∣∣∫

|u(t, x) − u(n)(t, x, ω)|2∣∣∣∣ � Ct1/22−c1m = Ct1/22−c1[nd/(2c1+d)].

COROLLARY 2. In Example 2,

E

∣∣∣∣∫

1

2a(n)|u(t, x) − u(n)(t, x, ω)|2 dx

∣∣∣∣ � c4t1/22−[n/(c1+d)],

where c4 is a constant.Proof. Taking m = [n/(c1 + d)] as the integer part of n/(c1 + d), we have

2mdq(m, n) = 2md2−n � 2−c1m


and thus

E

∣∣∣∣∫

1

2a(n)|u(t, x) − u(n)(t, x, ω)|2 dx

∣∣∣∣ � Ct1/22−[n/(c1+d)]. ✷However, if one perturbates the thermal diffusivity by the same random fields,

the situation is different. Let us consider the one-dimensional case

∂u

∂t(t, x) = ∂

∂x

(a(x)

∂u

∂x(t, x)

),

u(0, x) = u0(x)

(15)

and

∂uk

∂t(t, x, ·) = ∂

∂x

(a(k)(x, ·)∂u

(k)

∂x(t, x, ·)

),

u(k)(0, x) = u0(x),

(16)

where u0(x) is a differentiable function with compact support and {1/a(k)} is well-mixing with mean 1/a and rate less than q(m, k). We further assume that a(k)

satisfy the uniform ellipticity condition 1/λ � a(k) � λ, where λ is a positiveconstant.

Let e(k)(s, x) = u(s, x) − u(k)(s, x). Then e(k)(s, x) satisfies the followingstochastic PDE:

∂e(k)

∂s(s, x, ·) = ∂

∂x

(a(k)(x, ·)∂e(k)

∂x(s, x, ·)

)

+ ∂

∂x

((a(x) − a(k)(x, ·))∂u

∂x

). (17)

Multiply 2e(k)(s, x) to both sides of the above equation and integrate with respectto time. We obtain from the integral by parts formula that

(e(k)(t), e(k)(t))2

= −2∫ t

0

∫a(k)(x)

(∂e(k)

∂x(s, x)

)2

dx ds −

− 2∫ t

0

∫ 1

0(a(x) − a(k)(x))

∂u

∂x(s, x)

∂e(k)

∂x(s, x) dx ds. (18)

Using the uniform ellipticity of a(k)(x), we can derive from (18) that

E(e(k)(t), e(k)(t))2 + 2λ−1E

∫ t

0

∫D

(∂e(k)

∂x(t, x)

)2

dx ds

� 2E∫ t

0

∫(a(x) − a(k)(x))

(∂u

∂x

∂u(k)

∂x

)(s, x) dx ds −

− 2E∫ t

0

∫(a(x) − a(k)(x))

(∂u

∂x(s, x)

)2

dx ds. (19)


Let us estimate the last two terms.Take m < k, decompose R = ⋃

z Bm,z and select one point ym,z from each Bm,z.Since

∂

∂t

(a(k)(x)

∂u(k)

∂x

)= a(k)�

(a(k)(x)

∂u(k)

∂x

), (20)

(a(k)(x)(∂u(k)/∂x)) satisfies Lemma 1. Using a similar argument as used in (11),we get

E

∣∣∣∣∫ t

0

∫(a(x) − a(k)(x))

(∂u

∂x

∂u(k)

∂x

)(s, x) dx ds

∣∣∣∣= E

∣∣∣∣∫ t

0

∫a(x) − a(k)(x)

a(x)a(k)(x)

((a(x)

∂u

∂x

)(a(k)(x)

∂u(k)

∂x

))(s, x) dx ds

∣∣∣∣�

∑z

E

∣∣∣∣∫ t

0

∫Bm,z

a(x) − a(k)(x)

a(x)a(k)(x)

((a(x)

∂u

∂x

)×

×(a(k)(x)

∂u(k)

∂x

))(s, x) dx ds

∣∣∣∣�

∑z

CE

∣∣∣∣∫ t

0

∫Bm,z

a(x) − a(k)(x)

a(x)a(k)(x)

((a(ym,z)

∂u

∂x

)×

×(a(k)(ym,z)

∂u(k)

∂x

))(s, ym,z) dx ds

∣∣∣∣ +

+ Ct1/22−(c1+1)m∑z

supx∈Bm,z

{η(x)}

=∑z

CE

∣∣∣∣∫ t

0

∫Bm,z

[1

a(k)(x)− 1

a(x)

]((a(ym,z)

∂u

∂x

)×

×(a(k)(ym,z)

∂u(k)

∂x

))(s, ym,z) dx ds

∣∣∣∣ +

+ Ct1/22−(c1+1)m∑z

supx∈Bm,z

{η(x)}. (21)

By Aronson’s inequality (6), (a(x)(∂u/∂x)) is dominated by a deterministic func-tion with exponential tail. Following the same trail as in the proof of Theorem 1,we get

E

∣∣∣∣∫ t

0

∫(a(x) − a(k)(x))

(∂u

∂x

∂u(k)

∂x

)(s, x) dx ds

∣∣∣∣�

∑z

supx∈B1,z

η(x)∑

(z; mz∈B(m,z))

CE

∣∣∣∣∫ t

0

∫Bm,z

[1

a(x)− 1

a(k)(x)

]dx ds

∣∣∣∣ +

+ Ct1/22−c1m∑z

supx∈B1,z

{η(x)}


� 2m∑z

supx∈B1,z

η(x) sup(z; mz∈B(m,z))

CE

∣∣∣∣∫ t

0

∫Bm,z

[1

a(x)− 1

a(k)(x)

]dx ds

∣∣∣∣ +

+ Ct1/22−c1m∑z

supx∈B1,z

{η(x)}

� Ct2mq(m, k) + Ct1/22−c1m. (22)

The same bound stands for the second term of (19) as well. Thus we get

THEOREM 2. E|u(k) − u|2 � Ct2mq(m, k) + Ct1/22−c1m.

Acknowledgement

We sincerely thank the anonymous referee who pointed out a mistake in the previ-ous version of formula (22).

References

1. Aronson, D. G.: Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math.Soc. 73 (1967), 890–896.

2. Bourgeat, A. and Piatniski, A.: Estimates in probability of the residual between the randomand the homogenized solutions of one-dimensional second-order operator, Asymptot. Anal. 21(1999), 303–315.

3. Jikov, V. V. and Kozlov, S. M. and Oleinik, O. A.: Homogenization of Differential Operatorsand Integral Functionals, Springer-Verlag, New York, 1994.

4. Kleptsyna, M. and Piatniski, A.: Averaging of non-self-adjoint parabolic equations with randomevolution, Preprint, 2000.

5. Krylov, N. V.: Nonlinear Elliptic and Parabolic Equations of the Second Order, D. Reidel,Dordrecht, 1987.

6. Papanicolaou, G. C. and Varadhan, R. S.: Diffusion in regions with many small holes,In: Lecture Notes in Control and Inform. Sci. 25, Springer, New York, 1978, pp. 190–206.

7. Yurinski, V. V.: Averaging of symmetric diffusion in random media, Sibirsk. Mat. Zh. 27(4)(1986), 167–180.


125

Algebraic and Geometric Properties of MatrixSolutions of Nonlinear Wave Equations

V. V. GUDKOVInstitute of Mathematics and Computer Science, University of Latvia, Rainis Boulevard 29,Riga LV-1459, Latvia. e-mail: [email protected]

(Received: 20 November 2001, in final form: 15 May 2002)

Abstract. We improve the construction of exact matrix solutions for nonlinear wave equations byusing unitary anti-Hermitian and anticommuting matrices. We prove the theorem that constructs thematrix functions un satisfying the nonlinear wave equation for a set of special potentials. In this case,the graph of complex solution u1 has a soliton-like form with a finite number of coils. Exponentialrepresentation of matrix solutions un is associated with continuous rotations that can be used fordescribing intrinsic rotations and state changes of elementary particles. We also prove the theoremon the decomposition of continuous rotation (described by solution u2) onto three simultaneous rota-tions about coordinate vectors. Each of the three constructed matrix solutions u3 is also decomposedinto the triplet of elementary matrix solutions.

Mathematics Subject Classification (2000): 35Q58.

Key words: anticommuting algebra, decomposition of rotation, matrix solution, nonlinear Klein–Gordon equation, nonlinear wave equation, unitary anti-Hermitian matrix.

1. Introduction

In 1925 Dirac wrote [1] that the whole situation was suddenly changed by Heisen-berg who introduced the idea of noncommuting algebra into physics. This ideais also useful for constructing exact solutions of wave equations in the form oflinear combination of unitary anticommuting matrices as shown in [2–4]. In thepresent paper, we extend the construction of matrix solutions un for nonlinear waveequations by using unitary anti-Hermitian and anticommuting n× n-matrices.

In the case when n = 1, we obtain a complex travelling wave solution, u1,which has the form of a helical curve for the linear wave equation and a soliton-like form for the nonlinear Klein–Gordon equation. Our approach is essentiallysimplified with respect to that of, e.g., Bohun and Cooperstock [5]. They constructa soliton-like solution for the Dirac–Maxwell equation in 1+3D space. In turn, weconstruct exact solutions along some lines in space; more precisely, we choose anarbitrary direction in 1 + 3D space and pass on to the moving frame of reference.This simplified approach allows us to use the matrix solutions to describe intrinsicrotations and state changes of elementary particles.

126 V. V. GUDKOV

In the case n = 2, we construct a matrix solution u2 which belongs to thespecial unitary group, SU(2), and describes a rotation of a unit sphere S2 ∈ R3 onthe quaternion basis. Since there is a one-to-one correspondence between matrixsolution u2(φ, a) and a continuous rotation by angle 2φ of unit sphere S2 aboutradial vector a, we also denote this rotation by u2(φ, a). We prove the theorem ondecomposition of rotation u2(φ, a) into three simultaneous rotations about coordi-nate axes. Furthermore, we then state that two rotations about two radial vectorsgenerate a composite rotation. This result is associated with the interaction of themagnetic fields of two electric conductors.

We also focus on the case of n = 3 where an approach is proposed that isperfected with respect to [4] for the construction of a one-parameter set of matrixsolutions u3. Here we note another important point, namely, the decomposition ofeach of the three constructed solutions u3 onto three elementary matrix solutionswith numerical exponents such as ± 1

3 and ± 23 so that the sum of corresponding

three exponents is always +1 or −1. Hence, we have by analogy with quarks, threetriplets of elementary matrix solutions related to the number +1 and three tripletsrelated to −1. In the particular case of n = 4, we show that the matrix solution u4

describes the rotation of unit sphere that is twice faster than the solution u2.For the case n = 2k , we propose a uniform approach for the construction of

anticommuting matrices and describe, more precisely than in [3], an application ofClifford algebras for our constructions. In the last section, we will prove that thereis a one-to-one correspondence between a set of solutions u2 and a cross-sectionSφ of unit sphere S3 ∈ R4 and show that Sφ pulsates from S2 (with the right frameof reference) throught the origin to a turned inside out sphere with the left frameand inversely.

2. Matrix Solutions

Consider our basic construction of matrix functions

un(φ, a) = cos(φ)En + sin(φ)m∑j=1

ajMj = exp(φaM), n = 1, 2, . . . , (1)

where

M2j = −En and MiMj = −MjMi for i �= j. (2)

Equalities (2) are the main properties of the anticommuting algebra of matrices Mj .Here a = (a1, . . . , am),

∑mj=1 a

2j = 1, aj are real numbers, En is a unit n × n-

matrix, M = (M1, . . . ,Mm), Mj are unitary anti-Hermitian anticommuting n× n-matrices, m is the number of linear-independent matrices Mj which form a basis.To simplify calculations, we use such notations as �n = ∑m

j=1 ajMj = aM. Itfollows from the properties of matrices Mj that expansion �n satisfies the equality

MATRIX SOLUTIONS OF NONLINEAR WAVE EQUATIONS 127

�2n = −En and thus �n = exp(�nπ/2), hence �n, among other things, plays a

role of an imaginary unit. Therefore

(un(φ, a))k = (cosφ + sinφ�n)k = cos(kφ)+ sin(kφ)�n = un(kφ, a), (3)

where k is a real number. Owing to the properties of matrices Mj , expression (1)determines the relation between matrix solutions un and SU(n) matrix groups.

Now let us substitute the matrix function (1) into the nonlinear Klein–Gordonequation

∂2u

∂t2−�u + dQ

du= 0, Q(u) = η2

4(u2 − 1)2 (4)

taken for the natural system of units c = 1 = h. Under the notation used in [2–4];namely, the chosen direction x = ∑3

j=1 λjxj with∑3

j=1 λ2j = 1, where λj are real

numbers, and the moving frame of reference z = x − vt , where v is the velocity,Equation (4) is reduced to an ordinary differential equation (ODE)

(1 − v2)u

(d2φ

dz2�n −

(dφ

dz

)2

En

)= dQ

du. (5)

Equation (5) with respect to φ seems more complicated than (4). However, forsome potentials Q, we can find exact solutions of (5) as well as of (4). It followsfrom [2–4] that the matrix function un(φ, a), where

φ ≡ φ(αz) = arccot(− sinh(αz)), α = η√

2/(1 − v2), v2 < 1,

satisfies Equation (4). It is convenient to consider φ in the region 0 � φ � π if zvaries from −∞ to ∞. Note that if u(z) satisfies (4), then u(z+C) also satisfies (4)for any real C.

In the case where Q = ±η2u2/2, Equation (4) is reduced to a linear waveequation

∂2u

∂t2−�u ± η2u = 0 (6)

and ODE (5) is reduced to

d2φ

dz2= 0,

(dφ

dz

)2

= ±η2

v2 − 1.

Thus matrix function un satisfies Equation (6) for φ = βz, where β = η/√|1 − v2|

if η �= 0 and β is an arbitrary real number if η = 0 and v2 = 1.In the more complicated case when potential Q has the form

Qk(u) = k2η2

4(u2)

k−1k ((u2)

1k − 1)2, k ∈ (−∞,∞).

Equations (5) and (4) also can be integrated, as is proven below.

128 V. V. GUDKOV

THEOREM 1. A family of matrix functions un(kφ, a), where k ∈ (−∞,∞) n =1, 2, . . . , satisfies nonlinear wave equation

∂2u

∂t2−�u + dQk

du= 0. (7)

Proof. Let us substitute matrix function un(kφ, a) into (7), take into accountthe relation α2 = 2η2/(1 − v2) and (3) in the form un(kφ, a) = wk, where w =un(φ, a), then reduce (7) to the ODE

kd2φ

dz2�n − k2

(dφ

dz

)2

En = k2α2

4

(k + 1

kw2 + k − 1

kw−2 − 2En

). (8)

It is easy to find that

dφ

dz= α sinφ,

d2φ

dz2= α2 sin φ cosφ,

w2 = cos(2φ)En + sin(2φ)�n.

Thus (8) is converted into the equality

sin(φ) cos(φ)�n − k sin2(φ)En = k(cos(2φ) − 1)En/2 + sin(2φ)�n/2

which is identically fulfilled. ✷We consider potentials for the nonlinear wave equation as a response of a phys-

ical medium, therefore the representation of Q and Qk in matrix form is fully justi-fied by virtue of its dependence on matrix solutions. One of the steps in this direc-tion is a potential in the form of a ‘bottle bottom’ considered by Okun’ [6, p. 191]and Huang [7, p. 77] for a complex scalar field. Based on Penrose and Rindler’s[8, p. 41] statement on the rotational properties of spin-matrices and unit quater-nions, we conclude that u2(φ, a) describes a rotation of the unit sphere by theangle 2φ about vector a. The rotational properties of matrix solutions allow us torepresent the potentials as combined rotations in a physical medium. Indeed, dueto the property (3), we find for example,

Q1 = (λ/2)2(u(2φ) − 1)2, Q1/2 = (λ/2)2u(−2φ)(u(4φ) − 1)2,

where, for u = u2, the right sides of these equalities contain the matrices whichaccomplish the rotation of unit sphere.

In the complex plane, the graph of complex solution u1(φ) = exp(iφ) of non-linear wave equation looks as a half-circle for Q1, as a full circle for Q2, and as aquarter of a circle for Q1/2. For Qk in the complex space (�u, u, z), it looks as ahelix with k/2 coils.

To construct any worthy model, we should consider at least a pair of solutionsas a whole. Let u1 and u2 be the solutions of linear wave equation (6) and let us


put φ = β(z + C) for any real C. In the remaining frame of a particle wherez = x − vt = 0 solution u1 = exp(iβC) describes a unit circle, u1-circle, inthe range 0 � C � 2π/β. Let the center of the unit sphere be a point exp(iβC)which lies on the u1-circle, then while C is varying from 0 to 2π/β, the solutionu2 = exp(βC�2) rotates the sphere S2 and solution u1 moves the center of S2 insuch a way that the sphere describes a torus. If we take a set of positions of S2 forall C ∈ (0, 2π/β), then we find the entire torus. This torus can be considered asa torus wheel which is stationary for z = 0 (x = vt) and it rolls when z varies.This torus wheel can be proposed as a possible model of a photon due to the matrixsolution u2(φ, e3) = cos φE2 + sinφH3 being similar to the expression of neutralvector field W0 (see [4]) which is associated with a photon.

3. Simultaneous Rotations

In this section we consider the matrix solution

u2(φ, a) = cos(φ)E2 + sin(φ)3∑

j=1

ajHj = exp(φaH),

where a = (a1, a2, a3) is a unit vector, H = (H1,H2,H3) and unit quaternions are

H1 =(

0 i

i 0

), H2 =

(0 −11 0

), H3 =

(i 00 −i

).

Since the angular parameter φ is a continuous function of time, it is better to refor-mulate a rotational property of solution u2 in the following way: matrix solutionu2(φ, a) = exp(φaH) describes a continuous rotation of sphere S2 about radialvector a with angular velocity ω = 2(dφ/dt). We denote this continuous rotationby exp(φaH) in accordance with finite rotations (turns) which are often written inthe exponential form (see, e.g., [9, p. 15], [10, p. 31]).

It will be more suitable, at least in this section, to designate the finite rotations(turns) by finite turns. The point is that we consider continuous simultaneous rota-tions, a binary operation of a composition of these rotations we denote by symbol�, and note that simultaneous rotations commute with respect to this operation. Incontrast, a realization of sequential finite turns depends on the order in which theseturns should be performed (see, e.g., [9, p. 15], [10, p. 177]).

Let us take a local frame of reference e1, e2, e3 (frame ej) with the center of S2

as its origin. Put the frame ej in correspondence with H in such a way that e1 =(1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) in the quaternion basis. Now along withsolution u2(φ, a) of nonlinear wave equation (4), we will consider three matrixsolutions

u2(φaj , ej) = exp(φajHj) for j = 1, 2, 3

of nonlinear wave equation (7) with corresponding k = aj . It is clear that these so-lutions describe the rotations of S2 about the corresponding vector ej with angularvelocities ωj = ωaj , respectively.

130 V. V. GUDKOV

THEOREM 2. Continuous rotation exp(φaH) of unit sphere S2 about the unitvector a = (a1, a2, a3) with angular velocity ω = 2(dφ/dt) can be decomposedonto three simultaneous rotations exp(φajHj) (j = 1, 2, 3) of S2 about coordinatevectors ej with angular velocities ωj = ωaj , respectively, that can be written in thefollowing way:

exp(φaH) = exp(φa1H1)� exp(φa2H2)� exp(φa3H3). (9)

Proof. The proof is based on the correspondence between decomposition (9) anda decomposition of the vector of the linear velocity of an arbitrary point on S2 intothree vectors, each of them being orthogonal to corresponding coordinate axis. Letthe arbitrary point P on S2 be the endpoint of the radial unit vector b = (b1, b2, b3)

and let the projection of b be orthogonal to the ej plane denoted by rj. Then

r1 = (0, b2, b3), r2 = (b1, 0, b3), and r3 = (b1, b2, 0).

The rotation exp(φaH) defines a vector of linear velocity v = ω(a × b). It is clearthat this vector can be uniquely decomposed into the sum of vectors vj, each ofthem being orthogonal to the corresponding coordinate vector ej. If we take vj =ωaj(ej × rj) (j = 1, 2, 3), i.e. vectors of linear velocities of rotations exp(φajHj)

related to point P , then due to definition of rj, we find

3∑j=1

vj =3∑

j=1

ω(ajej × b) = v.

Thus vector v is precisely decomposed into the sum of vectors of linear velocitiesof rotations exp(φajHj), which means that rotation exp(φaH) is decomposed intothree rotations as in (9). ✷

The equality (9) can be also considered as a composition of three simultane-ous rotations exp(φajHj), i.e. the latter rotations generate a composite rotationexp(φaH). Furthermore, two simultaneous rotations exp(φaH) and exp(ψbH),where ψ may be different from φ on wave number β, with one and the sameorigin of unit vectors a and b generating a composite rotation exp(cH), wherec = φa + ψb. Really, each of the rotations about a and b can be decomposed intosimultaneous rotations about the coordinate axes. Two simultaneous rotations aboutone of the vectors ej give a composite rotation u2(φaj +ψbj , ej). The compositionof three such rotations gives the rotation exp(cH).

As is shown below in Section 6, matrix solution u2 can be used to also describethe rotation of a sphere of arbitrary radius. Thus, under rotation exp(φaH), we canalso understand a rotation of all points about the a-direction. Now if we associatethe circular path of each point about a-direction with a magnetic line, then thecomposition of two rotations exp(φaH) and exp(ψbH) is similar to an attractionof two conductors of an electric current.


4. Anticommuting Matrices

In this section we describe (more precisely than in [3]) an application of Cliffordalgebras for the construction of a set of matrix functions (1). Anticommuting ma-trices Mj satisfying property (2) are constructed in [3] for orders n = 4, 6, 8. Theauthor has not found anticommuting matrices for n = 5 and n = 7 but knows thatsuch anticommuting matrices do not exist for n = 3.

For n = 2, we, as usual, derive matrix solution u2 via unit quaternions. Forn = 2k (k = 3, 4, . . .), we propose the following set of matrices: Mj = Mj(n) forj = 1, 2, . . . , k + 2,

Mj(n) =(Mj(n/2) 0

0 −Mj(n/2)

), j < k + 2,

(10)

Mk+2(n) =(

0 −En/2

En/2 0

),

where Mj(2) = Hj for (j = 1, 2, 3).The fact that some Clifford algebras C(0,m) are realized in the form of the

direct sum of two sets of matrices, gives the possibility of constructing matrixsolutions in the form of pairs [un, un] and [un, u∗

n] as elements of the correspond-ing direct sum (see [3]). Moreover, the Clifford algebras C(0,m), where m is anumber of generators (i.e. linear independent matrices such as Mj which satisfyproperty (2)), give the possibility of forming a finite subset of solutions wn froma set of solutions un. This idea was expressed in [3] but was not realized therebecause of our inaccuracy in using indexes m and n. Now we can choose a finitesubset of solutions wn for m = 1, . . . , 8. A restriction by 8 is explained by theCoquereaux periodicity theorem [11] which states that algebras C(0,m + 8) areisomorphic to the direct product of C(0,m) and C(0, 8).

Let us construct this subset. First of all, we take w1 = u1, w2 = u2. Then, dueto algebra C(0, 3) being realized in the form of the direct sum H ⊕H , we can takew3 in the form of a pair, for example,

w3 = [u2, u∗2] = cosφ[E2, E2] + sin φ

3∑j=1

aj [Hj,H∗j ].

w4 = u4 can be taken for m = 4, where u4 is based on the matrices Mj(4) definedin (10). w5 = u8 can be taken for m = 5, where u8 is based on five matricesMj(8) from (10). For m = 6, the Clifford algebra C(0, 6) gives the possibility ofconstructing six 8 × 8-matrices Mj as a basis for w6 = u8. The Clifford algebraC(0, 7) is realized in the form of the direct sum of two sets of 8 × 8-matrices, sowe can take w7 = [u8, u

∗8]. The Clifford algebra C(0, 8) gives the possibility of

constructing eight 16 × 16-matrices Mj as a basis for w8 = u16.Now consider the matrix solution u4(φ, a) of the nonlinear wave equation. If u4

is based on matrices Mj(4) from (10) and a = (a1, a2, a3, 0), then we can show that

132 V. V. GUDKOV

u4(φ, a) rotates S2 about vector a by angle 4φ, i.e. two times faster than u2(φ, a).Really, due to the diagonal form of Mj(4) (j = 1, 2, 3), the solution u4 is writtenas

u4(φ, a) = diag(exp(φaH), exp(φaH ∗)).

It follows from the rotation exp(φaH) that the circular movement of each pointof S2 on the big circle which is orthogonal to a, can be described by the formulaexp(2iφ). Analogously, rotation exp(φaH ∗), in relation to the big circle, can bedescribed by exp(−2iφ). Thus the unitary matrix

diag(exp(2iφ), exp(−2iφ)) = exp(2φH3)

corresponds to the solution u4(φ, a) provided that the vector e3 associated with H3

coincides with a, i.e. the rotation exp(2φH3) about a by the angle 4φ correspondsto the matrix solution u4(φ, a).

5. Solution u3

In this section we give more complete approach than in [4] for constructing thesolution u3(φ, c). Here we calculate extreme positions of vector c in contrast to [4],where extreme values are found only for the third component of this vector. Herewe also show that solution u3 can be decomposed into three elementary solutions.

As is known, any unitary anti-Hermitian 3 × 3-matrix M has no anticommutingpartner so solution u3 looks as

u3(φ) = cos(φ)E3 + sin(φ)M.

Let us construct the basis using unit quaternions in the space of all the matrices M

µj =(Hj 00 −i

)and µj+6 =

( −i 00 Hj

)for j = 1, 2, 3,

µ4 = 0 0 i

0 −i 0i 0 0

, µ5 =

0 0 1

0 −i 0−1 0 0

,

µ6 = −i 0 0

0 −i 00 0 i

.

Each element of this basis differs by an imaginary unit from the corresponding onein the basis constructed for Hermitian matrices in [4]. The second triple µj+3 isobtainable from the first one and the third triple µj+6 is obtainable from the second


one by the transformations T µjT∗ = µj+3 and T µj+3T

∗ = µj+6, respectively,where

T = 0 1 0

0 0 11 0 0

, T ∗ =

0 0 1

1 0 00 1 0

.

An arbitrary unitary anti-Hermitian matrix M can be represented as an expan-sion of the matrices µj . A more interesting case is an expansion along any of thesetriples. Consider the expansion along the first triple:

M =3∑

j=1

cjµj = i

c3 c1 + ic2 0c1 − ic2 −c3 0

0 0 −c1 − c2 − c3

. (11)

The properties of matrix M lead to the following restrictions on cj

3∑j=1

c2j = 1 and

3∑j=1

cj = ±1. (12)

In spite of the fact that unitary matrices µj do not satisfy anticommuting prop-erty (2), solution u3 can be represented in the exponential form. It can be verifiedby expansion

exp(φcµ) = E3 + φ�3 + φ2

2�2

3 + · · · , where �3 =3∑

j=1

cjµj .

Due to

2(c1c2 + c1c3 + c2c3) =(

3∑j=1

cj

)2

−3∑

j=1

c2j = 0,

we find �23 = −E3. Thus the above expansion leads to exp(φcµ) = cosφ +

sinφ�3 = u3(φ, c).Equations (12) have one-parameter solutions cj . Let us write these solutions in

the form

c1 = 1 − ρ sin2 θ, c2 = 1 − ρ cos2 θ, c3 = ρ − 1,

where ρ = 1/(1 + cos θ sin θ). Due to the periodicity of cj with period π , it issufficient to consider cj in the range 0 � θ < π . It follows from the positivenessof ρ (2/3 � ρ � 2) that the maximum values of cj are the following: c1 = 1for θ = 0, c2 = 1 for θ = π/2, c3 = 1 for θ = 3π/4. So the combination ofcoordinates of vector c = (c1, c2, c3) at the points θ = 0, π/2, 3π/4 looks like oneof the cyclic permutation of (1, 0, 0), i.e. vector c coincides with the coordinatevectors ej at these points.

134 V. V. GUDKOV

To find the minimum values of cj , let us write the derivatives of cj with respectto θ as follows:

c′1 = −ρ2(sin(2θ) + sin2 θ), c′

2 = ρ2(sin(2θ) + cos2 θ),

c′3 = ρ2(sin2 θ − cos2 θ).

The first equality leads to c′1 = 0 if sin θ1 = −2 cos θ1. It can be fulfilled at the point

where cos θ1 = −1/√

5, sin θ1 = 2/√

5 due to sin2 θ + cos2 θ = 1. At this point,c1 = − 1

3 , c2 = c3 = 23 . Then, c′

2 = 0 if cos θ2 = −2 sin θ2. This can occur at thepoint where sin θ2 = 1/

√5, cos θ2 = −2/

√5. At this point c2 = −1/3, c1 = c3 =

2/3. Finally, c′3 = 0 at the point θ3 = π/4 where c3 = −1/3, c1 = c2 = 2/3. Thus,

during its transposition, vector c also coincides with the following vectors: h1 =( 2

3 ,23 ,− 1

3 ) at θ3 = π/4; h2 = (− 13 ,

23 ,

23 ) at θ1 ≈ 116◦34′ and h3 = ( 2

3 ,− 13 ,

23 ) at

θ2 ≈ 153◦26′.It is clear that the frame ej, being turned by the angle π/3 around line (1, 1, 1),

coincides with triple hj. In other words, vector c in the time when θ varies from 0to π describes an upper cone with the axis (1, 1, 1) and the vertex angle π/2 + ψ

where ψ = arctan(√

2/4) ≈ 19◦30′. The case∑3

j=1 cj = −1 in (12) leads to alower cone with axis (−1,−1,−1). The vectors −ej and −hj are located here.

Now let us show that solution u3 can be decomposed into three elementarysolutions by the formula

exp(φcµ) = exp(φc1µ1)� exp(φc2µ2)� exp(φc3µ3). (13)

Really, solution u3 has the form exp(φcµ) = diag(exp(φcH), exp(−iφ∑ cj ));rotation exp(φcH) obeys the property of decomposition (9); for scalar exponentialfunction the operation of decomposition is equivalent to the usual product of threefunctions exp(−iφcj ) (j = 1, 2, 3); for diagonal matrices the operation � pos-sesses a property diag(A�B,C �D) = diag(A,C)� diag(B,D). Thus, we findthat decomposition (13) is valid. It means that the existence of solution exp(φcµ)is equivalent to the existence of three simultaneous rotations about coordinate axeswhich correspond to µ1, µ2, µ3. It is remarkable that coefficients cj have the ex-tremal values ± 1

3 or ± 23 , so that their sum is equal to +1 or −1. We can consider∑

cj as a charge to obtain a more complete association with a model of quarks.Now let us represent a geometric scheme of interaction of solutions u2 and u3.

Solution u3(φ, c) depends on the vector c which is located only on the upper orlower cone because of

∑cj = ±1. Solution u2(φ, a) depends on the vector a

which is an arbitrary unit vector. Consider a neutral disk,∑

aj = 0, among twocones. In the moving frame at point z = 0 (φ(αz) = π/2), solution u2(φ, a) turnsS2 by the angle π . Thus u2 transfers the upper cone to the lower one or, inversely,if the vector a is located on the neutral disk. If Q = Q1/2, then the correspondingsolution u2 transfers the two vectors located on the cone’s opposite sides to theneutral disk.

Note that the movement along one of the cones (for example, the transpositionh1 to h2, h2 to h3 and h3 to h1) conserves the common charge but changes the


property of the fractional charge (2/3 or −1/3) to one or another coordinate vector,these changes can be associated with the colour change of quarks. All the aboveconsiderations related to the first triple µj can be applied to the second and thirdtriples.

It is significant to note that the derivative of u3 with respect to θ gives a linearcombination of Gell-Mann [12] matrices λj (j = 1, . . . , 8). Indeed, for the firsttriple µj , one finds

du3

dθ= sin(φ)

3∑j=1

c′jµj , where c′

j = dcjdθ

. (14)

Taking into account the evident equality∑3

j=1 c′j = 0, one obtains

3∑j=1

c′jµj = ic′

1λ1 − ic′2λ2 + ic′

3λ3.

If we replace the first triple in (14) by the second triple and then by the third one, thematrices diag(1, 0,−1) from µ6 and diag(0, 1,−1) from µ9 appear to be lacking inthe set of Gell-Mann matrices. However, their linear combination with coefficients1/

√3 gives the eighth Gell-Mann matrix λ8 = diag(1, 1,−2)/

√3.

6. Pulsating Sphere

In this section we show that rotation exp(φaH) can be applied to spheres of arbi-trary radius. Then we put in one-to-one correspondence a set of matrix solutionsu2(φ, a) and cross-section Sφ of sphere S3 and describe the behaviour of Sφ.

As is known (see, e.g., [8, p. 32], [10, p. 32]) a turn of the arbitrary point P ∈ S2

about coordinate axis ej corresponds to a turn of point ζ (a stereographic projectionof P onto a complex plane). A finite turn of point ζ is described by linear-fractionaltransformation

ζ → ζ ′ = ζf + g

−ζ g + f, where

(f g

−g f

)≡ U ∈ SU(2). (15)

Note that there is a one-to-one correspondence between unitary matrices U andmatrix solutions u2(φ, a). Furthermore, if we take the sphere of arbitrary radius,then the transformation (15) remains valid, i.e. one and the same matrix solutionu2(φ, a) is fit to describe a rotation exp(φaH) of a sphere of arbitrary radius.

To describe the correspondence between u2 and Sφ , we start with a matrixrepresentation of solution u2:

u2(φ, a) =(

cosφ + ia3 sinφ (−a2 + ia1) sin φ(a2 + ia1) sinφ cosφ − ia3 sinφ

)≡(

f g

−g f

).

136 V. V. GUDKOV

This matrix, as in (15), has the unit determinant |f |2 + |g|2 = 1. Note that u2(φ +π, a) = −u2(φ, a), so the whole region 0 � φ � 2π should be considered.

Let us set f = x0 + ix3 and g = −x2 + ix1, then the relation

x0 = cosφ, xj = aj sinφ for j = 1, 2, 3

holds. This means that for each φ from the region 0 � φ < 2π and each unitvector a there exists only one point (x0, x1, x2, x3) on S3 satisfying the relation. Onthe other hand, for an arbitrary point (x0, x1, x2, x3) ∈ S3 with x0 �= ±1 there existsonly one pair (φ, a) satisfying the relation. Finally, if x0 = 1 or −1 then φ = 0or π , respectively, and u2 = x0E2. Thus one-to-one correspondences u2 ↔ SU(2)and u2 ↔ S3 are established as an intermediate result. It follows from the latter thatthe same correspondence is valid between the set u2(φ, S

2) of solutions u2(φ, a)for all vectors a and the cross-section, call it Sφ, of S3 by hyperplane x0 = cosφ.

Now consider sphere Sφ . Arbitrary point (x1, x2, x3) on Sφ satisfies relationxj = aj sinφ; the radius of Sφ is ρ = |sinφ|; Sφ is simply a contraction of S2

along each radius vector a. Sphere Sφ with the right frame is called a right orientedsphere and is denoted by S+

φ , while Sφ with the left frame is called a left orientedsphere and is denoted by S−

φ .Consider the behavior of Sφ in time when φ varies from 0 to 2π . Starting from

the point (1, 0, 0, 0) at φ = 0 the sphere S+φ expands to S2, at φ = π/2 then it con-

tracts to the point (−1, 0, 0, 0) at φ = π . After φ passes the value π , coordinatesxj change their signs according to xj = −ajρ. This means that each radial vectorx ∈ S+

φ is transferred to its opposite x− ∈ S−φ at φ = π and the same holds for

coordinate vectors ej. In other words, S+φ passing through the origin turns inside out

to S−φ with a simultaneous changing of the right frame ej to the left one e−

j = −ej.Further, S−

φ expands to the left oriented unit sphere at φ = 3π/2 and after that, itcontracts to the starting point at φ = 2π . Passing through the starting point, sphereS−φ turns inside out to S+

φ . Thus we have proven the following statement:

THEOREM 3. The following one-to-one correspondences:

u2(φ, a) ↔ SU(2), u2(φ, a) ↔ S3, u2(φ, S2) ↔ Sφ

are valid. The sphere Sφ pulsates as S+φ on 0 � φ � π and as S−

φ on π � φ � 2πin time when φ varies from 0 to 2π .

We hope that the proposed approaches and results in this paper will be usefulfor modelling in the particle physics.

Acknowledgement

This research has been accomplished with the financial support of the ScienceCouncil of Latvia, Grant No. 01.0033.


References

1. Dirac, P. A. M.: The relativistic electron wave equation, Europhysics 8 (1977), 1–4.2. Gudkov, V. V.: Torus as a geometrical image of matrix solutions of wave equations, Nonlinear

Anal. 47 (2001), 5945–5953.3. Gudkov, V. V.: Matrix solutions of wave equations and Clifford algebras, J. Phys. A: Math.

Gen. 33 (2000), 6975–6979.4. Gudkov, V. V.: Geometrical properties of matrix solutions of the Klein–Gordon equation,

J. Phys. A: Math. Gen. 32 (1999), L281–L284.5. Bohun, C. S. and Cooperstock, F. I.: Dirac–Maxwell solitons, Phys. Rev. A 60 (1999), 4291–

4300.6. Okun’, L. B.: Leptons and Quarks, Nauka, Moscow, 1990.7. Huang, K.: Quarks, Leptons and Gauge Fields, Mir, Moscow, 1985.8. Penrose, R. and Rindler, W.: Spinors and Space-time. Vol. 1. Two-spinor Calculus and

Relativistic Fields, Mir, Moscow, 1987.9. Greiner, W. and Maruhn, J. A.: Nuclear Models, Springer, Berlin, 1996.

10. Biedenharn, L. C. and Louck, J. D.: Angular Momentum in Quantum Physics. Theory andApplication, Vol. 1, Mir, Moscow, 1984.

11. Coquereaux, R.: Modulo 8 of periodicity of real Clifford algebras and particle physics, Phys.Rev. B 115 (1982), 389–395.

12. Gell-Mann, M.: Symmetries of baryons and mesons, Phys. Rev. 125 (1962), 1067–1084.


139

Separation of Variables for Bi-Hamiltonian Systems

GREGORIO FALQUI1 and MARCO PEDRONI2

1SISSA, Via Beirut 2/4, I-34014 Trieste, Italy. e-mail: [email protected] di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 Genova, Italy.e-mail: [email protected]

(Received: 1 April 2002; in final form: 29 October 2002)

Abstract. We address the problem of the separation of variables for the Hamilton–Jacobi equationwithin the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special classof bi-Hamiltonian manifolds, called ωN manifolds, to give intrisic tests of separability (and Stäckelseparability) for Hamiltonian systems. The separation variables are naturally associated with thegeometrical structures of the ωN manifold itself. We apply these results to bi-Hamiltonian systemsof the Gel’fand–Zakharevich type and we give explicit procedures to find the separated coordinatesand the separation relations.

Mathematics Subject Classifications (2000): 70H20, 70H06, 37K10.

Key words: Hamilton–Jacobi equations, separation of variables, Nijenhuis structures, bi-Hamil-tonian manifolds.

1. Introduction

The technique of additive separation of variables for solving by quadratures theHamilton–Jacobi (HJ) equation is a very important tool in analytical mechanics,initiated by Jacobi and others back in the nineteenth century (see, e.g., [35, 9]).Following these classical works, an n-tuple (H1, . . . , Hn) of functionally indepen-dent Hamiltonians will be said to be separable in a set of canonical coordinates(q1, . . . , qn, p1, . . . , pn) if there exist n relations, called separation relations, ofthe form

φi(qi, pi,H1, . . . , Hn) = 0, i = 1, . . . , n, with det

[∂φi

∂Hj

]�= 0. (1.1)

The reason for this definition is that the stationary Hamilton–Jacobi equationsfor the Hamiltonians Hi can be collectively solved by the additively separatedcomplete integral

W(q1, . . . , qn;α1, . . . , αn) =n∑

i=1

Wi(qi;α1, . . . , αn), (1.2)

where the Wi are found by quadratures as the solutions of ordinary differentialequations.

140 GREGORIO FALQUI AND MARCO PEDRONI

One of the first systematic results was found by Levi-Civita, who provided, in1904, a test for the separability of a given Hamiltonian in a given system of canon-ical coordinates. Stäckel and Eisenhart concentrated on Hamiltonians quadratic inthe momenta and orthogonal separation variables. In particular, Stäckel consideredthe Hamiltonian

H(q, p) = 1

2

∑gii(q)pi

2 + V (q)

and showed that H is separable in the coordinates (q, p) if there exist an invertiblematrix S(q) and a column vector U(q) such that the ith rows of S and U dependonly on the coordinate qi , and H is among the solutions (H1, . . . , Hn) of the linearsystem

n∑j=1

Sij (qi)Hj = 12p

2i − Ui(qi).

These equations provide the separation relations for the (commuting) Hamiltonians(H1, . . . , Hn).

With the works of Eisenhart, the theory of separation of variables was insertedwithin the context of global Riemannian geometry, and this still represents an activearea of research, where the notions of Killing tensor and Killing web play a keyrole ([44, 25, 4]).

Starting from the study of algebraic-geometric solutions of (stationary reduc-tions of) soliton equations and the introduction of the concept of algebraic com-pletely integrable system (see, e.g., [3, 9, 43]), separation of variables has receiveda renewed attention ([16, 1, 20, 23, 39]). This research activity, also connected withthe theory of quantum integrable systems, deals with Hamiltonian systems admit-ting a Lax representation with spectral parameter and an r-matrix formulation. Inthis case, the separation relations are provided by the spectral curve

det(µI − L(λ)) = 0

associated with the Lax matrix L(λ). Indeed, one can often find canonical coor-dinates (λ1, . . . , λn, µ1, . . . , µn) on the phase space such that every pair (λi, µi)

belongs to the spectral curve. Since the Hamiltonians are defined by the spectralcurve, they are separable in these coordinates.

The two classes of separable systems briefly recalled above strongly suggestthat a ‘theory of separability’ should start from the following data:

(1) a class of symplectic manifolds M;(2) a class of canonical coordinates on M;(3) a class of Hamiltonian functions on M,

and should provide

SEPARATION OF VARIABLES FOR BI-HAMILTONIAN SYSTEMS 141

(a) separability test(s) to ascertain whether the HJ equations associated with theselected Hamiltonians admit a complete integral which is additively separatedin the chosen coordinates;

(b) algorithms to compute the separation coordinates and to exhibit the separationrelations, so that the HJ equations can be explicitly solved.

Within the context of Riemannian geometry, the manifolds are cotangent bundlesof Riemannian manifolds, the coordinates are (fibered) orthogonal coordinates, andthe Hamiltonians are quadratic in the momenta. For Lax systems, roughly speaking,the manifolds are suitable coadjoint orbits in loop algebras, the coordinates arethe so-called spectral Darboux coordinates ([1]), possibly to be found using the‘Sklyanin magic recipe’ ([39]), and the separable Hamiltonians are the spectralinvariants.

The point of view herewith presented is the following. The class of manifoldswe will consider are particular bi-Hamiltonian manifolds, to be termed ωN mani-folds, where one of the two Poisson brackets is nondegenerate and thus defines asymplectic form ω and, together with the other one, a recursion operator N . Theclass of coordinates, called Darboux–Nijenhuis (DN) coordinates, are canonicalwith respect to ω and diagonalize N .

The first result is that an n-tuple (H1, . . . , Hn) of Hamiltonians on M (wheren = 1

2 dimM) is separable in DN coordinates if and only if they are in involutionwith respect to both Poisson brackets. This condition is clearly intrinsic, i.e., it canbe checked in any coordinate system. A second result of the present paper is that ex-amples of separable systems on ωN manifolds are provided by suitable reductionsof bi-Hamiltonian hierarchies, called Gel’fand–Zakharevich systems. They are bi-Hamiltonian systems defined on a bi-Hamiltonian manifold (M, {·, ·}, {·, ·}′) by thecoefficients of the Casimir functions of the Poisson pencil {·, ·}λ := {·, ·}′ − λ{·, ·}.Such coefficients are in involution with respect to both Poisson brackets, and aresupposed to be enough to define integrable systems on the symplectic leaves of{·, ·}. If there exists a foliation of M, transversal to these symplectic leaves andcompatible with the Poisson pencil (in a suitable sense), then every symplectic leafof {·, ·} becomes an ωN manifold, and the (restrictions of the) GZ systems naturallyfall in the class of systems which are separable in DN coordinates. For this reason,we can say that the Poisson pencil separates its Casimirs.

The third result concerns the Stäckel separability. With a slight extension of theclassical notion, we say that (H1, . . . , Hn) are Stäckel separable if the separationrelations (1.1) are affine in the Hi:

n∑j=1

Sij (qi, pi)Hj − Ui(qi, pi) = 0, i = 1, . . . , n. (1.3)

In this case, the collection (H1, . . . , Hn) is called a Stäckel basis. We give an intrin-sic test for the Stäckel separability in DN coordinates, which has a straightforwardapplication to GZ systems. This goes as follows. We notice that if (H1, . . . , Hn)


are in involution with respect to both Poisson brackets (and therefore separable inDN coordinates), then there exists a matrix F (depending on the choice of the Hi)such that

N∗ dHi =n∑

j=1

Fij dHj.

We prove that (H1, . . . , Hn) is a Stäckel basis if and only if

N∗ dFij =n∑

k=1

Fik dFkj .

The geometric theory of separability we present in this paper may be, in ouropinion, regarded as an effective bridge between the ‘classical’ and the ‘mod-ern’ aspects of the theory of separability. More evidence of this claim will begiven in [10], where we will also show how to frame Eisenhart’s theory withinour approach, and discuss the problem of associating a Lax representation to GZsystems.

This paper is organized as follows. The first part (Section 2 to 5) is devoted to thegeometry of separability on ωN manifolds. In Section 2 we will introduce the no-tion of ωN manifold and we will study the DN coordinates. Section 3 contains themain results about separability on ωN manifolds, whereas in Section 4 the Stäckelseparability is considered. In Section 5 we will come back to DN coordinates,pointing out some algorithms for their explicit computation.

In the second part of the paper we will turn our attention to GZ systems. Sec-tion 6 deals with the particular case where there is only one Casimir of the Poissonpencil (i.e., one bi-Hamiltonian hierarchy), and contains the example of the 3-particle open Toda lattice. This section is intended for an introduction to Section 7,where the general case is treated. We will give conditions under which a bi-Hamil-tonian manifold is foliated in ωN manifolds, and we will show that the GZ systemsare separable in DN coordinates. Subsection 7.3 is devoted to the Stäckel separa-bility of such systems. In Section 8 we will show an efficient way to determine,in the Stäckel separable case, the separation relations for GZ systems. Finally, wepresent an example in the loop algebra of sl(3).

2. ωN Manifolds

In this section we describe the manifolds where our (separable) systems will bedefined. They are called ωN manifolds, since they are Poisson–Nijenhuis (PN)manifolds ([26, 28, 30]) such that the first Poisson structure is nondegenerate, andtherefore defines a symplectic form. In turn, PN manifolds are particular instancesof bi-Hamiltonian manifolds, i.e., smooth (or complex) manifolds M endowed witha pair of of compatible Poisson brackets, {·, ·} and {·, ·}′. This means that everylinear combination of them is still a Poisson bracket.


DEFINITION 2.1. AnωN manifold is a bi-Hamiltonian manifold (M, {·, ·}, {·, ·}′)in which one of the Poisson brackets (say, {·, ·}) is nondegenerate.

Therefore, M is endowed with a symplectic form ω defined by

{f, g} = ω(Xf ,Xg), (2.1)

where Xf is the Hamiltonian vector field associated with f by means of {·, ·}.In terms of the Poisson tensor P corresponding to {·, ·}, viewed as a section ofHom(T ∗M,TM), this simply means that P is invertible and ω is its inverse. Us-ing also the Poisson tensor P ′ associated with {·, ·}′, one can construct the tensorfield N := P ′P−1, of type (1, 1), to be termed recursion operator of the ωN

manifold M.

PROPOSITION 2.2. The Nijenhuis torsion of N ,

T (N)(X, Y ) := [NX,NY ] − N([NX,Y ] + [X,NY ] − N[X,Y ]), (2.2)

vanishes as a consequence of the compatibility between P and P ′.

A proof of this well-known fact can be found in [28].There are two main sources of examples of ωN manifold. The first one comes

from classical mechanics. Let Q be an n-dimensional manifold endowed with a(1, 1) tensor field L with vanishing Nijenhuis torsion, and let us consider its cotan-gent bundle T ∗Q with the canonical Poisson bracket {·, ·}. As shown in [24], thevanishing of the Nijenhuis torsion of L entails that one can use it to define a secondPoisson bracket {·, ·}′ on T ∗Q as

{qi, qj }′ = 0, {qi, pj }′ = −Lij , {pi, pj }′ =

(∂Lk

j

∂qi− ∂Lk

i

∂qj

)pk,

where (pi, qi) are fibered coordinates. This Poisson bracket is compatible with{·, ·}, so that the phase space T ∗Q becomes an ωN manifold, whose recursionoperator N is the complete lifting ([45]) of L.

The second class of examples of ωN manifolds can be obtained by reduc-tion from a bi-Hamiltonian manifold (M,P, P ′) where both Poisson tensors aredegenerate ([14]).

This happens, in particular, in the following situation. Suppose that P has con-stant corank k, that dimM = 2n+k, and that one can find a k-dimensional foliationZ of M with the properties:

(1) The foliation Z is transversal to the symplectic foliation of P ;(2) The functions which are constant along Z form a Poisson subalgebra of

(C∞(M), {·, ·}) and of (C∞(M), {·, ·}′), i.e., if f and g are constant along Z,then the same is true for {f, g} and {f, g}′.


Then any symplectic leaf S of {·, ·} inherits a bi-Hamiltonian structure from M.Moreover, the reduction of the first Poisson structure coincides with the sym-plectic form of S, so that S is an ωN manifold. Such a procedure is one of themain topics of the paper, and will be fully discussed in Section 7, where we willalso show that bi-Hamiltonian systems on M give rise to separable systems on S.The corresponding variables of separation are going to be introduced in the nextsubsection.

2.1. DARBOUX–NIJENHUIS COORDINATES

In this subsection we will describe a class of canonical coordinates on ωN mani-folds, called Darboux–Nijenhuis coordinates. They will play the important role ofvariables of separation for (suitable) systems on ωN manifolds.

DEFINITION 2.3. A set of local coordinates (xi, yi) on an ωN manifold is calleda set of Darboux–Nijenhuis (DN) coordinates if they are canonical with respect tothe symplectic form ω,

ω =n∑

i=1

dyi ∧ dxi ,

and put the recursion operator N in diagonal form,

N =n∑

i=1

λi

(∂

∂xi⊗ dxi + ∂

∂yi⊗ dyi

). (2.3)

This means that the only nonzero Poisson brackets are

{xi, yj } = δij , {xi, yj }′ = λiδij .

The assumption, contained in (2.3), that the eigenvalues λi of N are (at least)double is not restrictive, since its eigenspaces have even dimension, equal to thedimension of the kernel of P ′ − λiP . For the ωN manifold T ∗Q described in theprevious section, it is easy to check that the eigenvalues of L (if they are inde-pendent) and their conjugate momenta are DN coordinates. In order to ensure theexistence of DN coordinates on more general ωN manifolds, we give the following

DEFINITION 2.4. A 2n-dimensional ωN manifold M is said to be semisimple ifits recursion operator N has, at every point, n distinct eigenvalues λ1, . . . , λn. It iscalled regular if the eigenvalues of N are functionally independent on M.

It can be shown ([18, 27, 41]) that every point of a semisimple ωN manifoldhas a neighborhood where DN coordinates can be found, and that, if the ωN man-ifold M is regular, one half of these coordinates are ‘canonically’ provided bythe recursion operator. Indeed, as a consequence of the vanishing of the Nijenhuistorsion of N , the eigenvalues λi always satisfy N∗ dλi = λi dλi , where N∗ is theadjoint of N , and one has


PROPOSITION 2.5. In a neighborhood of a point of a regular ωN manifold wherethe eigenvalues of N are distinct it is possible to find by quadratures n functionsµ1, . . . , µn that, along with the eigenvalues λ1, . . . , λn, are DN coordinates.

DEFINITION 2.6. The coordinates (λ1, . . . , λn, µ1, . . . , µn) described in the pre-vious proposition will be called a set of special Darboux–Nijenhuis (sDN) coordi-nates.

Such coordinates will often be used in the sequel, because the λi are simplythe roots of the minimal polynomial of N . Proposition 2.5 means also that everyregular ωN manifold is locally equal to the ‘lifted’ ωN manifold T ∗Q we haveseen in Section 2.

The distinguishing property of the pairs of DN coordinates (xi, yi), and, a for-tiori, of the ‘special’ pairs (λi, µi), is that their differentials span an eigenspaceof N∗, that is, satisfy the equations

N∗ dxi = λi dxi, N∗ dyi = λi dyi , i = 1, . . . , n. (2.4)

This leads us to the following definition:

DEFINITION 2.7. A function f on an ωN manifold is said to be a Stäckelfunction (relative to the eigenvalue λi of N) if

N∗ df = λi df. (2.5)

The following property of Stäckel functions, which also explains their name,will be used many times in the rest of the paper.

PROPOSITION 2.8. Let M be a semisimple ωN manifold. A function f on M isa Stäckel function relative to λi if and only if, in any (some) system (x1, . . . , yn) ofDN coordinates, f depends only on xi and yi .

Proof. It is obvious that if f = f (xi, yi) then N∗ df = λi df . Conversely,if (2.5) holds, then df belongs to the λi-eigenspace of N∗, so that df is a linearcombination of dxi and dyi and therefore f depends only on xi and yi . ✷

3. Separability on ωN Manifolds

In Section 2 we have introduced a class of (symplectic) manifolds and we haveselected a class of (canonical) coordinates on such manifolds. Now we are going tocharacterize, from a geometric point of view, those integrable Hamiltonian systemson ωN manifolds which are separable in DN coordinates. In the next section wewill consider the same problem for Stäckel separability.

We recall that an n-tuple (H1, . . . , Hn) of functionally independent Hamil-tonians on an ωN manifold M is said to be separable in the DN coordinates


(x1, . . . , xn, y1, . . . , yn) if there exist relations of the form

φi(xi, yi,H1, . . . , Hn) = 0, i = 1, . . . , n, with det

[∂φi

∂Hj

]�= 0. (3.1)

It can be easily shown (e.g., via the Hamilton–Jacobi method) that this entails theinvolutivity of the Hi . Obviously enough, the separability property is not peculiarof the specific choice of the functions Hi . If Ki = Ki(H1, . . . , Hn) are functions ofthe Hi , they are also separable according to (3.1). So we see that the property (3.1)concerns the geometrical features of an integrable system, i.e., is to be regardedas a property of the Lagrangian distribution defined by the mutually commutingfunctions Hi . Thus one can say that the Hi define a separable foliation of M.According to the following theorem, that will be proved during this section, theseparability property can be formulated in terms of the geometric objects ω and N ,or {·, ·} and {·, ·}′, of the ωN manifold M.

THEOREM 3.1. Let M be a semisimple ωN manifold (so that DN coordinateslocally exist, see the comment after Definition 2.4) and let (H1, . . . , Hn) be a set ofn functionally independent Hamiltonians on M. Then the following statements areequivalent:

(a) The foliation defined by (H1, . . . , Hn) is separable in DN coordinates (andtherefore Lagrangian with respect to ω);

(b) The distribution tangent to the foliation defined by (H1, . . . , Hn) is Lagrangianwith respect to ω and invariant with respect to N ;

(c) The functions (H1, . . . , Hn) are in bi-involution, i.e., {Hi,Hj } = 0 and{Hi,Hj }′ = 0 for all i, j .

We will often refer to property (c) by saying that the foliation defined by theHi is bi-Lagrangian. This is a fundamental property in our approach to separabil-ity, and will be exploited especially in Sections 6 and 7. Incidentally, we noticethat bi-Lagrangian foliations play an important role in the study of special Kählermanifolds [22].

Throughout the rest of the section M will be a semisimple ωN manifold,(λ1, . . . , λn) the eigenvalues of the recursion operator N , and (xi, yi) DN coor-dinates on M. We begin with showing that the invariance with respect to N is anecessary condition for separability.

PROPOSITION 3.2. Let (H1, . . . , Hn) be functions on M that are separable inDN coordinates. Then the subspace spanned by (dH1, . . . , dHn) is invariant withrespect to N∗. More precisely, there exists a (simple) matrix F with eigenvalues(λ1, . . . , λn) such that

N∗ dHi =n∑

j=1

Fij dHj, i = 1, . . . , n. (3.2)


Consequently, the Lagrangian distribution defined by (H1, . . . , Hn), which isspanned by the Hamiltonian vector fields XHi

, is invariant with respect to N .Proof. Differentiate the relations (3.1),

∂φi

∂xidxi + ∂φi

∂yidyi +

n∑j=1

∂φi

∂Hj

dHj = 0, (3.3)

then apply N∗ to obtain

∂φi

∂xiλi dxi + ∂φi

∂yiλi dyi +

n∑j=1

∂φi

∂Hj

N∗ dHj = 0. (3.4)

It follows thatn∑

j=1

∂φi

∂Hj

N∗ dHj = −λi

(∂φi

∂xidxi + ∂φi

∂yidyi

)= λi

n∑j=1

∂φi

∂Hj

dHj , (3.5)

that is, in matrix form,

JN∗ dH = )J dH, (3.6)

where

Jij = ∂φi

∂Hj

, dH = (dH1, . . . , dHn)T ,

N∗ dH = (N∗ dH1, . . . , N∗ dHn)

T ,

and ) = diag(λ1, . . . , λn). Therefore (3.2) is satisfied with F = J−1)J , and theeigenvalues of F are (λ1, . . . , λn). The final assertion easily follows. ✷

The matrix F will be called the control matrix, with respect to the basis(H1, . . . , Hn), of the separable foliation.

PROPOSITION 3.3. If (H1, . . . , Hn) define a distribution which is invariant withrespect to N , that is,

N∗ dHi =n∑

j=1

Fij dHj, i = 1, . . . , n, (3.7)

and the eigenvalues of F are distinct, then the Hi are separable in DN coordinates.Proof. Since the eigenvalues of F are distinct, they are the eigenvalues

(λ1, . . . , λn) of N , so that there exists a matrix S such that F = S−1)S. With S

we define the 1-forms θi := ∑nj=1 Sij dHj , for i = 1, . . . , n. They are eigenvectors

of N∗, since

N∗θi =n∑

j=1

SijN∗ dHj =

n∑j,k=1

SijFjk dHk =n∑

k=1

λiSik dHk = λiθi. (3.8)


Then there exist functions Li and Mi such that θi = Li dxi + Mi dyi , that is,

n∑j=1

Sij dHj − Li dxi − Mi dyi = 0. (3.9)

This means that dim〈dH1, . . . , dHn, dxi, dyi〉 � n + 1, so that there exists arelation of the form (3.1), i.e., the functions (H1, . . . , Hn) are separable in DNcoordinates. ✷

In order to complete the proof of the equivalence between statements (a) and (b)of Theorem 3.1, we need the following:

LEMMA 3.4. If (H1, . . . , Hn) are independent functions in involution with respectto ω such that (3.2) holds, then the eigenvalues of F are distinct.

Proof. Suppose that N∗ dHi = ∑nj=1 Fij dHj , with {Hi,Hj } = 0 for all i, j .

Since F represents the restriction of N∗ to 〈dH1, . . . , dHn〉, it is diagonalizable.Thus, if λi would be a double eigenvalue of F , the span 〈dH1, . . . , dHn〉 would con-tain the two-dimensional eigenspace spanned by dxi and dyi . But the involutivityof the Hi would entail that {xi, yi} = 0, which is false. ✷

Relations (3.2) may be called generalized Lenard relations (and the functionsHi fulfilling them a Nijenhuis chain, as in [13]), as enlightened by the followingexample.

EXAMPLE 3.5. If Hk := 1/2k trNk = ∑nj=1 λ

kj , then N∗ dHk = dHk+1 for

k = 1, . . . , n− 1, that is, the Lenard relations P ′ dHk = P dHk+1 hold. Moreover,dHn+1 = ∑n

j=1 cj dHn+1−j , where λn −∑n−1j=0 cn−j λ

j is the minimal polynomialof N . Therefore, condition (3.2) is satisfied with

F =

0 1 0 · · · 00 0 1 · · · 0...

......

......

0 0 · · · · · · 1cn cn−1 · · · · · · c1

. (3.10)

Remark 3.6. It is well known that functions Hi satisfying the Lenard relationsare in involution with respect to both Poisson brackets, and so they provide afirst instance of correspondence between invariant distributions and bi-involutivity,which is at the same time trivial and paradigmatic.

Indeed, it is trivial from the point of view of the theory of separation of vari-ables, since such Hamiltonians are easily seen to depend only on (λ1, . . . , λn) if theωN manifold M is regular and semisimple. Then the Hamilton–Jacobi equationsassociated with the Hi are trivially separable in the sDN coordinates (λ1, . . . , λn,

µ1, . . . , µn). Nevertheless, it is paradigmatic with respect to the issues of this


paper. Indeed, the Hi (that is, the λi) define a distinguished bi-Lagrangian fo-liation, called principal foliation, which coincides with the canonical fibrationπ : T ∗Q → Q of classical phase spaces when T ∗Q is the ωN manifold consideredin Section 2. However, there are in general bi-Lagrangian foliations which aredifferent from the principal one, as we will explicitly see in Section 6. We aregoing to show that such foliations are characterized by the invariance with respectto N , so that they give rise to separable systems. This means that our theory dealswith cases in which the Hamiltonians are not simply the traces of the recursionoperator. In other words, we will deal with cases in which the control matrix F ofEquation (3.2) need not be a companion matrix of the form (3.10). Accordingly,the separable vector fields we will consider are tangent to a bi-Lagrangian foliation,but they are not, in general, bi-Hamiltonian.

PROPOSITION 3.7. Let (H1, . . . , Hn) be independent functions on M. Then (3.2)holds, with a matrix F with distinct eigenvalues, if and only if the functions Hi arein bi-involution:

{Hi,Hj } = {Hi,Hj }′ = 0, for all i, j = 1, . . . , n. (3.11)

Proof. We know from Proposition 3.3 that condition (3.2), with a simple ma-trix F , implies separability and therefore involutivity with respect to {·, ·}. More-over,

{Hi,Hj }′ = 〈dHi, P′ dHj 〉 = 〈dHi,NP dHj 〉 = 〈N∗ dHi, P dHj 〉

=n∑

k=1

Fik{Hk,Hj } (3.12)

showing that {Hi,Hj }′ vanishes as well.Conversely, suppose that {Hi,Hj } = {Hi,Hj }′ = 0 for all i, j . Then the

foliation H defined by the Hi is Lagrangian with respect to {·, ·}, and

〈N∗ dHi, P dHj 〉 = 〈dHi,NP dHj 〉 = 〈dHi, P′ dHj 〉 = 0.

Thus, N∗ dHi belongs, for every i, to the annihilator of 〈P dH1, . . . , P dHn〉, whichis tangent to H , since H is Lagrangian. This shows that (3.2) holds, and Lemma 3.4entails that F has distinct eigenvalues. ✷

Thus we have proved also the equivalence between (b) and (c) of Theorem 3.1.

Remark 3.8. One could also prove that a function H is separable in DN coor-dinates if and only if its Hamiltonian vector field XH is tangent to a bi-Lagrangianfoliation H . The ‘if’ part of this statement is a simple corollary of Theorem 3.1.Indeed, let H be defined by the functions (H1, . . . , Hn). Then H is a functionof the Hi , since the distribution is Lagrangian, and one can find other (n − 1)functions K2, . . . , Kn such that H is defined by (H,K2, . . . , Kn). The ‘only if’part of this statement is deeper, and essentially gives rise to the intrinsic picture ofthe Levi-Civita conditions for separability, to be fully discussed in [10].


Summing up, we have proved a criterion for the separability in DN coordinates,which can be tested without knowing explicitly these coordinates. Indeed, the state-ment (3.11) can be checked in any coordinate system. An important application ofthis criterion will be given in Section 7, where we will show that the bi-Hamiltonianhierarchies on a bi-Hamiltonian manifold admitting a transversal distribution withthe properties mentioned at the end of Section 2 give rise to separable Hamiltonianvector fields on the reduced ωN manifolds.

4. Stäckel Separability on ωN Manifolds

The separability criteria of the previous section do not give explicit informationon the form of the separating relations (3.1). For this reason, in this section wewill concentrate on the more stringent notion of Stäckel separability. Recall that(H1, . . . , Hn), independent functions on an ωN manifold, were defined to beStäckel separable in the DN coordinates (x1, . . . , yn) if there exist relations of theform (3.1), given by affine equations in the Hj , that is,

n∑j=1

Sij (xi, yi)Hj − Ui(xi, yi) = 0, i = 1, . . . , n, (4.1)

with S an invertible matrix. In this case, we say that the Hi are a Stäckel basis ofthe (separable) foliation. The entries Sij and Ui depend only on xi and yi , i.e., theyare Stäckel functions according to Proposition 2.8. Usually, S is called a Stäckelmatrix, and U a Stäckel vector. Notice that the definition of Stäckel separabilitydepends on the choice of the Hi defining the Lagrangian distribution. Indeed, if(H1, . . . , Hn) are Stäckel-separable, then Ki = Ki(H1, . . . , Hn), for i = 1, . . . , n,will not, in general, fulfill relations of the form (4.1). A natural problem, thatwill not be discussed in this paper, is to give a geometrical characterization ofthe Lagrangian foliations admitting a set of defining functions for which Stäckelseparability holds. Some results in this direction will be presented in [10].

Now we will give a necessary and sufficient condition for the Stäckel sepa-rability in DN coordinates of a given n-tuple (H1, . . . , Hn) of functions on anωN manifold. We will also show that in this case one can explicitly find the re-lations (3.1) and has useful information to algebraically determine the separationvariables.

Suppose (H1, . . . , Hn) to be independent functions on a regular semisimple ωNmanifold that are Stäckel separable in the DN coordinates. Then we know fromProposition 3.2 that there exists a control matrix F , with eigenvalues (λ1, . . . , λn),such that N∗ dH = F dH . Since Proposition 2.8 entails that N∗ dS = ) dS andN∗ dU = ) dU , we can show:

PROPOSITION 4.1. In the above-mentioned hypotheses, the matrix F satisfies

N∗ dF = F dF, that is, N∗ dFij =n∑

k=1

Fik dFkj , ∀i, j = 1, . . . , n. (4.2)


Proof. First we show that F = S−1)S. Indeed,

SF dH = SN∗ dH = N∗[d(SH) − (dS)H ] = N∗ dU − (N∗ dS)H

= ) dU − ) dS H = )S dH. (4.3)

Then we have

N∗ dF = N∗d(S−1)S) = N∗(−S−1 dS S−1)S + S−1 d)S + S−1) dS)

= −S−1) dS S−1)S + S−1) d)S + S−1)2 dS

= S−1)S(−S−1 dS S−1)S + S−1 d)S + S−1) dS) = F dF, (4.4)

and the proof is complete. ✷Condition (4.2) is also sufficient for the Stäckel separability, as shown in the

following:

THEOREM 4.2. Let (H1, . . . , Hn) be independent functions, defining a bi-La-grangian foliation on a regular semisimple ωN manifold. If the control matrix F

fulfills (4.2), then:

(1) The left eigenvectors of F , if suitably normalized, form a Stäckel matrix. Moreprecisely, if S is a matrix such that F = S−1)S, and such that in every rowof S there is an entry equal to 1, then S is a Stäckel matrix in DN coordinates(x1, . . . , yn);

(2) The functions (H1, . . . , Hn) are Stäckel separable in DN coordinates.

Proof. From (4.2) we have that

N∗(−S−1 dS S−1)S + S−1 d)S + S−1) dS)

= S−1)S(−S−1 dS S−1)S + S−1 d)S + S−1) dS), (4.5)

that is,

N∗(−dS S−1)S + ) dS) = )(−dS S−1)S + ) dS), (4.6)

or (−N∗ dS + ) dS)F = )(−N∗ dS + ) dS). Hence, the j th row of (−N∗ dS +) dS) is a left eigenvector of F , relative to λj . This entails that it is proportional tothe j th row of S, i.e., there exists a 1-form αj such that

ej (−N∗ dS + ) dS) = αjejS, (4.7)

where ej is the j th row vector of the standard basis. Multiplying Equation (4.7) byeTk , where Sjk = ejSe

Tk = 1, we obtain αj = 0, so that

N∗ dS = ) dS. (4.8)


In components, this reads N∗ dSjk = λj dSjk , which implies (see Proposition 2.8)that Sjk depends only on xj and yj , i.e., S is a Stäckel matrix. Finally, the fact thatU := SH is a Stäckel vector follows from

N∗ dU = N∗(dS H + S dH) = ) dS H + SF dH

= )(dS H + S dH) = ) dU. (4.9)

This completes the proof. ✷The results obtained so far can be summarized in the following statements. An

n-tuple of functions (H1, . . . , Hn) in involution is separable (in DN coordinates) ifand only if the span of their differentials is invariant for N∗. Let F be the matrixwhich represents (the restriction of) N∗ on such a span. Then Equation (4.2) repre-sents a test for the Stäckel separability of theHi . Once this test is passed, the Stäckelmatrix is easily constructed as a (suitably normalized) matrix that diagonalize F ,and the separation procedure can be quite explicitly performed. Therefore, in oursetting the Hamiltonians provide their Stäckel matrix as well as the separationrelations (4.1).

We end this section with the following comment on the intrinsic meaning of theStäckel separability conditions (4.2). It is known [17] that, as a consequence of thevanishing of the Nijenhuis torsion of N , the de Rham complex of M is endowedwith a second derivation dN , which is defined to be the unique (anti)derivation withrespect to the wedge product extending

dNf (X) = df (NX) = N∗ df (X),(4.10)

dNθ(X, Y ) = X(θ(NY )) − Y (θ(NX)) − θ([X,Y ]N),where f is a function, θ is a 1-form, X, Y are vector fields on M, and

[X,Y ]N = [NX,Y ] + [X,NY ] − N[X,Y ].This differential is a cohomology operator (dN

2 = 0) and anticommutes withthe usual exterior derivative d. One notices that the invariance condition (3.2) canequally be written, in matrix notation, as

dNH = F dH. (4.11)

Imposing the condition dN2 = 0 on this equation, taking into account the anticom-

mutativity of d and dN , and translating back dNf = N∗ df if f is a function on M,one gets

(N∗ dF − F dF) ∧ dH = 0. (4.12)

So we see that the Stäckel separability conditions (4.2) imply the Equation (4.12)imposed on the control matrix F by the cohomological condition dN

2 = 0.


5. Special DN Coordinates

In this section we will discuss the problem of explicitly finding sets of special DNcoordinates on an ωN manifold M. We assume that M be regular and complex,so that the eigenvalues (λ1, . . . , λn) of N can be used as (half of the) coordinateson M. We know that in a neighborhood of a point where the λi are distinct thereexist functions (µ1, . . . , µn) forming with the eigenvalues a system of DN coordi-nates, and that the µi can be computed by quadratures. However, they can often befound in an algebraic way, as we will see below. We divide our argument in threemain points.

We start remarking that there are simple conditions to be checked the µi inorder to ensure that they form with the λi a set of DN coordinates. To this aim, weobserve that the µi must fulfill two kinds of requirements:

(1) They have to be Stäckel functions, that is, they must satisfy N∗ dµi = λi dµi ;(2) They have to fulfill the canonical commutation relations with respect to the

first Poisson bracket: {λi, µj } = δij , {µi, µj } = 0.

In principle, these conditions require the computation of the λi . We will showthat this can be avoided, and that a smaller number of equations must be checked.The first step is to notice that, once conditions 1 are satisfied, conditions 2 can bereplaced with the n equations

{λ1 + · · · + λn, µi} = 1, (5.1)

which do not require the computation of the λi , but only of their sum, that is,c1 := 1

2 trN and, consequently, of the Hamiltonian vector field

Y := −P dc1 =n∑

i=1

∂

∂µi

. (5.2)

Indeed, suppose that µj be a Stäckel function, and observe that

λi{λi, µj } = λi〈dλi, P dµj 〉 = 〈λi dλi, P dµj 〉 = 〈N∗ dλi, P dµj 〉= 〈 dλi,NP dµj 〉 = 〈 dλi, PN

∗ dµj 〉 = 〈 dλi, λjP dµj 〉= λj {λi, µj }, (5.3)

so that {λi, µj } = 0 if i �= j . Then Equation (5.1) becomes {λi, µi} = 1. In thesame way one shows that {µi, µj } = 0. Hence, in order to find the µi coordinatewe have to look for a Stäckel function (relative to λi) such that (5.1) holds.

The second point starts from the following idea, which will be extensively usedin the part of the paper dealing with Gel’fand–Zakharevich systems. Let us considerthe minimal polynomial

/(λ) = λn − (c1λn−1 + c2λ

n−2 + · · · + cn) (5.4)


of N . Using the Newton formulas relating the traces of the powers of N and thecoefficients ci of /(λ), one easily verifies that the latter satisfy

N∗ dci = dci+1 + ci dc1, i = 1, . . . , n − 1,

N∗ dcn = cn dc1.(5.5)

These relations are equivalent to the following equation for the polynomial /(λ),

N∗ d/(λ) = λ d/(λ) + /(λ) dc1. (5.6)

Relations of this kind are very interesting for our purposes. For instance, it holds:

PROPOSITION 5.1. Let f (x;λ) be a function defined on M, depending on anadditional parameter λ. Suppose that there exists a 1-form αf such that

N∗ d(f (x;λ)) = λ d(f (x;λ)) + /(λ)αf . (5.7)

Then, the function fi defined by fi(x) := f (x;λi(x)), i.e., the evaluation of f (x;λ)on λ = λi , is a Stäckel function relative to λi .

Proof. The differential of fi equals

dfi(x) = df (x;λ)∣∣λ=λi

+ ∂f (x;λ)∂λ

∣∣λ=λi

dλi, (5.8)

where, in the term d(f (x;λ))|λ=λj , one treats λ as a parameter. Applying theadjoint of the recursion operator we get

N∗ d(f (x;λi)) = N∗ d(f (x;λ))∣∣λ=λi

+ λi∂f (x;λ)

∂λ

∣∣λ=λi

dλi, (5.9)

whence the assertion, since /(λi) = 0. ✷DEFINITION 5.2. We will call a function on M depending on the additionalparameter λ a Stäckel function generator if it satisfies (5.7) with a suitable 1-form αf .

LEMMA 5.3. The space of Stäckel function generators is closed under sum andmultiplication, and is invariant with respect to the action of the vector field Y

defined by (5.2). If f is a Stäckel function generator and g is a function of onevariable, then g ◦ f is a Stäckel function generator.

Proof. The only assertion whose proof is not straightforward is the invariancewith respect to Y . This follows from the fact, already noticed in Example 3.5,that Y is a bi-Hamiltonian vector field; hence LY(N

∗) = LY (P−1P ′) = 0 and,

consequently, LY (λi) = 0. ✷It is clear that if fi is a Stäckel function relative to λi for i = 1, . . . , n, then

there exists a Stäckel function generator f (x;λ) such that fi = f (x;λi), e.g., the


interpolating polynomial. In terms of the generator, condition (5.1) can be writtenas

Y (f (x;λ)) = 1 for λ = λi , i = 1, . . . , n. (5.10)

For further use, we state and prove the following:

PROPOSITION 5.4. Let f be a Stäckel function generator, and suppose that forn � 1 the action of Y on f closes, that is, the relation

Y n(f ) =n−1∑j=0

ajYj (f ), (5.11)

with Y (aj ) = 0, holds. Then Equation (5.10) can be algebraically solved.Proof. There are two cases: in a first instance, suppose that, actually, Y is nilpo-

tent, that is, Y n(f ) = 0 is satisfied for some n � 1 (whilst Y n−1(f ) �= 0). Then it iseasily seen that Y n−2(f )/Y n−1(f ) is a Stäckel function generator fulfilling (5.10).

On the contrary, if (a0, . . . , an−1) �= (0, . . . , 0), then the matrix A represent-ing the action of Y on 2 := (f, Y (f ), . . . , Y n−1(f ))T has at least one nonzeroeigenvalue ν, which is a solution of νn = ∑n−1

i=0 ajνj . Let w = (w0, . . . , wn−1)

be a (left) eigenvector of A relative to ν, e.g., the one given by wn−1 = 1 andwk = νn−k−1 −∑n−k−2

j=0 ak+j+1νj for k = 0, . . . , n − 2. Then

Y

(1

νlog

n−1∑j=0

wjYj (f )

)= 1. (5.12)

Indeed,∑n−1

j=0 wjYj (f ) = w2 and

Y (w2) = wA2 = νw2,

implying (5.12). ✷These arguments reveal a further important aspect of Stäckel separability within

our approach to separation of variables. Indeed, the condition of Stäckel sepa-rability, whose intrinsic form is given by Equation (4.2), entails that the matrixof the (suitable normalized) eigenvectors of the control matrix F is a Stäckelmatrix, that is, its columns are Stäckel functions of N∗. Since we have shownthat a way to algebraically find the µi coordinates is to find Stäckel functions(or Stäckel function generators) and to combine them in order to fulfill Equa-tion (5.10), we see that, in the Stäckel case, the Hamiltonians themselves mayalgebraically provide the coordinates in which the corresponding flows can beseparated.


6. Separability on Odd-Dimensional Bi-Hamiltonian Manifolds

This section starts the second (and more applicative) part of the paper, in which wewill use the results of Sections 3 and 4 to discuss the separability of a specific fam-ily of integrable systems. They are defined on a class of bi-Hamiltonian manifolds,known in the literature as complete torsionless bi-Hamiltonian manifolds of pureKronecker type (see [19, 34] and the references quoted therein).

In this section we will consider the simplest case, corresponding to generic odd-dimensional bi-Hamiltonian manifolds (while in Section 3 we studied the case ofregular ωN manifolds, which are generic even-dimensional bi-Hamiltonian man-ifolds). Their Poisson tensors have maximal rank. The more general case will betreated (with detailed proofs) in the next section.

Let (M,P, P ′) be a (2n+ 1)-dimensional bi-Hamiltonian manifold, and let therank of P be equal to 2n. Suppose that the Poisson pencil Pλ := P ′ − λP has apolynomial Casimir function

H(λ) =n∑

i=0

Hiλn−i .

This amounts to saying that the functions (H0, . . . , Hn), which we assume to befunctionally independent, form a bi-Hamiltonian hierarchy, starting from a CasimirH0 of P and terminating with a Casimir of P ′,

P dH0 = 0, P dHi+1 = P ′ dHi, P ′ dHn = 0. (6.1)

In particular, they are in involution with respect to {·, ·} and {·, ·}′. If dH0 �= 0 atevery point of M, then the symplectic foliation of P is simply given by the levelsurfaces ofH0. The restrictions of (H1, . . . , Hn) to a symplectic leaf S of P form anintegrable system (in the Arnold–Liouville sense). The corresponding Hamiltonianvector fields are the restrictions to S of Xi := P dHi , where i = 1, . . . , n.

At this point it is natural to wonder whether the bi-Hamiltonian structure of Mcan give information on the separability of the (restrictions of the) Hamiltonians(H1, . . . , Hn). More concretely, one can try to induce an ωN structure on S inorder to apply the separability theorems of Sections 3 and 4. As anticipated inSection 2, this can be done if there exists a vector field Z which is transversal tothe symplectic foliation of P and fulfills the following condition:

(C) if F , G are functions on M which are invariant for Z, that is, Z(F) =Z(G) = 0, then {F,G} and {F,G}′ are also invariant.

In this case, any symplectic leaf of P inherits a bi-Hamiltonian structurefrom M. Clearly, the first reduced bracket is the one associated with the symplecticform of S, so that S is an ωN manifold.

In the following section we will prove that, if Z is normalized in such a waythat Z(H0) = 1, condition (C) takes the infinitesimal form

LZP = 0, LZP′ = Y ∧ Z, (6.2)


for a suitable vector field Y . In this case there is a useful form of the reducedPoisson brackets {·, ·}S and {·, ·}′

S on the symplectic leaf S. If f and g are func-tions on S, by definition {f, g}S and {f, g}′

S should be computed by taking localextensions of f and g which are invariant along Z. But one can avoid the use ofinvariant functions and consider arbitrary extensions F , G. Then

{f, g}S = {F,G},{f, g}′

S = {F,G}′ + X′(F )Z(G) − X′(G)Z(F),

where X′ := P ′ dH0 = P dH1 and the right-hand sides of the previous equa-tions are implicitly understood to be restricted to S. These equations show thatthe restrictions of (H1, . . . , Hn) to S are in bi-involution, and then separable inDN coordinates because of Theorem 3.1. We are going to show that they are evenStäckel separable, by computing their control matrix F and checking that it satisfiesthe condition N∗ dF = F dF .

To this purpose, we notice that the Lenard relations (6.1) on M give rise to theequations

N∗ dHi = dHi+1 − Z(Hi) dH1, i = 1, . . . , n − 1, (6.3)

N∗ dHn = −Z(Hn) dH1, (6.4)

where N is the recursion operator of the ωN manifold S and ˆ denotes the re-striction to S. Therefore, the control matrix of (H1, . . . , Hn) is given by a singleFrobenius block:

F =

−Z(H1) 1 0 · · · 0−Z(H2) 0 1

.... . .

... 1−Z(Hn) 0

. (6.5)

So we see that the (restriction to the symplectic leaf S of the) functions ci =−Z(Hi) are the coefficients of the characteristic polynomial of the matrix F , thatis, the coefficients of the minimal polynomial of the recursion operator N , /(λ) =λn − (c1λ

n−1 +· · ·+ cn). Recalling that the coefficients of the minimal polynomialof N satisfy

N∗ dci = dci+1 + ci dc1, N∗ dcn = cn dc1, (6.6)

we see that the condition N∗ dF = F dF for the Stäckel separability of the Hamil-tonians is automatically verified. Hence we have proven

THEOREM 6.1. The Hamiltonians of a corank-1 torsionless GZ system are Stäc-kel separable in DN coordinates.


It is worthwhile to notice that the examples previously considered in the lit-erature within the theory of quasi-bi-Hamiltonian systems ([5, 33, 46]) fall intothis class. The link with the classical Stäckel–Eisenhart theory of separation ofvariables is discussed in [24].

We remark that the vector field Y appearing in (6.2) can be chosen to be tangentto S. In this case, Y = P d(Z(H1)) = −P dc1, so that its restriction to S is thevector field we used in the previous section to determine the µi coordinates. (Thisexplains why we made use of the same notation.)

Now we will write the separation equations for the GZ Hamiltonians. TheStäckel matrix S, being the (normalized) matrix of the left eigenvectors of F , iseasily seen to be the Vandermonde-like matrix

S = λn−1

1 · · · λ1 1... · · · ...

...

λn−1n · · · λn 1

,

where the λi are the eigenvalues of N , i.e., the roots of /(λ). Therefore, theseparation relations take the form

H1λn−1i + H2λ

n−2i · · · + Hn = Ui(λi, µi), (6.7)

where (λ1, . . . , λn, µ1, . . . , µn) are special DN coordinates on S and the Ui are theentries of the Stäckel vector. Such entries can be explicitly computed once we havethe map sending the DN coordinates to the corresponding point of S, as we willcheck in the example of the 3-particle nonperiodic Toda lattice.

Another way to arrive at the separation equations is to multiply (6.3) by λn−i

and then to add to (6.4). The result is

N∗ dH (λ) = λ dH (λ) − /(λ) dH1,

meaning that H (λ) := ∑ni=1 Hiλ

n−i is a Stäckel function generator according toProposition 5.1. Thus, in DN coordinates, H (λi) = Ui(λi, µi), which coincideswith (6.7). We stress that H (λ), being a Stäckel function generator, can be in somecases used to determine the µi coordinates. Instances of this situation are providedby the Toda lattice, as discussed in [12], and by the stationary reductions of theKdV hierarchy [11]. Here we will present the example of the 3-particle nonperiodicToda lattice.

EXAMPLE 6.2. The Hamiltonian of the system is

HToda = 1

2

3∑i=1

pi2 +

2∑i=1

exp(qi − qi+1). (6.8)

As usual (see, e.g., [15], and [16] for the separability), one introduces the‘Flaschka–Manakov coordinates’ (a1, a2, b1, b2, b3), where

bi = pi, ai = − exp(qi − qi+1),


and consider the manifold M = (C∗)2 ×C3, or M = R>02 ×R3. We endow it with

the Poisson pencil Pλ = P ′ − λP given by ([32] and references cited therein)

Pλ =

0 −a1a2 (b1 − λ)a1 (λ − b2)a1 0

a1a2 0 0 (b2 − λ)a2 (λ − b3)a2

(λ− b1)a1 0 0 a1 0

(b2 − λ)a1 (λ− b2)a2 −a1 0 a2

0 (b3 − λ)a2 0 −a2 0

. (6.9)

It has a polynomial Casimir H(λ) = H0λ2 + H1λ+ H2, where

H0 = b1 + b2 + b3,

H1 = −(b1b2 + b2b3 + b3b1 + a1 + a2),

H2 = b1b2b3 + a1b3 + a2b1.

The Hamiltonian (6.8) is related to the coefficients of H(λ) by HToda = H1 + 12H0.

There are two nontrivial flows, given by:

X1 = P0 dH1 = a1(b1 − b2)∂

∂a1+ a2(b2 − b3)

∂

∂a2+ a1

∂

∂b1+

+ (a2 − a1)∂

∂b2− a2

∂

∂b3,

X2 = P0 dH2 = a1[a2 + b3(b2 − b1)] ∂

∂a1+ a2[a1 + b1(b3 − b2)] ∂

∂a2−

− a1b3∂

∂b1+ (a1b3 − a2b1)

∂

∂b2+ a2b1

∂

∂b3.

The symplectic leaves of P are the level surfaces of H0, so that they can be parame-trized by (a1, a2, b1, b2). A possible choice for the normalized transversal vectorfield is Z = ∂/∂b3, because Z(H0) = 1 and

LZP = 0, LZP′ = Y ∧ Z,

with Y = a2∂/∂a2. Since Y (H0) = 0, we know that Y = P d(Z(H1)) = −P dc1.If S is a symplectic leaf of P , the reduced bi-Hamiltonian structure on S is simplyobtained by removing the last row and the last column of Pλ:

PS =

0 0 a1 −a1

0 0 0 a2

−a1 0 0 0a1 −a2 0 0

,

P ′S =

0 −a1a2 a1b1 −a1b2

a1a2 0 0 a2b2

−a1b1 0 0 a1

a1b2 −a2b2 −a1 0

.


For completeness, we display recursion operator

N = P ′SP

−1S =

b1 a1(b1 − b2)/a2 a1 a1

0 b2 −a2 00 a1/a2 b1 0

−1 −a1/a2 0 b2

,

whose minimal polynomial is

/(λ) = λ2 + Z(H1)λ + Z(H2) = λ2 − (b1 + b2)λ + a1 + b1b2.

The coordinates λ1, λ2 are its roots.The restrictions of H1 and H2 to the symplectic leaf H0 = c are

H1 = −c(b1 + b2) + b12 + b2

2 + b1b2 − a1 − a2,

H2 = c(a1 + b1b2) − (a1 + b1b2)(b1 + b2) + a2b1.

We know that H (λ) := H1λ + H2 is a Stäckel function generator, and that theseparation equations are H (λi) = U(λi, µi), for i = 1, 2. To write them explicitly,we need the form of the µi . They can be found using Proposition 5.4 and thefact that Y 2(H (λ)) = Y (H(λ)). This entails that f (λ) := log Y (H (λ)) satisfiesY (f (λ)) = 1, so that, according to the results of Section 5,

µi = log Y (H(λi)) = log(a2b1 − a2λi), i = 1, 2,

form with the eigenvalues λ1 and λ2 of N a set of (special) DN coordinates.Finally, using the expression of (a1, a2, b1, b2) in terms of the DN coordinates

one can easily find the separation relations

H (λi) = λi3 + expµi − cλi

2, i = 1, 2,

leading to the solution by quadratures of the Hamilton–Jacobi equations for H1

and H2.We notice that the ‘change of variables’ (ai, bi) �→ (λi, µi) is not the lift of a

point trasformation on the configuration space; thus, there is no contradiction withthe results of [6], stating that it is impossible to separate the 3-particle Toda latticewith point tranformations.

7. Separability of Gel’fand–Zakharevich Systems

In this section we will generalize (and give proofs of) the results of the previoussection to the case of corank k. As we will see, the picture outlined in the previoussection still holds good. The only relevant difference concerns the Stäckel separa-bility, which is no longer valid in general, but requires an additional assumption onthe Hamiltonians.


We consider a bi-Hamiltonian manifold (M,P, P ′) admitting k polynomialCasimir functions of the Poisson pencil Pλ = P ′ − λP ,

H(a)(λ) =na∑i=0

H(a)i λna−i , a = 1, . . . , k, (7.1)

such that n1 + n2 + · · · + nk = n, with dimM = 2n + k, and such that thedifferentials of the coefficients H(a)

i are linearly independent on M. The H(a)i , for

a fixed a, form a bi-Hamiltonian hierarchy and, in particular, H(a)0 (resp. H(a)

na) is

a Casimir of P (resp. P ′). We assume that the corank of P is exactly k, so thatthe H

(a)0 , for a = 1, . . . , k, are a maximal set of independent Casimirs of P . The

collection of the n bi-Hamiltonian vector fields

X(a)i = P dH(a)

i = P ′ dH(a)i−1, i = 1, . . . , na, k = 1, . . . , a, (7.2)

associated with the Lenard sequences defined by the Casimirs H(a) is called theGel’fand–Zakharevich (GZ) system, or the axis, of the bi-Hamiltonian manifold M.Since standard arguments from the theory of Lenard–Magri chains show that all thecoefficients H(a)

i pairwise commute with respect to both {·, ·} and {·, ·}′, we have

PROPOSITION 7.1. Let S be a symplectic leaf of P , that is, a 2n-dimensionalsubmanifold defined by H

(1)0 = c1, . . . , H

(k)0 = ck . Then the vector fields X(a)

i ofthe Lenard sequences associated with the polynomial Casimirs (7.1) of {·, ·}λ onM define a completely integrable Hamiltonian system on S.

We call the family {H (a)i | i = 1, . . . , na, k = 1, . . . , a} of the restrictions to S

of the coefficients of the H(a) the GZ basis of the symplectic leaf S. The Lagrangianfoliation defined by the GZ basis will be referred to as the GZ foliation of S.

In the following subsection we will give sufficient conditions so that a sym-plectic leaf S of P inherits an ωN structure from the bi-Hamiltonian structureof M. Then we will come back to the integrable system described in the previousproposition and we will discuss its separability in DN coordinates.

7.1. THE INDUCED ωN STRUCTURE

Our strategy to induce on a symplectic leaf S of P a second Poisson bracket whichis compatible with the ‘canonical’ one is based on the geometrical considera-tions already mentioned at the end of Section 2. We suppose that there exists ak-dimensional foliation Z of M such that

(C1) the foliation Z is transversal to the symplectic foliation of P ;

(C2) the functions that are constant on Z form a Poisson subalgebra with respectto both {·, ·} and {·, ·}′.


Thus S has a (projected) bi-Hamiltonian structure. The projection of {·, ·} coincideswith the symplectic structure {·, ·}S of S, while the projection of {·, ·}′ defines asecond Poisson bracket {·, ·}′

S on S. Since the compatibility between {·, ·}S and{·, ·}′

S is guaranteed by the fact that the whole pencil {·, ·}λ is projectable on S,we have endowed S with an ωN structure. We will suppose it to be a regular ωNmanifold, in order to apply (in the open dense set where the eigenvalues of N aredistinct) the results of Section 3 and 4, leaving the discussion of the problem offinding the conditions on (M,P, P ′) and Z ensuring the regularity of S for futurework.

Let (Z1, . . . , Zk) be local vector fields spanning the distribution tangent to Z.Because of the transversality condition, we can always normalize these vectorfields with respect to the Casimirs H(a)

0 of P :

Zb(H(a)

0 ) = δab . (7.3)

In terms of these generators, the projectability condition takes a very concise form,as shown in

PROPOSITION 7.2. (1) The normalized vector fields Za locally generating Z aresymmetries of P ,

LZa(P ) = 0, (7.4)

and satisfy

LZaP ′ =

∑b

Y ba ∧ Zb, (7.5)

where Y ba = P d(Za(H

(b)1 )) = [Za, P

′ dH(b)0 ] = [Za,X

(b)1 ].

(2) Vice versa, suppose that there exists a k-dimensional integrable distribu-tion on M which is transversal to the symplectic leaves of P and such that (7.4)and (7.5) hold for a suitable local basis (Z1, . . . , Zk) of the distribution (and forsuitable vector fields Y b

a ). Then the integral foliation of the distribution satisfiesthe projectability requirements (C1) and (C2), so that every symplectic leaf of Pbecomes an ωN manifold. Moreover, if the Za are normalized, then they commute.

Proof. First of all, we recall ([42], p. 54) that the condition that the functionsconstant along Z form a Poisson subalgebra with respect to {·, ·} is equivalent tothe assertion that the following equations hold,

LZaP =

k∑b=1

Wba ∧ Zb, (7.6)

for some vector fields Wba . This entails the validity of assertion (2), except the com-

mutativity of the vector fields Za , normalized according to (7.3), that can be proved


as follows. The integrability of the distribution implies that there are functions φcab

such that

[Za,Zb] =k∑

c=1

φcabZc,

and evaluating this relation on the Casimirs H(d)

0 of P we easily see that φcab = 0.

In order to prove assertion (1), we notice that the vector fields Wba are not

uniquely defined, and can be taken to be tangent to the symplectic leaves of P .This is accomplished by changing

Wba �→ Wb

a −k∑

c=1

Wba (H

(c)0 )Zc.

Indeed,

k∑b=1

(Wb

a −k∑

c=1

Wba (H

(c)0 )Zc

)∧ Zb

=k∑

b=1

Wba ∧ Zb −

k∑b,c=1

Wba (H

(c)0 )Zc ∧ Zb

=k∑

b=1

Wba ∧ Zb

since equations LZa〈dH(c)

0 , P dH(d)0 〉 = 0 and (7.6) imply that

Wdb (H

(c)

0 ) = Wcb (H

(d)

0 ).

Thus the vector fields Wba in (7.6) can be chosen in such a way that Wb

a (H(c)

0 ) = 0.Now, deriving the relation P dH(c)

0 = 0 along Za one obtains that the normalizedvector fields Wb

a vanish, so that, indeed, the vector fields Za are symmetries of P .As far as the second Poisson tensor P ′ is concerned, in the same way we can

show that there exist vector fields Y ba tangent to the symplectic leaves of P such

that

LZaP ′ =

k∑b=1

Y ba ∧ Zb. (7.7)

By deriving the relation P ′ dH(c)0 = X

(c)1 with respect to Za, one has that

Y ca = [Za,X

(c)

1 ] = LZa(P dH(c)

1 ) = P d(Za(H(c)

1 )).

This completes the proof. ✷


In the sequel we will always suppose that the normalization conditions (7.3) onthe transversal vector fields Za and the tangency conditions on the Y b

a are satisfied.For the sake of simplicity, we will also assume that the Za are defined on the wholemanifold M, or at least in a tubular neighborhood of S. Next we give a usefulformula for the (second) reduced Poisson bracket on S.

PROPOSITION 7.3. Let f , g be functions on a symplectic leaf S of P , and F , Garbitrary extensions of f , g to M. Then

{f, g}S = {F,G}, (7.8)

{f, g}′S = {F,G}′ +

k∑a=1

(X(a)1 (F )Za(G) − X

(a)1 (G)Za(F )), (7.9)

where X(a)1 = P ′ dH(a)

0 = {H(a)0 , ·}′.

Proof. The symplectic leaf S is given by the equations H(a)0 = ca , for a =

1, . . . , k, where the ca are suitable constants. The first formula simply says that{·, ·}S corresponds to the symplectic structure of S. The second formula followsfrom the remark that F := F − ∑k

a=1 Za(F )(H(a)

0 − ca) coincides with F andfulfills Zb(F ) = 0 on S. Hence it can be used to compute {f, g}′

S , giving (7.9). ✷Remark 7.4. The projectability conditions we have imposed in order to endow

a fixed symplectic leaf S with an ωN structure can be weakened in the followingway. We can consider a distribution transversal to T S and defined only at thepoints of S, generated by a family of vector fields (Z1, . . . , Zk), normalized asZa(H

(b)

0 ) = 〈dH(b)

0 , Za〉 = δba . Then we introduce, according to (7.9), a composi-tion law {·, ·}′

S on C∞(S) and we look for conditions ensuring that it is a Poissonbracket, compatible with {·, ·}S . One can show [14] that {·, ·}′

S is a Poisson bracketif and only if

k∑a=1

X(a)1 ∧

(LZa

(P ′) +k∑

b=1

[Za,X′b] ∧ Zb

)+

+ 1

2

k∑a,b=1

X(a)

1 ∧X(b)

1 ∧ [Za,Zb] = 0 (7.10)

at the points of S. In this case, the two Poisson brackets are compatible if and onlyif

k∑a=1

X(a)1 ∧ LZa

(P ) = 0 (7.11)

at the points of S. Hence, the requirements (7.4) and (7.5), on the whole mani-fold M, are very ‘strong’ solutions for (7.10) and (7.11). Finally, we mention thatthe reduction process presented in this remark does not fit into the Marsden–Ratiuscheme [31], whereas the one based on (C1) and (C2) clearly does.


7.2. SEPARABILITY AND THE CONTROL MATRIX

After endowing any symplectic leaf S of P with anωN structure, we can reconsiderthe GZ foliation of S and prove its separability in DN coordinates. Notice that (seealso below) the restrictions to S of the bi-Hamiltonian vector fields X(a)

i are not bi-Hamiltonian with respect to the ωN structure of S. This is due to the fact that thisstructure is obtained by means of a projection, while the Hamiltonian are restrictedto S.

We suppose that (Z1, . . . , Zk) are vector fields on M, fulfilling the hypothesesof part 2 of Proposition 7.2 and normalized, i.e., Za(H

(b)

0 ) = δba . Then the expres-sions (7.8) and (7.9) of the reduced Poisson brackets immediately show that therestrictions of H(a)

i to S are in bi-involution. Therefore, they are separable in DNcoordinates.

THEOREM 7.5. The GZ foliation of S is separable in DN coordinates.

Using once more Theorem 3.1, we can conclude that the distribution tangentto the GZ foliation is invariant with respect to the recursion operator N . We aregoing to describe the form of the associated control matrix, which will be neededto discuss the Stäckel separability of the GZ basis.

Let g be any function on S and let G be an extension of g to M. Using (7.9) andthe Lenard relations on the H(a)

i , we have

{H (a)i , g}′

S = {H(a)i ,G}′ +

k∑b=1

(X(b)

1 (H(a)i )Zb(G) − X

(b)

1 (G)Zb(H(a)i ))

= {H(a)i+1,G} −

k∑b=1

Zb(H(a)i ){H(b)

1 ,G}, (7.12)

where we have put H(a)

na+1 := 0. Therefore, for all g ∈ C∞(S),

{H (a)i , g}′

S = {H (a)i+1, g}S −

k∑b=1

Zb(H

(a)i ){H (b)

1 , g}S, (7.13)

or, in terms of the (reduced) Poisson tensors PS and P ′S ,

P ′S dH (a)

i = PS dH (a)

i+1 −k∑

b=1

Zb(H

(a)i )PS dH (b)

1 . (7.14)

Hence, we can conclude that

N∗ dH (a)i = dH (a)

i+1 −k∑

b=1

Zb(H

(a)i ) dH (b)

1 (7.15)


and read the form of the control matrix F associated with the GZ basis. Indeed, ifwe order the n functions of the GZ basis as

H(1)1 , H

(1)2 , . . . , H (1)

n1, H

(2)1 , . . . , H (k)

nk, (7.16)

then we realize that F has a k × k block form,

F =

F1 C1,2 · · · C1,k

C2,1 F2 · · · C2,k...

...

Ck,1 Fk

, (7.17)

with Fa an na × na square matrix of Frobenius type of the form

Fa =

− Za(H

(a)

1 ) 1 0 · · · 0

− Za(H

(a)

2 ) 0 1...

. . .... 1

− Za(H

(a)na ) 0

(7.18)

and Ca,b a rectangular matrix with na rows and nb columns where only the firstcolumn is nonzero:

Ca,b =

− Zb(H

(a)

1 ) 0 · · · 0

− Zb(H

(a)

2 ) 0 · · · 0...

......

− Zb(H

(a)na ) 0 · · · 0

. (7.19)

Remark 7.6. The vector field Y , defined in Section 5 as the Hamiltonian vectorfield associated with − 1

2 trN by the first Poisson structure, can be obtained in the

present setting by restricting to S the vector field∑k

a=1 Yaa . Indeed,

k∑a=1

Y aa = P d

(k∑

a=1

Za(H(a)

1 )

),

and using (7.17) we have∑k

a=1 Za(H(a)1 ) = − trF = − 1

2 trN .

Thus, we have seen that GZ systems on bi-Hamiltonian manifolds admitting asuitable transversal foliation provide examples of non trivial (but still somewhatspecial) Hamiltonian systems for which the separability condition in DN coordi-nates holds, that is, they provide interesting examples of control matrices, discussed


in Section 3. Such matrices were introduced (in the specific example of a stationaryreduction of the Boussinesq equation) in [13].

7.3. STÄCKEL SEPARABILITY OF GZ SYSTEMS

Let us now consider the Stäckel (i.e., linear) separability of GZ systems. We haveseen that the invariance with respect to N of the Lagrangian distribution defined bythe restricted Hamiltonians H (a)

i is a consequence of the Lenard recursion relationson M, and that the nontrivial coefficients in F are given by the deformations ofthe polynomial Casimirs along the normalized generators Za of the foliation Z. Onthe other hand, in Section 6 we have proved that, in the corank 1 case, the controlmatrix F automatically satisfies the condition for Stäckel separability, N∗ dF =F dF . The next proposition shows that, in order to ensure this condition in thegeneral case, one has to require that the Hamiltonians H(a)

i be affine with respectto the vector fields Za .

PROPOSITION 7.7. The GZ basis, formed by the H (a)i , is a Stäckel basis (i.e., it

is Stäckel separable in DN coordinates) if and only if Zb(Zc(H(d)j )) = 0 on S, for

all b, c, d = 1, . . . , k and for all j = 1, . . . , nd .Proof. Stäckel separability is equivalent to N∗ dF = F dF , where F is the

control matrix (7.17). Since dF has nonvanishing entries only in the columns1, n1 + 1, n2 + 1, . . . , nk−1 + 1, this condition takes the form

N∗ d( Za(H

(b)i )) = d(

Za(H(b)

i+1)) −k∑

c=1

Zc(H

(b)i ) d(

Za(H(c)

1 )), (7.20)

where, as usual, we have put H(b)nb+1 := 0. In order to compute the left-hand side

of (7.20), we observe that (7.9) implies

P ′S df =

(P ′ +

k∑c=1

Zc ∧ X(c)1

)dF,

where f ∈ C∞(S) and F is any extension of f . Moreover, we have that

LZa

(P ′ +

k∑c=1

Zc ∧ X(c)

1

)=

k∑c=1

[Za,Zc] ∧ X(c)

1 = 0, (7.21)

since the Zb commute. Hence,

P ′S d(

Za(H(b)i )) =

(P ′ +

k∑c=1

Zc ∧ X(c)

1

)d(Za(H

(b)i ))

= LZa

[(P ′ +

k∑c=1

Zc ∧X(c)

1

)dH(b)

i

]


= LZa

(P dH(b)

i+1 −k∑

c=1

Zc(H(b)i )P dH(c)

1

)

= P d(Za(H(b)

i+1)) −k∑

c=1

Zc(H(b)i )P d(Za(H

(c)

1 )) −

−k∑

c=1

Za(Zc(H(b)i ))P dH(c)

1 , (7.22)

so that

N∗ d( Za(H

(b)i )) = d(

Za(H(b)

i+1)) −k∑

c=1

Zc(H

(b)i ) d(

Za(H(c)

1 )) −

−k∑

c=1

Za(Zc(H

(b)i ))dH

(c)

1 . (7.23)

A comparison with (7.20) completes the proof. ✷Thus, the GZ basis is Stäckel separable if (and only if) the second derivatives

of the Hamiltonians along the transversal vector fields vanish. This condition isautomatically verified in the case of corank k = 1. This ‘discrepancy’ between thegeneric and the rank 1 case can be understood as follows. Since, by assumption,the transversal distribution Z is integrable, the tubular neighborhood in which it isdefined is equipped with a fibered structure, in which the fibers are the symplecticleaves of P . The conditions

LZa(P ) = 0; LZa

(P ′ +

k∑c=1

Zc ∧ X(c)1

)= 0

of Equations (7.4) and (7.21) imply that the recursion operator (to be seen, in thispicture, as an endomorphism of the vertical tangent bundle to the local fibration) isinvariant along all the Za. So its eigenvalues and hence its minimal polynomial areinvariant with respect to the Za . In the case k = 1, as we have seen in Section 6, thecoefficients of the minimal polinomial are the derivatives of the Casimir with re-spect to the (single) transversal vector field Z, but this is not necessarily true in thehigher corank case. Notice that, whenever the second derivatives of the Casimirsvanish, our separated variables are ‘invariant’ with respect to the Casimirs, as theones considered in [40].

Still under the assumptions of the above proposition, the results of Section 4 tellus how to construct the Stäckel matrix and, in principle, the separation relations.We also know that the entries of the Stäckel matrix and of the Stäckel vector canbe used (under additional hypotheses) to explicitly find the separation coordinates,i.e., the DN coordinates. In the next section we will exploit the special properties


of the GZ foliation in order to determine the separation relations and, eventually,the DN coordinates without computing the Stäckel matrix.

8. Separation Relations for GZ Systems

Let us consider the GZ foliation (on the symplectic leaf S) studied in Subsec-tion 7.2. The aim of this section is to write, in the Stäckel separable case, theseparation relations for the Hamiltonians of the GZ basis. To simplify the notations,we will not use anymore the symbol ˆ to denote the restriction to S.

First of all, we notice that the relevant information contained in the n×n controlmatrix F is actually encoded in the k × k polynomial matrix F(λ), which is the Ja-cobian matrix of the Casimirs H(a)(λ) with respect to the transversal (normalized)vector fields Zb, that is, the matrix

F(λ) = Z1(H

(1)(λ)) · · · Zk(H(1)(λ))

......

Z1(H(k)(λ)) · · · Zk(H

(k)(λ))

. (8.1)

We can translate the results about separability and Stäckel separability of GZ sys-tems, based on the n × n matrix equations

N∗ dH = F dH, (8.2)

N∗ dF = F dF, (8.3)

into corresponding equations for the polynomial matrix F(λ). To this end we de-note by H(λ) = (H (1)(λ),H (2)(λ), . . . , H (k)(λ))T the k-component vector of thepolynomial Casimir functions, and by

H 1 = (H(1)1 ,H

(2)1 , . . . , H

(k)

1 )T and F1 = [Zb(H(a)

1 )]the analogs of the vector H(λ) and of the matrix F(λ), constructed by using thecoefficients H(a)

1 instead of the full Casimir functions H(a)(λ).

LEMMA 8.1. The polynomial control matrix F(λ) satisfies the equation

(N∗ − λ) dH(λ) = −F(λ) dH 1, (8.4)

which is the counterpart of the matrix equation (8.2).Proof. The λna−i-coefficient of the ath row of (8.4) is exactly (7.15). ✷In complete analogy, we obtain the ‘polynomial form’ of the Stäckel separabil-

ity condition (8.3).

LEMMA 8.2. The GZ basis is a Stäckel basis iff F(λ) satisfies the condition

(N∗ − λ) dF(λ) = −F(λ) dF1. (8.5)


Proof. The simplest way to prove this lemma is to expand both sides in powersof λ. We first write (8.5) in componentwise form as

N∗ dFba(λ) = λ dFb

a(λ)−k∑

c=1

Fbc(λ) d(F1)

ca,

and then expand in powers of λ, getting

N∗ d(Za(H(b)i )) = d(Za(H

(b)

i+1)) −k∑

c=1

Zc(H(b)i ) d(Za(H

(c)

1 )),

which are exactly the Stäckel conditions (7.20) for the GZ basis. ✷The following lemma shows that the eigenvalues of N can be easily obtained

from the matrix F(λ).

LEMMA 8.3. The determinant of F(λ) is the characteristic polynomial of F . Inparticular, it coincides with the minimal polynomial /(λ) of the recursion opera-tor N , that is,

det F(λ) = det(λI − F) = /(λ). (8.6)

Proof. Let λi be an eigenvalue of F . Then one can check that the relative (left)eigenvectors have the form

vi = (σ i1λ

n1−1i , σ i

1λn1−2i , . . . , σ i

1, σi2λ

n2−1i , . . . , σ i

kλnk−1i , . . . , σ i

k ),

where σi := (σ i1, . . . , σ

ik ) is a nonzero vector such that σiF(λi) = 0. This shows

that det F(λi) = 0. Since det F(λ) is a monic degree n polynomial and the λi aredistinct, we can conclude that (8.6) holds. ✷

The next step is to introduce the adjoint (or cofactor) matrix F∨(λ), satisfyingthe equation

F∨(λ)F(λ) = F(λ)F∨(λ) = det F(λ)I. (8.7)

We will show that the rows of F∨(λ), after a suitable normalization, provide Stäckelfunction generators and play the role of the Stäckel matrix. If σ (λ) := ek F∨(λ) isa row of the adjoint matrix, then, obviously,

σ (λ)F(λ) = /(λ)ek. (8.8)

Let σj (λ) be a nonvanishing entry of σ (λ) and let us consider the normalized row

ρ(λ) = 1

σj (λ)σ, (8.9)

which satisfies the equation

ρ(λ)F(λ) = /(λ)

σj(λ)ek. (8.10)


PROPOSITION 8.4. Suppose that the component ρa(λ) of ρ(λ) is defined for λ =λi , i = 1, . . . , n. Then it is a Stäckel function generator, that is, it verifies theequation

(N∗ − λ) dρa(λ) = 0, for λ = λi , i = 1, . . . , n. (8.11)

Proof. It is convenient to consider the full vector ρ(λ). From Equation (8.10),we have

(N∗ − λ) dρ(λ) · F(λ) + ρ(λ) · (N∗ − λ) dF(λ) = 0, for λ = λi. (8.12)

Using Lemma 8.2 we can write the second summand in this equation as

ρ(λ) · (N∗ − λ) dF(λ) = −ρ(λ)F(λ) dF1, for λ = λi, (8.13)

so that we finally obtain

(N∗ − λi) dρ(λi) · F(λi) = 0, for i = 1, . . . , n. (8.14)

But the kernel of F(λi) is one-dimensional, due to the fact that the λi are distinct.Indeed, from (8.7) we have that

det F∨(λ) = (det F(λ))k−1 =n∏

i=1

(λ − λi)k−1. (8.15)

If dim ker F(λi) � 2 for some i, then the rank of F(λi) would be less than k− 1, sothat F∨(λi) = 0 and, therefore, F∨(λ) = (λ− λi)F(λ) for some polynomial matrixF(λ). But then det F∨(λ) = (λ − λi)

k det F(λ), contradicting (8.15).Coming back to (8.14), we can assert that there exist 1-forms νi such that

(N∗ − λi) dρ(λi) = νiρ(λi), for i = 1, . . . , n. (8.16)

Since the j th component of ρ(λi) is 1, we have that the νi vanish, and this closesthe proof. ✷

Now we are ready to show how to compute the separation equations for GZsystems.

PROPOSITION 8.5. Let the ρa(λ) be as in the previous proposition and supposethat they are defined for λ = λi . Then

∑ka=1 σa(λ)H

(a)(λ) is a Stäckel functiongenerator.

Proof. Let us write compactly

k∑a=1

ρ(λ)H (a)(λ) = ρ(λ) · H(λ)


and compute

(N∗ − λ) d(ρ(λ) ·H(λ)) = (N∗ − λ) dρ(λ) ·H(λ) ++ ρ(λ)(N∗ − λ) dH(λ). (8.17)

For λ = λi the first summand vanishes thanks to Proposition 8.4, while the secondequals (according to Lemma 8.1)

−ρ(λi) F(λi) dH 1,

and so vanishes as well. ✷Therefore we have shown that the separation relations of the GZ basis (in the

Stäckel case) are given by

k∑a=1

ρa(λi)H(a)(λi) = 2i(λi, µi), i = 1, . . . , n, (8.18)

that in the corank 1 case boils down to Equation (6.7).We end this section with the following remark. Let us suppose that the mul-

tipliers ρa(λ) and the coordinates µ1, . . . , µn be related by a ‘simple’ algebraicexpression, e.g., that there exist integer numbers p1 = 0, p2, . . . , pk such that

µpai = ρa(λi), i = 1, . . . , n and a = 1, . . . , k.

This means, according to the results of Section 5, that the path root ρa(λ) is aStäckel function generator satisfying the Equation (5.10), i.e., Y (pa

√ρa(λ)) = 1,

for λ = λi . Then the separation relations (8.18) ‘degenerate’ to a single one, thatis, they can be read as the vanishing of the two-variable function

k∑a=1

µpaH (a)(λ)− 2(λ,µ) (8.19)

evaluated at the points (λi, µi), for i = 1, . . . , n. Hence, in such an instance, wecan associated with the GZ system a ‘spectral curve’ over which the separationcoordinates lie.

This is an indication, which is verified in several concrete examples, that thetheory herewith presented may provide an effective bridge between the classi-cal theory of the Hamilton–Jacobi equation and its modern outsprings, related toalgebraic integrability. In this respect, several questions naturally arise, namely,

(1) Can the degeneration property of the separation relations be characterized interms of the bi-Hamiltonian structure?

(2) In this case, what can one say about the algebraicity of the separation relation?


We will further address these problems in [10]. In this paper we limit ourselves togive an example related to loop algebras, where all these features are present.

9. An Example Related to sl(3)

Applications of the scheme we have presented in this paper have already appearedin the literature. Namely, in [13] a preliminary picture of these ideas has beenapplied to the t5-stationary reduction of the Boussinesq hierarchy. Subsequently,in [11] we have shown how to frame all stationary reductions of the KdV theoryinside this picture, and in [12] the classical An-Toda lattices have been considered(see also [36] for the Neumann system, and [8]). In this final section we will illus-trate how our theoretical scheme concretely works in an example, which is relatedto the t5-stationary Boussinesq system, in the sense that the latter can be obtainedvia a reduction from the one we will present. Even if, for the sake of brevity, wewill stick to such a particular example, we claim that the same arguments hold fora wide class of integrable systems on finite-dimensional orbits of loop algebras,that are separable in the so-called spectral Darboux coordinates [1, 2]. (Thesecoordinates have recently been shown [21] to be DN coordinates for a suitableωN structure.)

The system we are going to study is defined on the space sl(3) × sl(3) of pairs(X0, X1) of 3 × 3 traceless matrices. The cotangent (and the tangent) space at apoint is identified with the manifold itself via the pairing

〈(V0, V1), (W0,W1)〉 = tr(V0W0 + V1W1),

so that the differential of a scalar function F is represented by a pair of matrices,

dF =(∂F

∂X0,∂F

∂X1

).

We introduce [29, 37] the two compatible Poisson tensors defined, at the point(X0, X1), by

P :

[V0

V1

]�→[ [X1, V0] + [A,V1]

[A,V0]],

P ′:[V0

V1

]�→[ −[X0, V0]

[A,V1]],

where

A = 0 0 0

1 0 00 1 0

.

One can easily see that the functions

C1(X0, X1) = tr(AX1) = (X1)12 + (X1)23,

C2(X0, X1) = tr(A2X1) = (X1)13,


are common Casimirs of P and P ′. Thus the bi-Hamiltonian structure can betrivially restricted to

M = {(X0, X1) ∈ sl(3) × sl(3) | (X1)12 + (X1)23 = 0, (X1)13 = 1},which is the 14-dimensional manifold where our GZ system will be defined. In-deed, it can be directly shown (see also [37]) that, if

L(λ) = λ2A + λX1 + X0,

then

H(1) = 12 trL(λ)2 and H(2) = 1

3 trL(λ)3 (9.1)

are Casimir functions of the Poisson pencil Pλ = P ′ − λP . One finds that

H(1) = λ3 + H(1)0 λ2 + H

(1)1 λ + H

(1)2 ,

H (2) = λ5 + H(2)0 λ4 + H

(2)1 λ3 + H

(2)2 λ2 + H

(2)3 λ + H

(2)4 ,

(9.2)

where

H(1)0 = tr(AX0 + 1

2X12), H

(1)1 = tr(X0X1), H

(1)2 = 1

2 trX02,

H(2)0 = tr(A2X0 + AX1

2), H(2)1 = tr( 1

3X13 + AX0X1 + AX1X0),

H(2)2 = tr(X1

2X0 + AX02), H

(2)3 = tr(X1X0

2), H(2)4 = 1

3 trX03.

Obviously, H(1)0 and H

(2)0 are Casimirs of P , whereas H(1)

2 and H(2)4 are Casimirs

of P ′. Since the differentials of the functions H(a)i are linearly independent on a

dense open subset of M, and the corank of P and P ′ is 2, we can conclude thatthe hypotheses of Section 7 are verified, with k = 2, n1 = 2, and n2 = 4. TheGZ system on M is given by the 6 bi-Hamiltonian vector fields associated with thecoefficients of the Casimirs (9.2). The first vector fields of the two bi-Hamiltonianhierarchy are, respectively,

X(1)1 = ([A,X0], [A,X1]), X

(2)1 = ([A2, X0], [A2, X1]). (9.3)

Let us fix a symplectic leaf S of P , defined by the constraints H(1)0 = c1,

H(2)0 = c2. According to Proposition 7.1, the six remaining Hamiltonians define

a Lagrangian foliation, called the GZ foliation, on the 12-dimensional symplecticmanifold S. The results of Section 7 entail that, in order to separate the GZ system,we need a distribution which is transversal to the symplectic leaves of P . Moreprecisely, let {·, ·} and {·, ·}′ be the Poisson brackets associated with P and P ′.Then we must find a pair of vector fields (Z1, Z2), spanning a two-dimensionalintegrable distribution on M, such that

Za(H(b)0 ) = δba (9.4)


and such that the functions invariant along the distribution form a Poisson sub-algebra with respect to both {·, ·} and {·, ·}′. It is not difficult to show that theserequirements are fulfilled by

Z1: X0 = E23, X1 = 0,

Z2: X0 = E13, X1 = 0,

where Eij is the matrix with 1 in the (i, j) entry and 0 elsewhere. In fact, a functionF ∈ C∞(M) is invariant with respect to both Z1 and Z2 if and only if(

∂F

∂X0

)31

=(∂F

∂X0

)32

= 0, (9.5)

and such functions form a Poisson subalgebra, because the matrices fulfilling (9.5)are a Lie subalgebra of sl(3). Moreover,

Z1(H(1)0 ) = tr(AE23) = 1 and Z1(H

(2)0 ) = tr(A2E23) = 0,

and the analogous equations for Z2 hold, so that we have the normalization (9.4).Then Proposition 7.2 implies that

LZaP = 0, LZa

P ′ =2∑

b=1

Y ba ∧ Zb,

with Y ba = [Za,X

(b)1 ]. The vector field (see Remark 7.6)

Y = Y 11 + Y 2

2 = [Z1, X(1)1 ] + [Z2, X

(2)1 ]

is given, on account of (9.3), by

X0 = −1 0 0

0 −1 00 0 2

, X1 = 0.

Hence, the symplectic leaf S has an ωN structure and Theorem 7.5 tells us that theabove-defined GZ foliation is separable in DN coordinates.

Now we will use the results of Section 7 and 8 to discuss the Stäckel separabilityand the separation relations of the GZ basis. Indeed, H(1)(λ) and H(2)(λ) are easilyseen to be affine with respect to the transversal vector fields,

Za(Zb(H(1)(λ))) = Za(Zb(H

(2)(λ))) = 0, for all a, b = 1, 2,

meaning that the GZ basis is Stäckel separable.A set of special DN coordinates (λi, µi)i=1,...,6 on S is determined as follows.

We write the polynomial matrix

F(λ) =[Z1(H

(1)(λ)) Z2(H(1)(λ))

Z1(H(2)(λ)) Z2(H

(2)(λ))

]=[

L(λ)32 L(λ)31

(L(λ)2)32 (L(λ)2)31

],


whose determinant gives the minimal polynomial of the recursion operator N of S:

det F(λ) = λ6 −6∑

i=1

ciλ6−i. (9.6)

Its roots are the eigenvalues (λ1, . . . , λ6) of N . The µi coordinates can be foundwith the strategy described in Section 5, which consists in looking for a Stäckelfunction generator f (λ) such that Y (f (λ)) = 1. In Section 8 we saw that thenormalized rows of the adjoint matrix F∨(λ) of F(λ) provide Stäckel functiongenerators. We have

F∨(λ) =[

(L(λ)2)31 −L(λ)31

−(L(λ)2)32 L(λ)32

],

so that f (λ) := −(L(λ)2)31/L(λ)31 is a Stäckel function generator. Since

Y ((L(λ)2)31) = −L(λ)31 and Y (L(λ)31) = 0,

we obtain Y (f (λ)) = 1, and therefore

µi = f (λi) = −(L(λi)2)31/L(λi)31, i = 1, . . . , 6, (9.7)

form with the λi a set of special DN coordinates.At this point we could, in principle, use (9.6) and (9.7) to explicitly write the

point (X0, X1) of S in terms of (λi, µi)i=1,...,6, and we could compute the functions2i in (8.18) in order to obtain the separation relations for the GZ basis:

ρ1(λi)H(1)(λi) + ρ2(λi)H

(2)(λi) = 2i(λi, µi),

with ρ1(λ) = f (λ) and ρ2(λ) = 1. Thus we have

µiH(1)(λi) + H(2)(λi) = 2i(λi, µi). (9.8)

However, we can directly show that these separation relations coincide with theones given by the spectral curves, i.e., det(µI − L(λ)) = 0. Since

det(µI − L(λ)) = µ3 − 12 tr(L(λ)2)µ− 1

3 tr(L(λ)3),

the points (λi, µi)i=1,...,6 given by (9.6) and (9.7) belong to the spectral curve if andonly if

−((L(λi)

2)31

L(λi)31

)3

+ 12 tr(L(λi)

2)(L(λi)

2)31

L(λi)31− 1

3 tr(L(λi)3) = 0 (9.9)

for all λi such that

L(λi)32(L(λi)2)31 − L(λi)31(L(λi)

2)32 = 0. (9.10)


Since it can be checked that Equation (9.9) holds for every traceless 3 × 3 matrixfulfilling (9.10), we have indeed shown that the separation relations (9.8) are givenby the spectral curve.

Acknowledgements

The results presented in this paper are a first account of a long-standing collabora-tion with Franco Magri, which we gratefully acknowledge. We wish to thank alsoSergio Benenti, Boris Dubrovin, and John Harnad for useful discussions. This workhas been partially supported by INdAM-GNFM and the Italian MURST under theresearch project Geometry of Integrable Systems.

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181

Extracting Linear and Bilinear Factorsin Feynman’s Operational Calculi

G. W. JOHNSON1 and B. S. KIM2

1Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln,NE 68588-0323, U.S.A. e-mail: [email protected] of Mathematics, Yonsei University, Seoul 120-749, Korea.e-mail: [email protected]

(Received: 2 July 2002; in final form: 14 December 2002)

Abstract. The operation of ‘disentangling’ is the key to Feynman’s operational calculi for non-commuting operators. Hence, formulas which simplify this procedure under various conditions arecentral to the subject. The main results here make it possible to carry out the disentangling in aniterative manner. The linear (alternatively, bilinear) term alluded to in the title of the paper refers tothe first (alternatively, first and second) step (or steps) of the process. These results have many furtherconsequences some of which are given in this paper.

Mathematics Subject Classifications (2000): Primary: 47A60; secondary: 46J15, 81Q30, 81S99.

Key words: Feynman’s operational calculus for noncommuting operators, disentangling, autono-mous brackets, extracting a linear factor.

1. Introduction

Feynman’s 1951 paper on the operational calculus for noncommuting operatorsarose out of his ingenious work on quantum electrodynamics and was inspired inpart by his earlier work on the Feynman path integral. Indeed, Feynman thought ofhis operational calculus as a kind of generalized path integral. Much surprisinglyvaried work on the subject has been done since by mathematicians and physi-cists. References can be found in the recent books of Johnson and Lapidus [7]and Nazaikinskii, Shatalov and Sternin [12]. Chapter 14 of [7] provides an intro-duction to the heuristic ideas of Feynman’s operational calculus whereas Chap-ters 15–18 show how Feynman’s operational calculus can be made rigorous in theheat and quantum mechanical settings via rigorous Wiener and Feynman integrals,respectively.

This paper is a further development of the recent mathematically rigorous ap-proach to Feynman’s operational calculi initiated by Jefferies and Johnson [2–5]and continued in [6, 8, 13, 14]. Basic properties of the theory were establishedin [2, 3] and a related spectral theory was begun in [4] and continued in [5]. Thepapers [2, 3] are most directly related to the present work.

182 G. W. JOHNSON AND B. S. KIM

Among the appealing features of the theory just referred to above is that onecan consider various operational calculi within a single framework, thus making itpossible, for example, to learn things about one operational calculus by regarding itas the limit of a sequence of simpler operational calculi; see [8, 14]. Further, the useof a variety of operational calculi enables one to solve a wide variety of evolutionproblems; see [6, Sect. 4] in conjunction with the results of DeFacio, Johnson andLapidus in [7, Chap. 19].

The most extensively studied and applied approach to Feynman’s operationalcalculus is the one initiated by Maslov; see [11, 12]. The main results of this paperare an extension to the Jefferies–Johnson theory of the technique of calculationassociated with Maslov’s ‘autonomous brackets’ notation. This technique is usedmany times in both [11] and [12] and is referred to in [12] as ‘the extraction of alinear factor’.

We turn now to reviewing the basic definitions and a few results from [2] afterwhich we will describe more precisely the results of this paper.

Given a positive integer n and n positive numbers r1, . . . , rn, let A(r1, . . . , rn)

be the space of complex-valued functions of n complex variables f (z1, . . . , zn),which are analytic at (0, . . . , 0), and are such that their power series expansion

f (z1, . . . , zn) =∞∑

m1,...,mn=0

cm1,...,mnzm11 . . . zmnn (1.1)

converges absolutely, at least on the closed polydisk |z1| � r1, . . . , |zn| � rn. Suchfunctions are, of course, analytic at least in the open polydisk |z1| < r1, . . . , |zn| <rn.

For f ∈ A(r1, . . . , rn) given by (1.1), we let

‖f ‖ = ‖f ‖A(r1,...,rn) :=∞∑

m1,...,mn=0

|cm1,...,mn |rm11 . . . rmnn . (1.2)

The function on A(r1, . . . , rn) defined by (1.2) makes A(r1, . . . , rn) into a Ba-nach space. Perhaps the easiest way to see this is to realize that A(r1, . . . , rn) can beidentified with a weighted l1-space, where the weight on the index (m1, . . . , mn) isrm11 . . . rmnn . (A(r1, . . . , rn), ‖ · ‖A(r1,...,rn)) is also a Banach algebra under pointwise

multiplication of the functions involved [2].Let X be a Banach space and let A1, . . . , An be nonzero operators from L(X),

the space of bounded linear operators acting on X. Except for the numbers‖A1‖, . . . , ‖An‖, which will serve as weights, we ignore for the present the natureofA1, . . . , An as operators and introduce a commutative Banach algebra consistingof ‘analytic functions’ f (A1, . . . , An), where A1, . . . , An are treated as purely for-mal commuting objects. It is natural for most purposes that A1, . . . , An be linearlyindependent, but we do not even require them to be distinct.

FEYNMAN’S OPERATIONAL CALCULI 183

Consider the collection D(A1, . . . , An) of all expressions of the form

f (A1, . . . , An) =∞∑

m1,...,mn=0

cm1,...,mnAm11 . . . Amnn , (1.3)

where cm1,...,mn ∈ C for all m1, . . . , mn = 0, 1, . . . , and

‖f (A1, . . . , An)‖= ‖f (A1, . . . , An)‖D(A1,...,An)

:=∞∑

m1,...,mn=0

|cm1,...,mn |‖A1‖m1 . . . ‖An‖mn <∞. (1.4)

Adding and scalar multiplying such expressions coordinatewise, we can easily seethat D(A1, . . . , An) is a vector space and that ‖ · ‖D(A1,...,An) defined by (1.4) isa norm. The normed linear space (D(A1, . . . , An), ‖ · ‖D(A1,...,An)) can readily beidentified with the weighted l1-space, where the weight at the index (m1, . . . , mn) is‖A1‖m1 . . . ‖An‖mn . It follows that D(A1, . . . , An) is a Banach space and(D(A1, . . . , An), ‖ ·‖D(A1,...,An)) is a commutative Banach algebra with identity [2].

We refer to D(A1, . . . , An) as the disentangling algebra associated with then-tuple (A1, . . . , An) of bounded linear operators acting on X.

The Banach algebras A(r1, . . . , rn) and D(A1, . . . , An) are clearly closely re-lated. In fact D(A1, . . . , An) is obtained from A(r1, . . . , rn) simply by renamingthe indeterminates z1, . . . , zn.

Let A1, . . . , An be nonzero operators from L(X) and let µ1, . . . , µn be contin-uous probability measures defined at least on B[0, T ], the Borel class of [0, T ].We wish to define the disentangling map

Tµ1,...,µn : D(A1, . . . , An)→ L(X) (1.5)

according to the rule determined by the measures µ1, . . . , µn. Putting it anotherway, given any analytic function f ∈ A(‖A1‖, . . . , ‖An‖), we wish to form thefunction fµ1,...,µn(A1, . . . , An) of not necessarily commuting operators A1, . . . , Anas directed by µ1, . . . , µn.

Given nonnegative integers m1, . . . , mn, we let

Pm1,...,mn(z1, . . . , zn) = zm11 . . . zmnn (1.6)

so that

Pm1,...,mn(A1, . . . , An) = Am11 . . . Amnn . (1.7)

For each m = 0, 1, . . . , let Sm denote the set of all permutations of the integers{1, . . . , m}, and given π ∈ Sm, we let

�m(π) = {(s1, . . . , sm) ∈ [0, T ]m : 0 < sπ(1) < · · · < sπ(m) < T }. (1.8)


For j = 1, . . . , n and all s ∈ [0, T ], we let

Aj (s) = Aj . (1.9)

Now for nonnegative integers m1, . . . , mn and m = m1 + · · · +mn, we define

Ci(s) :=

A1(s) if i ∈ {1, . . . , m1},A2(s) if i ∈ {m1 + 1, . . . , m1 +m2},...

An(s) if i ∈ {m1 + · · · +mn−1 + 1, . . . , m},

(1.10)

for i = 1, . . . , m and for all 0 � s � T .Our first proposition is easy to prove (see [2, Prop. 2.2]) but it provides a crucial

link between Feynman’s ideas and the theory initiated in [2–5].

PROPOSITION 1.1. We have, for any nonnegative integers m1, . . . , mn and m =m1 + · · · +mn,

Pm1,...,mn(A1, . . . , An)

=∑π∈Sm

∫�m(π)

Cπ(m)(sπ(m)) . . . Cπ(1)(sπ(1))×

×(µm11 × · · · × µmnn )(ds1, . . . , dsm). (1.11)

We now define the map Tµ1,...,µn . For j = 1, . . . , n and all s ∈ [0, T ], we let

Aj(s) = Aj (1.12)

and for i = 1, . . . , m, we define

Ci(s) :=

A1(s) if i ∈ {1, . . . , m1},A2(s) if i ∈ {m1 + 1, . . . , m1 +m2},...

An(s) if i ∈ {m1 + · · · +mn−1 + 1, . . . , m},(1.13)

for all 0 � s � T .

DEFINITION 1.2. For any nonnegative integers m1, . . . , mn andm = m1 +· · ·+mn,

Tµ1,...,µn(Pm1,...,mn(A1, . . . , An))

:=∑π∈Sm

∫�m(π)

Cπ(m)(sπ(m)) . . . Cπ(1)(sπ(1))×

×(µm11 × · · · × µmnn )(ds1, . . . , dsm). (1.14)


Then, for f (A1, . . . , An) ∈ D(A1, . . . , An) given by

f (A1, . . . , An) =∞∑

m1,...,mn=0

cm1,...,mnAm11 . . . Amnn , (1.15)

we set

Tµ1,...,µnf (A1, . . . , An)

:=∞∑

m1,...,mn=0

cm1,...,mnTµ1,...,µnPm1,...,mn(A1, . . . , An). (1.16)

We will often use the alternative notation:

Pm1,...,mnµ1,...,µn

(A1, . . . , An) := Tµ1,...,µnPm1,...,mn(A1, . . . , An) (1.17)

and

fµ1,...,µn(A1, . . . , An) := Tµ1,...,µnf (A1, . . . , An). (1.18)

The following proposition is proved in [2].

PROPOSITION 1.3.

(a) The series (1.16) converges absolutely in the uniform operator topology ofL(X) for all f (A1, . . . , An) ∈ D(A1, . . . , An).

(b) Tµ1,...,µn is a linear map from D(A1, . . . , An) into L(X).(c) Tµ1,...,µn is an operator of norm 1 in L(D(A1, . . . , An),L(X)).

Some additional information from [2, 3] will be recalled as needed in Sections 2and 3 below.

We are now ready to discuss more precisely the results of this paper. Theo-rem 2.1 deals with monomials Pm1,...,mn and is the key to the development. Picka subset of k operator-measure pairs from the pairs (A1, µ1), . . . , (An, µn). Weassume for notational convenience that we have picked out (A1, µ1), . . . , (Ak, µk).Let 0 < a < b < T and assume that the supports of µ1, . . . , µk are all containedin [a, b] whereas the supports of µk+1, . . . , µn are all contained in [0, a] ∪ [b, T ].Then the disentangling Pm1,...,mn

µ1,...,µn(A1, . . . , An) can be carried out by first computing

Km1,...,mk := Pm1,...,mkµ1,...,µk

(A1, . . . , Ak) followed by

P 1,mk+1,...,mnµ0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An),

where µ0 is any continuous probability measure on [a, b]. Note that the powerassociated with the operator-measure pair (Km1,...,mk , µ0) is one. It is this opera-tor that is regarded as the linear factor. Theorem 2.3 extends Theorem 2.1 to anappropriate larger class of analytic functions.


The later results, Theorems 3.1 and 3.3, deal with the case where there are twoisolated and ordered clusters of probability measures. We refer to formula (3.6) ofTheorem 3.3 as ‘the extraction of a bilinear factor’.

The autonomous bracket notation (or the extraction of a linear factor) is usedmany times in [11] and [12] to establish operator equations in the noncommutativesetting. Corollary 2.4 is designed to assist with that in the setting of the presenttheory. Example 2.5 is a simple illustration of the use of Corollary 2.4.

The main results of this paper can be thought of as partial Fubini theorems fordisentangling. Under additional assumptions on the supports of the measures, theanalogy with the Fubini theorem is strengthened; see Corollaries 2.7 and 2.8.

Various results from [2] and [3] can be combined with the main results here togive further simple corollaries. We settle for illustrating that fact in Proposition 2.9and Remark 2.10.

The relationship between the theory that began in [2–5] and the theory begunby Maslov (see [11, 12]) is not clear but we can say that neither one contains theother. We should remark that when the spectral theory [4, 5] is applicable, thepresent theory extends well beyond analytic functions.

2. Extraction of a Linear Factor

Let A1, . . . , An belong to L(X), where X is a Banach space and let µ1, . . . , µn becontinuous probability measures on B[0, T ].THEOREM 2.1. Suppose that the probability measures µ1, . . . , µk are supportedby [a, b] ⊂ [0, T ] and that the probability measures µk+1, . . . , µn are supportedby [0, a] ∪ [b, T ]. Let m1, . . . , mn be nonnegative integers. Let

Km1,...,mk := Pm1,...,mkµ1,...,µk

(A1, . . . , Ak) (2.1)

and let µ0 be any continuous probability measure supported by [a, b]. Then wehave

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

= P 1,mk+1,...,mnµ0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An). (2.2)

Proof. We take m = m1 + · · · + mn as usual. For any permutation π ∈ Sm forwhich any of sm1+···+mk+1, . . . , sm lies between any two of s1, . . . , sm1+···+mk in thelist sπ(1), . . . , sπ(m), we have

(µm11 × · · · × µmnn )(�m(π)) = 0.

Such permutations can be omitted from the sum which defines the left-hand sideof (2.2). In the integrands corresponding to the remaining set, say S ′

m, of permuta-tions, each of sm1+···+mk+1, . . . , sm comes before or after all of s1, . . . , sm1+···+mk inthe list sπ(1), . . . , sπ(m). Hence we can write

Pm1,...,mnµ1,...,µn

(A1, . . . , An)


=∑π∈S ′

m

∫�m(π)

Cπ(m)(sπ(m)) . . . Cπ(1)(sπ(1))×

×(µm11 × · · · × µmnn )(ds1, . . . , dsm).

But there is a unique correspondence between the permutations π ∈ S ′m and triples

(ρ, σ, p) where p ∈ {0, 1, . . . , mk+1 + · · · + mn}, ρ ∈ Sm1+···+mk and σ is apermutation of the mk+1 +· · ·+mn integers m1 +· · ·+mk+ 1, . . . , m. We denotethis last set of permutations by Sm1+···+mk+1,m. The triple (ρ, σ, p) corresponds tothe following ordering of the variables si ;

0 < sσ(m1+···+mk+1) < · · · < sσ(m1+···+mk+p) < sρ(1) < · · ·· · · < sρ(m1+···+mk) < sσ(m1+···+mk+p+1) < · · · < sσ(m) < T .

Using the correspondence just noted, we can write

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

=∑

ρ∈Sm1+···+mkσ∈Sm1+···+mk+1,m

mk+1+···+mn∑p=0

∫�m(ρ,σ,p)

Cσ(m)(sσ(m)) . . .

. . . Cσ(m1+···+mk+p+1)(sσ(m1+···+mk+p+1))××Cρ(m1+···+mk)(sρ(m1+···+mk)) . . . Cρ(1)(sρ(1))××Cσ(m1+···+mk+p)(sσ(m1+···+mk+p)) . . .. . . Cσ(m1+···+mk+1)(sσ(m1+···+mk+1)) (µ

m11 × · · · × µmnn )(ds1, . . . , dsm),

where

�m(ρ, σ, p) = {(s1, . . . , sm) : 0 < sσ(m1+···+mk+1) < · · ·· · · < sσ(m1+···+mk+p) < sρ(1) < · · ·· · · < sρ(m1+···+mk) < sσ(m1+···+mk+p+1) < · · ·· · · < sσ(m) < T }.

But from the supports of the measures µ1, . . . , µn, we know that

(µm11 × · · · × µmnn )(�m(ρ, σ, p)) = (µm1

1 × · · · × µmnn )(�′m(ρ, σ, p)),

where

�′m(ρ, σ, p) = {(s1, . . . , sm) : 0 < sσ(m1+···+mk+1) < · · ·

· · · < sσ(m1+···+mk+p) < a < sρ(1) < · · ·· · · < sρ(m1+···+mk) < b < sσ(m1+···+mk+p+1) < · · ·· · · < sσ(m) < T }.

Moreover it is easy to see that

�′m(ρ, σ, p) = �′

m1+···+mk (ρ)×�′m1+···+mk+1, m(σ, p), (2.3)


where

�′m1+···+mk (ρ) = {(s1, . . . , sm1+···+mk ) : a < sρ(1) < · · ·

· · · < sρ(m1+···+mk) < b}and

�′m1+···+mk+1,m(σ, p) = {(sm1+···+mk+1, . . . , sm) :

0 < sσ(m1+···+mk+1) < · · ·· · · < sσ(m1+···+mk+p) < a < b< sσ(m1+···+mk+p+1) < · · · < sσ(m) < T }.

Hence we obtain

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

=∑

ρ∈Sm1+···+mkσ∈Sm1+···+mk+1,m

mk+1+···+mn∑p=0

∫�′m1+···+mk (ρ)×�′

m1+···+mk+1,m(σ,p)

×

×Cσ(m)(sσ(m)) . . . Cσ(m1+···+mk+p+1)(sσ(m1+···+mk+p+1))××Cρ(m1+···+mk)(sρ(m1+···+mk)) . . . Cρ(1)(sρ(1))××Cσ(m1+···+mk+p)(sσ(m1+···+mk+p)) . . .. . . Cσ(m1+···+mk+1)(sσ(m1+···+mk+1)) (µ

m11 × · · · × µmnn )(ds1, . . . , dsm).

Note that

(µm11 × · · · × µmkk )(�′

m1+···+mk(ρ)) = (µm11 × · · · × µmkk )(�m1+···+mk (ρ))

and so if we integrate

Cρ(m1+···+mk)(sρ(m1+···+mk)) . . . Cρ(1)(sρ(1))

over the set �′m1+···+mk (ρ) and then sum with respect to ρ ∈ Sm1+···+mk , we obtain

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

=∑

σ∈Sm1+···+mk+1,m

mk+1+···+mn∑p=0

∫�′m1+···+mk+1,m(σ,p)

×

×Cσ(m)(sσ(m)) . . . Cσ(m1+···+mk+p+1)(sσ(m1+···+mk+p+1))××Km1,...,mkCσ(m1+···+mk+p)(sσ(m1+···+mk+p)) . . .. . . Cσ(m1+···+mk+1)(sσ(m1+···+mk+1))××(µmk+1

k+1 × · · · × µmnn )(dsm1+···+mk+1, . . . , dsm),


where Km1,...,mk is given by (2.1). But since µ0 is a continuous probability measurewhose support is contained within [a, b], we can write

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

=∑

σ∈Sm1+···+mk+1,m

mk+1+···+mn∑p=0

∫[a,b]×�′

m1+···+mk+1,m(σ,p)

×

×Cσ(m)(sσ(m)) . . . Cσ(m1+···+mk+p+1)(sσ(m1+···+mk+p+1))××Km1,...,mk (s0)Cσ(m1+···+mk+p)(sσ(m1+···+mk+p)) . . .. . . Cσ(m1+···+mk+1)(sσ(m1+···+mk+1))××(µ0 × µmk+1

k+1 × · · · × µmnn )(ds0, dsm1+···+mk+1, . . . , dsm). (2.4)

On the other hand,

P 1,mk+1,...,mnµ0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An)

=∑

π∈S1+mk+1+···+mn

∫�1+mk+1+···+mn(π)

×

×C ′π(1+mk+1+···+mn)(sπ(1+mk+1+···+mn)) . . . C

′π(1)(sπ(1))×

×(µ0 × µmk+1k+1 × · · · × µmnn )(ds1, . . . , ds1+mk+1+···+mn),

where

C ′i(s) :=

Km1,...,mk (s), if i = 1,Ak+1(s), if i ∈ {2, . . . , 1 +mk+1},...

An(s), if i ∈ {1 +mk+1 + · · · +mn−1 + 1, . . . ,1 +mk+1 + · · · +mn},

for i = 1, . . . , 1 + mk+1 + · · · +mn and for all 0 � s � T . But there is a uniquecorrespondence between the permutations π ∈ S1+mk+1+···+mn and pairs (σ ′, p)where σ ′ ∈ S2,1+mk+1+···+mn and p ∈ {0, 1, . . . , mk+1 + · · · +mn}. The pair (σ ′, p)corresponds to the following ordering of the variables si ;

0 < sσ ′(2) < · · · < sσ ′(p+1) < s1

< sσ ′(p+2) < · · · < sσ ′(1+mk+1+···+mn) < T .

Using the correspondence just noted, we can write

P 1,mk+1,...,mnµ0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An)

=∑

σ ′∈S2,1+mk+1+···+mn

mk+1+···+mn∑p=0

∫�1+mk+1+···+mn(σ ′,p)

×

×C ′σ ′(1+mk+1+···+mn)(sσ ′(1+mk+1+···+mn)) . . . C

′σ ′(p+2)(sσ ′(p+2))×


×Km1,...,mk (s1) C′σ ′(p+1)(sσ ′(p+1)) . . . C

′σ ′(2)(sσ ′(2))×

×(µ0 × µmk+1k+1 × · · · × µmnn )(ds1, . . . , ds1+mk+1+···+mn),

where

�1+mk+1+···+mn(σ′, p)

= {(s1, . . . , s1+mk+1+···+mn) : 0 < sσ ′(2) < · · ·· · · < sσ ′(p+1) < s1 < sσ ′(p+2) < · · ·· · · < sσ ′(1+mk+1+···+mn) < T }.

But from the supports of the measures µ0, µk+1, . . . , µn, we know that

(µ0 × µmk+1k+1 × · · · × µmnn )(�1+mk+1+···+mn(σ

′, p))= (µ0 × µmk+1

k+1 × · · · × µmnn )(�′1+mk+1+···+mn(σ

′, p)),

where

�′1+mk+1+···+mn(σ

′, p)= {(s1, . . . , s1+mk+1+···+mn) : 0 < sσ ′(2) < · · ·

· · · < sσ ′(p+1) < a < s1 < b < sσ ′(p+2) < · · ·· · · < sσ ′(1+mk+1+···+mn) < T }

and note that

�′1+mk+1+···+mn(σ

′, p) = (a, b)×�′2,1+mk+1+···+mn(σ

′, p),

where

�′2,1+mk+1+···+mn(σ

′, p)= {(s2, . . . , s1+mk+1+···+mn) : 0 < sσ ′(2) < · · ·

· · · < sσ ′(p+1) < a < b < sσ ′(p+2) < · · ·· · · < sσ ′(1+mk+1+···+mn) < T }.

Hence we can write

P 1,mk+1,...,mnµ0,µk+1,...,µn

Km1,...,mk , Ak+1, . . . , An)

=∑

σ ′∈S2,1+mk+1+···+mn

mk+1+···+mn∑p=0

∫[a,b]×�′

2,1+mk+1+···+mn(σ′,p)

×

×C ′σ ′(1+mk+1+···+mn)(sσ ′(1+mk+1+···+mn)) . . . C

′σ ′(p+2)(sσ ′(p+2))×

×Km1,...,mk (s1) C′σ ′(p+1)(sσ ′(p+1)) . . . C

′σ ′(2)(sσ ′(2))×

×(µ0 × µmk+1k+1 × · · · × µmnn )(ds1, . . . , ds1+mk+1+···+mn).


Define

σ (m1 + · · · +mk − 1 + j) = m1 + · · · +mk − 1 + σ ′(j) for

j = 2, . . . , 1 +mk+1 + · · · +mn.Then σ ∈ Sm1+···+mk+1, m and

Cσ(m1+···+mk−1+j)(sσ(m1+···+mk−1+j)) = C ′σ ′(j)(sσ ′(j)),

for j = 2, . . . , 1 + mk+1 + · · · + mn, where Ci(s) is given by (1.13). Hence weobtain

P 1,mk+1,...,mnµ0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An)

=∑

σ∈Sm1+···+mk+1,m

mk+1+···+mn∑p=0

∫[a,b]×�′

m1+···+mk+1,m(σ,p)

×

×Cσ(m)(sσ(m)) · · ·Cσ(m1+···+mk+p+1)(sσ(m1+···+mk+p+1))××Km1,...,mk (s1) Cσ(m1+···+mk+p)(sσ(m1+···+mk+p)) . . .. . . Cσ(m1+···+mk+1)(sσ(m1+···+mk+1))××(µ0 × µmk+1

k+1 × · · · × µmnn )(ds1, dsm1+···+mk+1, . . . , dsm). (2.5)

Comparing (2.4) and (2.5) (and renaming the variable s1), we see that the proof iscomplete. ✷LEMMA 2.2. For each choice of the nonnegative integers m2, . . . , mn, the map : L(X)→ L(X) defined by

(A) := P 1,m2,...,mnµ1,µ2,...,µn

(A,A2, . . . , An) (2.6)

is a bounded linear operator, that is, ∈ L(L(X)).Proof. Linearity is trivial. For any A ∈ L(X), inequality (2.16) from [2] yields

‖ (A)‖ = ‖P 1,m2,...,mkµ1,µ2,...,µk

(A,A2, . . . , An)‖ � ‖A‖‖A2‖m2 . . . ‖An‖mn,and so

‖ ‖L(L(X)) � ‖A2‖m2 . . . ‖An‖mn

as desired. ✷THEOREM 2.3 (Extraction of a linear factor). Let µ1, . . . , µn be given as in The-orem 2.1. Assume that g(A1, . . . , Ak) ∈ D(A1, . . . , Ak) and h(Ak+1, . . . , An) ∈D(Ak+1, . . . , An). Let

f (z1, . . . , zn) = g(z1, . . . , zk) h(zk+1, . . . , zn). (2.7)


LetK := Tµ1,...,µkg(A1, . . . , Ak) and let µ0 be any continuous probability measuresupported by [a, b]. Then f (A1, . . . , An) ∈ D(A1, . . . , An) and

Tµ1,...,µnf (A1, . . . , An) = Tµ0,µk+1,...,µnF (K, Ak+1, . . . , An), (2.8)

where F(z0, zk+1, . . . , zn) = z0h(zk+1, . . . , zn).

Proof. Suppose that g and h are given by

g(z1, . . . , zk) =∞∑

m1,...,mk=0

dm1,...,mkzm11 . . . z

mkk

and

h(zk+1, . . . , zn) =∞∑

mk+1,...,mn=0

emk+1,...,mnzmk+1k+1 . . . z

mnn .

Then we have

f (z1, . . . , zn) =∞∑

m1,...,mn=0

cm1,...,mnzm11 . . . zmnn ,

where

cm1,...,mn = dm1,...,mkemk+1,...,mn

for all m1, . . . , mn = 0, 1, 2, . . . . Now by Definition 1.2 and Theorem 2.1,

Tµ1,...,µnf (A1, . . . , An) =∞∑

m1,...,mn=0

cm1,...,mnPm1,...,mnµ1,...,µn

(A1, . . . , An)

=∞∑

m1,...,mn=0

dm1,...,mkemk+1,...,mn ×

× P 1,mk+1,...,mnµ0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An).

For each mk+1, . . . , mn = 0, 1, 2, . . . , we have

∞∑m1,...,mk=0

dm1,...,mkP1,mk+1,...,mnµ0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An)

(1)= limN→∞

N∑m1,...,mk=0

dm1,...,mkP1,mk+1,...,mnµ0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An)

(2)= P 1,mk+1,...,mnµ0,µk+1,...,µn

( ∞∑m1,...,mk=0

dm1,...,mkKm1,...,mk , Ak+1, . . . , An

).


Step (1) follows from the fact that the series in the left-hand side of the firstequality converges absolutely. By Lemma 2.2, the map defined by

(A) = P 1,mk+1,...,mnµ0,µk+1,...,µn

(A,Ak+1, . . . , An)

is linear and bounded. Moreover the series in the right hand side of the secondequality converges absolutely. Hence we obtain Step (2). But

∞∑m1,...,mk=0

dm1,...,mkKm1,...,mk = Tµ1,...,µkg(A1, . . . , Ak) = K

and so, we have

Tµ1,...,µnf (A1, . . . , An)

=∞∑

mk+1,...,mn=0

emk+1,...,mnP1,mk+1,...,mnµ0,µk+1,...,µn

(K,Ak+1, . . . , An)

= Tµ0,µk+1,...,µnF (K, Ak+1, . . . , An),

where

F(z0, zk+1, . . . , zn)

=∞∑

mk+1,...,mn=0

emk+1,...,mnz0zmk+1k+1 . . . z

mnn

= z0h(zk+1, . . . , zn)

as desired. ✷Theorem 2.3 extends to the theory initiated in [2–5] a computational technique

which is due to Maslov and which is used many times both in his book [11] andin the book by Nazaikinskii, Shatalov and Sternin [12]. The technique is referredto in [12] as ‘the extraction of a linear factor’ and is used in conjunction with the‘autonomous bracket’ notation of Maslov. Both books [11, 12] are restricted tothe case of discrete indices from R. This corresponds to the special case of thepresent theory where the supports S(µ1), . . . , S(µn) of the continuous probabilitymeasures µ1, . . . , µn are ordered, say, 0 � S(µ1) � S(µ2) � · · · � S(µn) � T .(Here, S(µj ) � S(µj+1) means that S(µj ) lies, except possibly for one commonpoint, entirely to the left of S(µj+1).) We note that Section 4 of [3] discusses theeffect of ordered supports on disentangling.

[We thank the referee for calling our attention to the paper of Karasev [10]where, in our terminology, the ordering measures µ1, . . . , µn are either copies ofLebesgue measure (as in the Weyl calculus) or are discrete. Karasev’s setting isrelated to that of this paper but is even more closely tied to a very recent paperof the first author and Lance Nielsen [9] which is entitled ‘Blending instantaneousand continuous phenomena in Feynman’s operational calculi’.]


The technique of extracting a linear factor is used in [11, 12] (and in relatedpapers) to establish equalities while doing calculations with noncommuting oper-ators. Of course, such calculations can be organized so that the goal is to showthat a related operator expression is equal to the zero operator. Our first corollarybelow will make it easy to apply Theorem 2.3 in the manner which we have justdiscussed.

COROLLARY 2.4. Let the hypotheses of Theorem 2.3 be satisfied and supposethat

K := Tµ1,...,µkg(A1, . . . , Ak) = O. (2.9)

Then Tµ1,...,µnf (A1, . . . , An) also equals the zero operator.

Proof. Formula (2.8) from Theorem 2.3 tells us that

Tµ1,...,µnf (A1, . . . , An) = Tµ0,µk+1,...,µnF (O, Ak+1, . . . , An), (2.10)

where F(z0, zk+1, . . . , zn) = z0 h(zk+1, . . . , zn) and where µ0 can be taken to beany continuous probability measure on [a, b]. Now the factor z0 appears in everyterm of the series expansion for F ; further, the operator associated with z0 and µ0 isthe zero operator O. Hence, the zero operator appears in every permutation associ-ated with every term of the series expansion for F . Then Tµ1,...,µnf (A1, . . . , An) =O as we wished to show. ✷

Next we give an example illustrating the use of Theorem 2.3 and Corollary 2.4.This example depends on a special relationship between A1, . . . , Ak.

EXAMPLE 2.5. Let bounded operators A1, . . . , A4 be given and suppose that

g(z1, z2) = z31 − z2

1z2 + z1z22 − z3

2 and h(z3, z4) ∈ A(‖A3‖, ‖A4‖).Further, assume that µ1, . . . , µ4 are continuous probability measures on [0, T ]with the supports of µ1 and µ2 being subsets of [a, b] and the supports of µ3

and µ4 being subsets of [0, a] ∪ [b, T ] where 0 < a < b < T . Finally, we takef (z1, . . . , z4) = g(z1, z2)h(z3, z4) and ask for the computation of Tµ1,...,µ4f (A1,

A2, A3, A4) in the case where A2 = A1. Since A2 = A1, A1 and A2 certainlycommute. Hence, by Proposition 3.1 of [2], all of the functional calculi Tν1, ν2 actingon (A1, A2) agree with the usual commutative functional calculus. Thus,

Tµ1,µ2g(A1, A2) = g(A1, A2) = A31 − A2

1A2 + A1A22 − A3

2 = O, (2.11)

where the last equality comes from the fact that A2 = A1. It follows immediatelyfrom Corollary 2.4 that Tµ1,...,µ4f (A1, . . . , A4) equals the zero operator.

The example above was certainly easy using Corollary 2.4. It can be done di-rectly from Definition 1.2. One would begin by finding the power series expansion


for f and then using formulas (1.16) and (1.14). The task would be tedious andmight be confusing (especially if one had not already worked out Example 2.5 inthe light of Theorem 2.3 and Corollary 2.4).

Remark 2.6. (a) Theorem 2.3 can be regarded as a kind of partial Fubinitheorem for disentangling. Corollary 2.7 below is, under additional restrictions,a somewhat fuller Fubini theorem for disentangling.(b) In our results so far as well as those below we have assumed that it is the first

k measures that are supported in the intermediate interval [a, b]. This assumptionis notationally convenient but is not necessary. Also, we have taken our interval tobe [0, T ], but it could just as well be any bounded subinterval of R.

Corollaries 2.7 and 2.8 below deal with a situation which is related to butdifferent from results in [2].

COROLLARY 2.7. Suppose that the probability measures µ1, . . . , µk are sup-ported by [0, a] and that the probability measures µk+1, . . . , µn are supported by[a, T ]. Then we have

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

= Pmk+1,...,mnµk+1,...,µn

(Ak+1, . . . , An)Pm1,...,mkµ1,...,µk

(A1, . . . , Ak). (2.12)

Proof. Applying Theorem 2.1 twice, we obtain

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

= P 1,mk+1,...,mnν0,µk+1,...,µn

(Km1,...,mk , Ak+1, . . . , An)

= P 1,1ν0,ν1(Km1,...,mk ,Kmk+1,...,mn), (2.13)

where ν0 and ν1 are continuous probability measures supported by [0, a] and [a, T ]respectively, and

Km1,...,mk = Pm1,...,mkµ1,...,µk

(A1, . . . , Ak)

and

Kmk+1,...,mn = Pmk+1,...,mnµk+1,...,µn

(Ak+1, . . . , An).

(Note: In applying Theorem 2.1 the first time, one can think of the support ofµk+1, . . . , µn as contained in [−1, 0]∪[a, T ]. This allows us to apply Theorem 2.1exactly as stated. A similar point of view can be adopted for the second applicationof Theorem 2.1.) Since S(ν0) � S(ν1), Proposition 4.5 from [3] implies that

Pm1,...,mnµ1,...,µn

(A1, . . . , An) = Kmk+1,...,mnKm1,...,mk

as desired. ✷


COROLLARY 2.8. Let µ1, . . . , µn be as in Corollary 2.7. Let f, g and h be givenas in Theorem 2.3. Then

Tµ1,...,µnf (A1, . . . , An)

= Tµk+1,...,µnh(Ak+1, . . . , An)Tµ1,...,µkg(A1, . . . , Ak). (2.14)

Proof. By Corollary 2.7, we have

Tµ1,...,µnf (A1, . . . , An)

=∞∑

m1,...,mn=0

cm1,...,mnPm1,...,mnµ1,...,µn

(A1, . . . , An)

=∞∑

m1,...,mn=0

dm1,...,mkemk+1,...,mn

Pmk+1,...,mnµk+1,...,µn

(Ak+1, . . . , An)Pm1,...,mkµ1,...,µk

(A1, . . . , Ak)

=[ ∞∑mk+1,...,mn=0

emk+1,...,mnPmk+1,...,mnµk+1,...,µn

(Ak+1, . . . , An)

]×

×[ ∞∑m1,...,mk=0

dm1,...,mkPm1,...,mkµ1,...,µk

(A1, . . . , Ak)

]

= Tµk+1,...,µnh(Ak+1, . . . , An)Tµ1,...,µkg(A1, . . . , Ak)

as desired. ✷Results in this paper can be combined with various results in [2, 3] to yield

further corollaries. We give an example and one variation of it below.

PROPOSITION 2.9. Under the assumptions of Theorem 2.1 and the additionalassumption that A1, . . . , Ak commute with one another, we have that Pm1,...,mk

µ1,...,µk(A1,

. . . , Ak) is given by the usual commutative functional calculus,

Pm1,...,mkµ1,...,µk

(A1, . . . , Ak) = Am11 . . . A

mkk , (2.15)

and so

Pm1,...,mnµ1,...,µn

(A1, . . . , An) = P 1,mk+1,...,mnµ0,µk+1,...,µn

(Am11 . . . A

mkk , Ak+1, . . . , An). (2.16)

Remark 2.10. (a) If the commutativity assumption above is reduced to the as-sumption that, say, A1 and A2 commute with all ofA1, . . . , Ak, then formula (2.16)becomes

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

= P 1,mk+1,...,mnµ0,µk+1,...,µn

(Am11 A

m22 P

m3,...,mkµ3,...,µk

(A3, . . . , Ak),Ak+1, . . . , An). (2.17)


(b) Not surprisingly, formulas (2.16) and (2.17) above lead to formulas forappropriate analytic functions fµ1,...,µn of A1, . . . , An.

(c) The case where µ1, . . . , µk have ordered supports (see Section 4 of [3])provides further examples of results which can be combined with the results of thispaper to give additional corollaries.

3. Extraction of a Bilinear Factor

Theorems 2.1 and 2.3 deal with the case where there is one cluster of k measurestaken as µ1, . . . , µk whose supports are all contained within an interval [a, b] withthe remaining measures all having supports contained within [0, a] ∪ [b, T ]. Thesetheorems extend to the case where there is two isolated clusters of measures. Wewill state results which correspond to Theorem 2.1, Lemma 2.2 and Theorem 2.3.Finally, we will comment on the most essential ideas involved in extending theproof of Theorem 2.1.

As before, let A1, . . . , An belong to L(X) where X is a Banach space and letµ1, . . . , µn be continuous probability measures on B[0, T ].

THEOREM 3.1. Let 0 < a < b < c < d < T and suppose that µ1, . . . , µk havesupports contained within [a, b] and that µk+1, . . . , µk+l have supports containedwithin [c, d]. Further suppose that µk+l+1, . . . , µn have supports contained within[0, a] ∪ [b, c] ∪ [d, T ]. Given nonnegative integers m1, . . . , mn, let

Km1,...,mk := Pm1,...,mkµ1,...,µk

(A1, . . . , Ak), (3.1)

and

Lmk+1,...,mk+l := Pmk+1,...,mk+lµk+1,...,µk+l (Ak+1, . . . , Ak+l), (3.2)

and let µ0 and ν0 be any continuous probability measures on [a, b] and [c, d],respectively. Then we have

Pm1,...,mnµ1,...,µn

(A1, . . . , An)

= P 1,1,mk+l+1,...,mnµ0,ν0,µk+l+1,...,µn

(Km1,...,mk , Lmk+1,...,mk+l , Ak+l+1, . . . , An). (3.3)

LEMMA 3.2. For each nonnegative integers m3, . . . , mn, the map : L(X) ×L(X)→ L(X) defined by

(A,B) := P 1,1,m3,...,mnµ1,µ2,µ3,...,µn

(A,B,A3, . . . , An) (3.4)

is a bounded bilinear form with

‖ ‖ � ‖A3‖m3 . . . ‖An‖mn. (3.5)


THEOREM 3.3 (Extraction of a bilinear factor). Let µ1, . . . , µn be as in Theo-rem 3.1. Assume that

g1(A1, . . . , Ak) ∈ D(A1, . . . , Ak),

g2(Ak+1, . . . , Ak+l) ∈ D(Ak+1, . . . , Ak+l )

and

h(Ak+l+1, . . . , An) ∈ D(Ak+l+1, . . . , An).

LetK := Tµ1,...,µkg1(A1, . . . , Ak), L := Tµk+1,...,µk+l g2(Ak+1, . . . , Ak+l) and letµ0 and ν0 be any continuous probability measures whose supports are containedwithin [a, b] and [c, d], respectively. Then

f (z1, . . . , zn) := g1(z1, . . . , zk) g2(zk+1, . . . , zk+l) h(zk+l+1, . . . , zn)

belongs to A(‖A1‖, . . . , ‖An‖) and we have

Tµ1,...,µnf (A1, . . . , An) = Tµ0,ν0,µk+l+1,...,µnF (K, L, Ak+l+1, . . . , An), (3.6)

where F(z0, ζ0, zk+l+1, . . . , zn) = z0ζ0h(zk+l+1, . . . , zn).

We finish by describing how the most essential ideas used in the proof of The-orem 2.1 can be extended to the case of Theorem 3.1 where there are two isolatedclusters of measures. In comments (a)–(c) below we try to communicate the ideaswith a minimum of notation. However, the notation seems to be needed for (d).

(a) The assumptions on the supports of the measures rule out certain orderings.More precisely, the measure µm1

1 × · · · × µmnn assigns 0 probability to certainsets �m(π) where the permutation π determines the order of the variables.(Conversely, the ordering of the variables determines the permutation. As wecontinue, we will think mainly in terms of the ordering of the variables.)Specifically, in the setting of Theorem 3.1, none of the variables associatedwith the measures µk+l+1, . . . , µn can lie between any two of the variables as-sociated with µ1, . . . , µk nor between any two of the variables associated withµk+1, . . . , µk+l . Further, all of the variables associated with µ1, . . . , µk mustcome before all of the variables associated with µk+1, . . . , µk+l . (These state-ments and related ones below all hold up to a set of µm1

1 × · · · ×µmnn -measure0 even though we will not always mention this.)

(b) Even for sets �m(π) having positive µm11 × · · · × µmnn measure, the assump-

tions on the supports constrain, up to a set of µm11 × · · · × µmnn measure 0,

the variables associated with µ1, . . . , µk to lie in (a, b), the variables asso-ciated with µk+1, . . . , µk+l to lie in (c, d) and the variables associated withµk+l+1, . . . , µn to lie in (0, a) ∪ (b, c) ∪ (d, T ).

(c) After (a) and (b) have been taken into account the permutations that remaincan be put into one-to-one correspondence with 5-tuples (ρ, σ, τ ;p, q) where


ρ is a permutation of the m1 +· · ·+mk variables in (a, b) and associated withµ1, . . . , µk , σ is a permutation of themk+1 +· · ·+mk+l variables in (c, d) andassociated with µk+1, . . . , µk+l , τ is a permutation of the mk+l+1 + · · · + mnvariables in (0, a) ∪ (b, c) ∪ (d, T ) and associated with µk+l+1, . . . , µn, p isa nonnegative integer between 0 and mk+l+1 + · · · + mn (which counts thenumber of variables in (0, a)) and q is a nonnegative integer between 0 andmk+l+1 + · · · + mn − p (which counts the number of variables in (b, c)). Ofcourse, the number of variables in (d, T ) will then be r = mk+l+1 + · · · +mn − p − q.

(d) The formula for Pm1,...,mnµ1,...,µn

will be the sum over all of the triples of permutations(ρ, σ, τ) described above and all triples of nonnegative integers (p, q, r) suchthat p+q+r = mk+l+1+· · ·+mn of integrals with respect to µm1

1 ×· · ·×µmnnover the sets

�′m(ρ, σ, τ ;p, q, r) = {(s1, . . . , sm) ∈ [0, T ]m :0 < sτ(m1+···+mk+l+1) < · · · < sτ(m1+···+mk+l+p) < a< sρ(1) < · · · < sρ(m1+···+mk) < b< sτ(m1+···+mk+l+p+1) < · · · < sτ(m1+···+mk+l+p+q) < c< sσ(m1+···+mk+1) < · · · < sσ(m1+···+mk+···+mk+l ) < d< sτ(m1+···+mk+l+p+q+1) < · · · < sτ(m1+···+mk+l+p+q+r)= sτ(m) < T }. (3.7)

Now a key point in the proof of Theorem 2.1 came in seeing that the set�′m(ρ, σ, p), the analogue of the set �′

m(ρ, σ, τ ;p, q, r) from (3.7), couldbe written as a product (see (2.3)). A similar step is the key here. We have

�′m(ρ, σ, τ ;p, q, r)= �′

m1+···+mk (ρ)×�′m1+···+mk+1,m1+···+mk+···+mk+l (σ )×

×�′m1+···+mk+l+1,...,m(τ ;p, q, r), (3.8)

where�′m1+···+mk (ρ)= {(s1, . . . , sm1+···+mk) ∈ (a, b)m1+···+mk :a < sρ(1) < · · · < sρ(m1+···+mk) < b}, (3.9)

�′m1+···+mk+1,m1+···+mk+···+mk+l (σ )= {(sm1+···+mk+1, . . . , sm1+···+mk+···+mk+l ) ∈ (c, d)mk+1+···+mk+l :c < sσ(m1+···+mk+1) < · · · < sσ(m1+···+mk+···+mk+l ) < d}, (3.10)

and�′m1+···+mk+l+1,...,m(τ ;p, q, r)= {(sm1+···+mk+l+1, . . . , sm) ∈ [(0, a) ∪ (b, c) ∪ (d, T )]p+q+r :

0 < sτ(m1+···+mk+l+1) < · · · < sτ(m1+···+mk+l+p) < a,b < sτ(m1+···+mk+l+p+1) < · · · < sτ(m1+···+mk+l+p+q) < c,d < sτ(m1+···+mk+l+p+q+1) < · · · < sτ(m1+···+mk+l+p+q+r) < T }. (3.11)


The detailed proof in the case of one isolated cluster of measures along with thediscussion in (a)–(d) above should make the essential ideas of the proof clear.

References

1. Feynman, R.: An operator calculus having applications in quantum electrodynamics, Phys. Rev.84 (1951), 108–128.

2. Jefferies, B. and Johnson, G. W.: Feynman’s operational calculi for noncommuting operators:Definitions and elementary properties, Russian J. Math. Phys. 8 (2001), 153–178.

3. Jefferies, B. and Johnson, G. W.: Feynman’s operational calculi for noncommuting operators:Tensors, ordered supports and disentangling an exponential factor, Math. Notes 70 (2001), 744–764.

4. Jefferies, B. and Johnson, G. W.: Feynman’s operational calculi for noncommuting operators:Spectral theory, Infinite Dimensional Anal. Quantum Probab. 5 (2002), 171–199.

5. Jefferies, B. and Johnson, G. W.: Feynman’s operational calculi for noncommuting operators:The monogenic calculus, Adv. Appl. Clifford Algebra 11 (2002), 233–265.

6. Jefferies, B., Johnson, G. W. and Nielsen, L.: Feynman’s operational calculi for time dependentnoncommuting operators, J. Korean Math. Soc. 38 (2001), 193–226.

7. Johnson, G. W. and Lapidus, M. L.: The Feynman Integral and Feynman’s OperationalCalculus, Oxford Math. Monographs, Oxford Univ. Press, Oxford, 2000.

8. Johnson, G. W. and Nielsen, L.: A stability theorem for Feynman’s operational calculus, In:F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Röckner and S. Scarlatti (eds), ConferenceProceedings Canadian Math. Soc.: Conference in Honor of Sergio Albeverio’s 60th Birthday29 (2000), 351–365.

9. Johnson, G. W. and Nielsen, L.: Blending instantaneous and continuous phenomena inFeynman’s operational calculi, submitted for publication.

10. Karasev, M. V.: The Weyl calculus and the ordered calculus for noncommuting operators, Mat.Z. 26 (1979), 885–907 (Russian), Eng. transl.: Math. Notes 26 (1979), 945–958.

11. Maslov, V. P.: Operational Methods, Mir, Moscow, 1976.12. Nazaikinskii, V. E., Shatalov, V. E. and Sternin, B. Yu.: Methods of Noncommutative Analysis,

Stud. in Math. 22, Walter de Gruyter, Berlin, 1996.13. Nielsen, L.: Effects of absolute continuity in Feynman’s operational calculi, Proc. Amer. Math.

Soc., 131 (2003), 781–791.14. Nielsen, L.: Stability properties of Feynman’s operational calculus for exponential functions of

noncommuting operators, Acta Appl. Math. 74 (2002), 265–292.


201

Self-Similarity, Operators and Dynamics

LEONID MALOZEMOV1 and ALEXANDER TEPLYAEV2,�1Countrywide Securities Corporation, 4500 Park Granada, Calabasas, CA 91302, U.S.A.e-mail: [email protected] of Mathematics, University of California, Riverside, CA 92521, U.S.A.e-mail: [email protected]

(Received: 19 December 2001; accepted in final form: 10 April 2003)

Abstract. We construct a large class of infinite self-similar (fractal, hierarchical or substitution)graphs and show, under a certain strong symmetry assumption, that the spectrum of the Laplacian canbe described in terms of iterations of an associated rational function (so-called ‘spectral decimation’).We prove that the spectrum consists of the Julia set of the rational function and a (possibly empty)set of isolated eigenvalues which accumulate to the Julia set. In order to obtain our results, we startwith investigation of abstract spectral self-similarity of operators.

Mathematics Subject Classifications (2000): 05C99, 28A80, 47A10.

Key words: infinite graphs, self-similar graphs, fractal graphs, hierarchical graphs, substitutiongraphs, Laplacian, spectral decimation, self-similar spectrum, Julia set, complex dynamics.

1. Introduction

In this paper we construct a large class of infinite self-similar graphs for whichthe spectrum of the Laplacian is self-similar in the sense that it can be completelydescribed in terms of iterations of a rational function. This phenomenon, usuallycalled ‘spectral decimation’, was first observed for Sierpinski lattice in the physicsliterature [16, 17], and later a mathematical theory was developed [2, 4, 19, 20, 23].Other fractals were considered in [7, 12, 14, 21], and the complex multi-dimensionalcase in [18]. Graphs of certain fractal groups studied in [1, 5, 6] have a fractalspectrum, although they seem to be different from the graphs considered here. Formore information on analysis on fractals, see [9] and references therein.

We show that the spectrum of self-similar graphs satisfying a certain strongsymmetry assumption consists of the Julia set of a rational function and a (pos-sibly empty) set of isolated eigenvalues which accumulate to the Julia set. Theseeigenvalues, typically, have infinite multiplicity.

The graphs we consider can be called fractal, hierarchical or substitution graphs.These graphs are often related to nested fractals [13, 22]. A class of such graphs

� Research supported by the National Science Foundation through a Mathematical SciencesPostdoctoral Fellowship. Current address: Department of Mathematics, University of Connecticut,Storrs, CT 06269, U.S.A.

202 LEONID MALOZEMOV AND ALEXANDER TEPLYAEV

with only two boundary points, two-point self-similar graphs (TPSG), was con-sidered in [15], where completeness of localized eigenfunctions was proved undermild conditions.

Our construction of a self-similar graph is based on a finite symmetric M-pointmodel graph. The model graph determines a self-similar sequence of finite graphswhich then defines an infinite self-similar graph. This infinite graph always haspolynomial growth and is assumed to be connected. There is no spectral gap that iszero (or one in the probabilistic statement), is a point of the spectrum, and is not anisolated point. The graph random walk is recurrent. The spectrum of the Laplacianis described in terms of rational functions R(z), ϕ1(z) and ϕ0(z)which depend onlyon the model graph. These functions are difficult to read directly from the structureof the model graph, but they can be computed effectively by (3.2). By (3.3), ϕ1(z)

and ϕ0(z) are two Green’s functions, and R(z) is the ratio of them.The necessary and sufficient conditions for spectral similarity on fractals were

first considered in [21]. However, these conditions are algebraic in nature and dif-ficult to verify. Our work gives sufficient conditions that are slightly different thanthose of [21], where only nested fractals were considered. We also give certain sim-ple necessary conditions for spectral similarity. The main difference between ourpaper and [21] is that we analyze the spectrum of infinite graphs, while [21] studiedthe fractals, which are compact sets and give rise to discrete spectrum. We note thatour main results were obtained independently of [21]. Another independent studyof the spectrum of self-similar graphs has appeared recently in [10, 11].

2. Abstract Spectral Self-Similarity

Let H and H0 be Hilbert spaces, and U : H0 → H be an isometry. Suppose that Hand H0 are operators on H and H0, respectively, and that ϕ0 and ϕ1 are complex-valued functions defined on a set � ⊆ C. Here and in what follows, an operatoralways means a bounded linear operator unless stated otherwise.

DEFINITION 2.1. We call an operator H spectrally similar to an operator H0

with functions ϕ0 and ϕ1 if

U ∗(H − z)−1U = (ϕ0(z)H0 − ϕ1(z))−1 (2.1)

z ∈ �0, where �0 is the set of those z ∈ � for which the two sides of (2.1) arewell defined. We always assume that �0 is open and not empty.

In particular, if ϕ0(z) �= 0 and R(z) = ϕ1(z)/ϕ0(z), then

U ∗(H − z)−1U = 1

ϕ0(z)(H − R(z))−1. (2.2)

Remark 2.2. The functions ϕ0 and ϕ1 are defined uniquely by (2.1) if and onlyif H0 is not a multiple of the identity operator on H0. In what follows, we alwaysassume that this is the case.

SELF-SIMILARITY, OPERATORS AND DYNAMICS 203

We also often assume that H0 is a closed subspace of H . In this case, U is theinclusion operator, which we will omit, and U ∗ = P0, the orthogonal projectoronto H0.

EXAMPLE 2.3. Suppose H = R3 and H0 = R2 in the decomposition R3 =R2 ⊕ R1. Then the operators H and H0 given by

H =( 1 1 1

1 1 11 1 1

)and H0 =

(1 11 1

)are spectrally similar. In view of (2.1) or (3.2) below, it is easy to compute thatϕ1(z) = z/(z − 1), ϕ0(z) = z and R(z) = z − 1.

EXAMPLE 2.4. Suppose again that H = R3 and H0 = R2 in the decompositionR3 = R2 ⊕ R1. Then the operators H and H0 given by

H = −1 0 1/2

0 −1 1/21/2 1/2 −1

and H0 =( −1 1

1 −1

)are spectrally similar, with ϕ1(z) = 1 + z − 1/(z + 1), ϕ0(z) = 1/2(z + 1) andR(z) = 2z2 + 4z by (2.1), or by (3.2) below.

DEFINITION 2.5. We call an operator H spectrally self-similar with functions ϕ0

and ϕ1 if there exists a co-isometry D on H , that is, DD∗ = I , such that

D(H − z)−1D∗ = (ϕ0(z)H − ϕ1(z))−1 (2.3)

for any z ∈ � for which the two sides are well defined.

We note that Equation (2.3) was used by J. Béllissard [2] for a particular caseof the Laplacian on Sierpinski lattice.

PROPOSITION 2.6. The operator H is spectrally self-similar if and only if His spectrally similar to an operator H0 on a closed subspace H0 such that H =UH0U

−1 for an isometry U : H0 → H .

3. Properties of Spectrally Similar Operators

Most of the results in this section appeared in [23] and [21] in a slightly differentsetting.

In this section we assume that H0 is a closed subspace of H . Then U is theinclusion operator, which we will omit, and U ∗ = P0, the orthogonal projectoronto H0. (If H0 is not a subspace of H one can use U to identify H0 with ImU

which is a closed subspace of H .)


NOTATION 3.1. Let H1 be the orthogonal complement to H0 and P1 = I − P0

be the orthogonal projector onto H1. The operators S: H0 → H0, X: H0 → H1,X: H1 → H0 and Q: H1 → H1 are defined by S = P0H , X = P1H , X = P0H

and Q = P1H . This means that H has the following block structure with respectto the representation H = H0 ⊕ H1:

H =(S X

X Q

). (3.1)

For i = 1, 2, Ii denotes the identity operator on Hi . The resolvent set of an operatorA is denoted by ρ(A).

Remark 3.2. Note that the operators S and X, as well as H0, are defined on H0,and the operators Q and X are defined on H1. Throughout this paper, we slightlyabuse the notation by omitting the operators of inclusion of H0 and H1 into H in allformulas. Also, we often write a constant instead of an identity operator multipliedby this constant, and we do not use any the notation for a restriction of an operatorto a particular subspace. It allows us to simplify many expressions and, we hope,will not confuse the reader.

The next lemma will be an important tool for dealing with spectrally similaroperators.

LEMMA 3.3. For z ∈ ρ(H) ∩ ρ(Q) relation (2.1) holds if and only if

S − zI0 − X(Q− zI1)−1X = ϕ0(z)H0 − ϕ1(z)I0. (3.2)

Proof. We have that

P0(H − z)−1(S − z − X(Q− z)−1X)P0

= P0(H − z)−1P0(H − z)P0 −− P0(H − z)−1P0(H − z)(Q− z)−1P1(H − z)P0

= P0(H − z)−1P0(H − z)P0 + P0(H − z)−1P1(H − z)P0 = P0.

Therefore (2.1) is equivalent to (3.2). ✷COROLLARY 3.4. Suppose H is spectrally similar to H0 on H0. Then there existsa unique analytic continuation of ϕ0 and ϕ1 from the set �0 ∩ρ(Q) to its connectedcomponent in ρ(Q) such that (3.2) holds.

In particular, if the spectrum of Q consists only of isolated eigenvalues, then ϕ0

and ϕ1 have a meromorphic continuation to all of C, with all the poles containedin σ (Q).

Proof. By Remark 2.2, there are two vectors e, e′ ∈ H0 such that 〈e, e′〉 = 0 but〈H0e, e

′〉 �= 0, then, by (3.2),


ϕ0(z) = 〈S − X(Q− z)−1X)e, e′〉〈H0e, e

′〉 ,

ϕ1(z) = 〈(S − z − X(Q− z)−1X − ϕ0(z)H0)e, e〉〈e, e〉

(3.3)

for any z ∈ ρ(Q). These formulas give the required unique analytic continuation. ✷By this corollary, without loss of generality, we can assume that ϕ0(z) and ϕ1(z)

are defined on ρ(Q) by (3.3).

DEFINITION 3.5. If H is spectrally similar to H0, then the set

E(H,H0) = {z ∈ C : z /∈ ρ(Q) or ϕ0(z) = 0}is called the exceptional set of the operators H and H0.

THEOREM 3.6. Let H be spectrally similar to H0 on H0 and z /∈ E(H,H0).Then

(1) R(z) ∈ ρ(H0) if and only if z ∈ ρ(H).(2) R(z) is an eigenvalue of H0 if and only if z is an eigenvalue of H . Moreover,

there is a one-to-one map

f0 �→ f = f0 − (Q − z)−1Xf0

from the eigenspace of H0 corresponding to R(z) onto the eigenspace of Hcorresponding to z.

Proof of (1). Suppose z ∈ ρ(H). Then the operator, ϕ0(z)H0 − ϕ1(z) has abounded inverse, as it follows from the proof of Lemma 3.3 and corollary follow-ing it. Hence R(z) ∈ ρ(H0) by the definition of R(z), see Definition 2.5, sinceϕ0(z) �= 0.

Suppose now that z /∈ ρ(H). Then there exists a sequence {gn}∞n=1 of elements

of H such that ‖gn‖ = 1 and (H − z)gn → 0 as n → ∞. That is

(Q− z)P1gn + XP0gn → 0 (3.4)

and

(S − z)P0gn + XP1gn → 0 (3.5)

as n → ∞. Since z /∈ ρ(Q), relation (3.4) implies that P1gn + (Q− z)−1XP0gn→ 0 and so P0gn �→ 0. At the same time, by (3.4) and (3.5), (S − z − X(Q −z)−1X)P0gn → 0 as n → ∞. Therefore, R(z) /∈ ρ(H0) by Lemma 3.3. ✷

Proof of (2). Suppose that R(z) is an eigenvalue of H0 and f0 is acorresponding eigenvector. Then, by the definition of R(z), ϕ0(z)H0f0−ϕ1(z)f0 =0 and so, by (3.2),

zf0 = (S − X(Q− z)−1X)f0.


Define f1 = −(Q−z)−1Xf0 and f = f0+f1, then P0HP1f = −X(Q−z)−1Xf0.Hence

(P0HP0 + P0HP1)f = S − X(Q − z)−1Xf0 = zf0. (3.6)

Also we have (Q− z)f1 = −Xf0 and so

(P1HP1 + P1HP0)f = Qf1 + Xf0 = zf1. (3.7)

Combining relations (3.6) and (3.7), we get

Hf = (P1HP1 + P1HP0 + P0HP0 + P0HP1)f = zf1 + zf0 = zf.

Therefore the required map is well defined. It is, obviously, one-to-one.Suppose now that z is an eigenvalue of H and f is a corresponding eigenvector.

Define f0 = P0f, f1 = P1f . Then, clearly,

zf0 = Sf0 + Xf1, zf1 = Qf1 + Xf0.

Therefore f1 = −(Q−z)−1Xf0. On the one hand, this implies zf0 = Sf0 −X(Q−z)−1Xf0, and so H0f0 = R(z)f0 by (3.2). On the other hand, f = f0 + f1 =f0 − (Q− z)−1Xf0. Therefore, the map under consideration is onto. ✷PROPOSITION 3.7. If H is spectrally similar to H0 on H0 then S = aH0 + bI0

for some a, b ∈ C, and for any k � 0 there are coefficients ak, bk ∈ C such that

XQkX = akH0 + bkI0. (3.8)

If for any k � 0 there are coefficients ak, bk, ck ∈ C such that

XQkX = akH0 + bkI0 + ckS, (3.9)

then H is spectrally similar to H0 on H0.Proof. Suppose there exists a continuous linear functional λ on the space B(H0)

of bounded operators on H0 such that λ(H0) = λ(I0) = 0 but λ(S) �= 0. Then (3.2)implies that λ(S − X(Q − z)−1X) = 0. This is impossible since ‖X(Q − z)−1X‖→ 0 as |z| → ∞. Therefore, such λ does not exist. By the Hahn–Banach theorem,S belongs to the linear span of H0 and I0.

If |z| > ‖Q‖ then X(Q − z)−1X = ∑∞k=0 z

−k−1XQkX. Hence, the resultfollows if one considers the expansion of the right-hand side of (3.2) into Laurentseries. ✷

The next proposition describes a few examples of trivial situations when wehave spectral similarity.

PROPOSITION 3.8. (1) Let H = H0 ⊕ H1 be an orthogonal sum of operatorsHi: Hi → Hi , i = 0, 1. Then H is spectrally similar to H0 on H0 if and only if


H0 = aH0 + b for some a, b ∈ C. In this case Equation (2.1) holds with ϕ0(z) = a

and ϕ1(z) = z − b.(2) Suppose H is spectrally similar to H0 on H0. Then ϕ0(z) = a and ϕ1(z) =

z − b if and only if XQkX = 0 for any k � 0.(3) If S = aH0 + b and XQkX = 0 for any k � 0 then H is spectrally similar

to H0.Proof. (1) We see that in this case Equation (2.1) has the form (H0 − z)−1 =

(ϕ0(z)H0 − ϕ1(z))−1 or H0 − z = ϕ0(z)H0 − ϕ1(z). The statement follows since

H0 does not depend on z.(2) Note that for |z| > r0, r0 is large enough, (3.2) is equivalent to

S − z +∞∑k=0

z−k−1XQkX = ϕ0(z)H0 − ϕ1(z). (3.10)

If (2.1) holds with ϕ0(z) = a and ϕ1(z) = z − b, then

S +∞∑k=0

z−k−1XQkX = aH0 + b

by Lemma 3.3 for |z| > r0. Therefore XQkX = 0 for any k � 0.Conversely, if XQkX = 0 for any k � 0 then, by (3.10),

S − z = ϕ0(z)H0 − ϕ1(z), |z| > r0.

Hence, ϕ0(z) = a and ϕ1(z) = z − b (see Remark 2.2).(3) Follows from Lemma 3.3 and (3.10). ✷

EXAMPLE 3.9. Let operators H and H have the following block structure withrespect to the decomposition H = H0 ⊕ H1

H =(aH0 + bI0 B

0 C

), H =

(aH0 + bI0 0

B C

).

Then both H and H are spectrally similar to H0 with functions ϕ0(z) = a andϕ1(z) = z − b.

The next lemma allows us to construct a pair of spectrally similar operatorsfrom a given family of spectrally similar operators. It will play an important rolein the next section. It allows us, in particular, to prove the existence of spectralsimilarity when a graph is not symmetric as in Example 4.8.

For each α ∈ A, let Hα be a closed subspace of H , Hα = Hα0 ⊕ Hα

1 , Pαi be

the orthogonal projector onto Hαi , i = 0, 1. Also assume that Pα

i Pβ

1 = 0 if α �= β,i = 0, 1. Then P1 = ∑

α∈A Pα1 is the orthogonal projector onto H1 = ⊕

α∈A Hα1 .

We define H0 as the orthogonal complement to H1.


For α ∈ A let Hα and Hα0 be operators on Hα and Hα

0 respectively. Supposethat for given functions ϕ1(z) and ϕ0(z) defined on

⋂α∈A ρ(Qα), we have

Sα − z − Xα(Qα − z)−1Xα = ϕ0(z)Hα0 − ϕ1(z)P

α0

for each α where Sα , Xα , Xα, Qα are defined in the same way as S, X, X, Q inLemma 3.3. This implies that each Hα is spectrally similar to Hα

0 with functionsϕ0(z) and ϕ1(z) which do not depend on α.

LEMMA 3.10. Let for families of operators {Lα}α∈A, {Rα}α∈A we have that P0=∑α∈A LαP α

0 Rα and for each α, P1L

α = RαP1 = Pα1 , P0L

α = LαP α0 , RαP0 =

Pα0 R

α. Then the operators H = ∑α∈A LαHαRα and H0 = ∑

α∈A LαHα0 R

α arespectrally similar with functions ϕ0(z), ϕ1(z).

Proof. We have

SP0 = P0

∑α∈A

LαHαRαP0 =∑α∈A

LαP α0 H

αPα0 R

α =∑α∈A

LαSαRα.

Also

P0HPα1 =

∑β∈A

P0LβHβRβP α

1 =∑β∈A

P0LβHβRβP1P

α1

= P0LαHαP α

1 = LαP α0 H

αPα1 = LαXαP α

1 ,

and similarly Pα1 HP0 = Pα

1 HαPα

0 Rα = XαPα

0 Rα. In addition, on H1

(Q− z)−1 =(∑α∈A

P1LαHαRαP1 − z

)−1

=(∑α∈A

Pα1 H

αPα1 − z

)−1

=∑α∈A

(P α1 H

αPα1 − z)−1Pα

1 =∑α∈A

(Qα − z)−1Pα1 .

Thus

S − X(Q − z)−1X

=∑α∈A

(LαSαRα + LαXα(Qα − z)−1XαRα)

=∑α∈A

(z − ϕ1(z))LαP α

0 Rα + ϕ0(z)L

αHα0 R

α

= (z − ϕ1(z))P0 + ϕ0(z)H0. ✷

4. Symmetric Graphs and Spectral Similarity of Laplacians

For a graph G we denote by V (G) and E(G) the set of its vertices and edges,respectively. By +(V ) we denote the linear space of functions on V . We always


assume that a graph is locally finite, that is each vertex is contained in a finitenumber of edges. A complete graph is a graph which has one edge between anytwo vertices, and does not have any loops or multiple edges. A graph isomorphismis a bijective map from one graph to another which preserves the graph structure(vertices are mapped into vertices, edges are mapped into edges, and the verticescontained in an edge are mapped into the vertices contained in the image of thisedge). A graph automorphism is an isomorphism onto itself.

DEFINITION 4.1. Let G be a graph and V0 ⊆ V (G). We say that G is sym-metric with respect to V0 if any bijection σ : V0 → V0 can be extended to agraph automorphism ψσ : G → G. We denote the set of these automorphisms by-(G,V0).

For the next lemma we assume that the inner product on H = +(V (G)) isinvariant under the action of the symmetries. If V0 ⊆ V (G) then we define H0 tobe a subset of H of the functions vanishing on V (G) \ V0.

LEMMA 4.2. Let G be a graph symmetric with respect to V0 ⊆ V (G) and H0 bean operator on H0 invariant under any permutation σ : V0 → V0. If an operatorH on H is invariant under -(G,V0) then H is spectrally similar to H0.

The proof of this lemma follows easily from Lemma 4.3.

LEMMA 4.3. Suppose for a family of operators {Tσ }σ∈- on H the following twoassumptions hold:

(1) An operator H and an orthogonal projector P0 commute with each Tσ .(2) There exists an operator H0 on H0 = ImP0 such that an operator H0 on H0

commutes with each Tσ if and only if H0 = aH0 + b for some a, b ∈ C.

Then H and H0 are spectrally similar.Proof. Let the operators S,X, X,Q be defined as in the Lemma 3.3. Assump-

tion (1) implies that, for any z �∈ ρ(Q), operator S − z− X(Q− z)−1X commuteswith each Tσ . Then assumption (2) implies that for a fixed z there are two complexnumbers, say ϕ0(z) and ϕ1(z), such that S− z− X(Q− z)−1X = ϕ0(z)H0 −ϕ1(z).Hence, H and H0 are spectrally similar by Lemma 3.3. ✷

Remark 4.4. Indeed, the lemma above is used in a situation when {Tσ }σ∈S is arepresentation of a group of symmetries. The conditions (1) and (2) can be writtenin the language of representation theory, but we will not use it here.

In the next definition and throughout this paper, ‘Laplacian’ is always a discretedifference operator.

DEFINITION 4.5. The graph or probabilistic Laplacian of a function f ∈+(V (G)) is defined by


/Gf (x) = −f (x) + 1

deg(x)

∑(x,y)∈E(G)

f (y), (4.1)

where deg(x) is the degree of the vertex x. The Markov operator (generator of thesimple random walk) is defined by

/Mf (x) = 1

deg(x)

∑(x,y)∈E(G)

f (y) (4.2)

and the adjacency matrix (combinatorial) Laplacian is defined by

/Af (x) = − deg(x)f (x) +∑

(x,y)∈E(G)f (y). (4.3)

It is easy to see that the graph Laplacian and the Markov operator are boundedand symmetric with respect to the graph inner product

〈f, g〉G =∑

x∈V (G)f (x)g(x) deg(x). (4.4)

The Hilbert space with this inner product will be denoted by H(G) and the corre-sponding norm ‖ · ‖G. The adjacency matrix Laplacian is symmetric in +2(V (G))

and may be unbounded on an infinite graph. Note that the graph Laplacian is theMarkov operator minus the identity operator and so any information on the firstone is easy to translate into information on the second one. If the graph is regular,as in the case of a group, then the adjacency matrix Laplacian is a multiple of thegraph Laplacian, and the same is true about its spectrum.

While Lemma 4.2 gives only sufficient conditions for spectral similarity, thenext simple lemma gives a weaker necessary condition (but strong enough to beapplicable to an example in Section 6).

For the next lemma, we assume that the inner product on H = +(V (G)) is oneof the two defined above. If V0 ⊆ V (G) then again we define H0 to be a subset ofH of the functions vanishing on V (G) \ V0.

LEMMA 4.6. Let G be a locally finite graph, and G0 be a finite complete graphwith V (G0) = V0 ⊆ V (G). Let H and H0 be Laplacians on G and G0 respectively(defined in Definition 4.5). If H is spectrally similar to H0, then distG(a, b) =distG(a, c) for any three distinct points a, b, c ∈ V0, where distG is the usual graphdistance in G.

Proof. First note that all the nondiagonal entries of the matrix of H0 are strictlypositive. Suppose that distG(a, b) > distG(a, c). If distG(a, c) = 1 then somenondiagonal entries of the matrix of S are positive while some are zero, whichcontradicts (3.2). If distG(a, c) � 2 then consider the term in the left-hand sideseries in (3.10) corresponding to k = distG(a, c) − 2. Then again, some nondi-agonal entries of the matrix of XQkX are positive while some are zero whichcontradicts (3.10) for |z| large enough. ✷


Let G be a graph symmetric with respect to V0 ⊆ V (G) and G0 be a completegraph with V (G0) = V0. By Lemma 4.2, /G and /G0 are spectrally similar withsome functions ϕ1(z) and ϕ0(z). If A is a set then G × A is a naturally definedgraph with the set of vertices V (G)× A and the set of edges E(G)× A. Let ∼ bean equivalence relation on the set V0 × A such that each element is equivalent to afinite set of elements. Let G = {G × A}/∼. The graph G × A can be consideredas a disjoint union of |A| copies of G and G is a union of |A| copies of G joined atthe equivalent vertices. Similarly we define G0 = {G0 × A}/∼. The Hilbert spaceunder consideration is H = H(G).

LEMMA 4.7. The graph Laplacian on G is spectrally similar to the graph Lapla-cian on G0 with the same functions ϕ1(z) and ϕ0(z).

Proof. Let H = H(G × A). Note that we can identify H with the subspace{f ∈ H : f (x) = f (y) if x ∼ y} of H , and the norms coincide by (4.4). DenoteV α = V (G × α) and V α

0 = V (G0 × α). Let for each α ∈ A we define Hα =H ∩ +(V α) and Hα

0 = H ∩ +(V α0 ). Let Hα and Hα

0 , α ∈ A, be the correspondingLaplacians on Hα and Hα

0 , respectively, that is Hα = /G×α and Hα0 = /G0×α .

Then Hα and Hα0 are spectrally similar for each α ∈ A with ϕ1(z) and ϕ0(z) by

Lemma 4.2 (clearly ϕ1(z), ϕ0(z) and ρ(Qα) do not depend on α).In order to apply Lemma 3.10 we need to define families of operators {Lα}α∈A,

{Rα}α∈A with the required properties. We define Rα = Pα and Lα = PHPα . Then

H = ∑α∈A LαHαRα = /G, H0 = ∑

α∈A LαHα0 R

α = /G0and the conditions of

Lemma 3.10 are easy to verify. ✷EXAMPLE 4.8. Let G0 be any graph and G be a graph obtained from G0 bysubstituting each edge with two consecutive edges. Then by this proposition, thegraph Laplacian on G is spectrally similar to the graph Laplacian on G0 withϕ1(z) = 1 + z − 1/(z + 1), ϕ0(z) = 1/2(z + 1) and R(z) = 2z2 + 4z (seeExample 2.4).

More generally, let G0 and G0 be any graphs, and suppose that G0 is symmetricwith respect to a two-point set V0 = {v1, v2}. Let G be a graph obtained from G0

by substituting each edge (x1, x2) with a copy of G0 in such a way that, after thesubstitution, v1 coincides with x1 and v2 coincides with x2. Then by the propositionabove the graph Laplacian on G is spectrally similar to the graph Laplacian on G0.Two-point self-similar graphs defined in [15] fall into this example.

We emphasize that in this example the graphs G may have no symmetries in anysense. However the spectral similarity holds due to symmetries in the ‘substituting’graph G0.

LEMMA 4.9. Let G and G0 be finite graphs with V (G0) = V0 � V (G), andsuppose the graph Laplacians /G and /G0 are spectrally similar. If G is connectedthen R(0) = 0. If G0 is also connected then, in addition, R′(0) > 1.

Proof. First note that 0 is not in the spectrum of Q because G is connectedand finite (even if the graphs are infinite, in the examples we consider Q can be


represented as a product of identical finite-dimensional matrices and so 0 is not inthe spectrum of Q).

We have /G1 = 0 where 1 stands for a function on V (G) that is identicallyone. Then by (3.1) we have XP01 + QP11 = 0 and so P11 = −Q−1XP01. Alsoby (3.1) we have XP11 + SP01 = 0, which implies

SP01 − XQ−1XP01 = 0.

By (3.2) we have ϕ1(0) = 0 since /G0P01 = 0. It is known that Q−1 has nonpos-itive matrix entries (see Lemma 2.7.1 in [8]) and XQ−1X has at least one strictlynegative nondiagonal entry since G is connected. Hence, ϕ0(0) �= 0 that impliesR(0) = 0.

By the same argument every diagonal matrix entry of XQ−1X is strictly nega-tive that implies 0 < ϕ0(0) < 1. By differentiating (3.2), we have

−I0 − XQ−2X = ϕ′0(z)H0 − ϕ′

1(z)I0.

Then ϕ′0(0) � 0 since at least some nondiagonal entries of H0 are positive if G0 is

connected, and all the entries if XQ−2X are nonnegative. Hence, ϕ′1(0) � 1 by the

comparison of the diagonal matrix entries. Thus R′(0) = ϕ′1(0)/ϕ0(0) > 1. ✷

5. Symmetric Self-Similar Graphs and Self-Similar Spectrum

DEFINITION 5.1. An M-point model graph G is a finite connected graph sym-metric (Definition 4.1) with respect to an M point set ∂G = V0 ⊂ V (G) if

(1) there are complete graphs Gs of M vertices such that G = ⋃s∈S G

s where Sis a finite set and |S| � M � 2;

(2) we have Gs ∩ Gs ′ = V (Gs) ∩ V (Gs ′) for all distinct s, s′ ∈ S, and thisintersection is either empty or has only one point;

(3) we have |Gs ∩ ∂G| � 1 for any s ∈ S;(4) any bijection σ : ∂G → ∂G has an extention (see Definition 4.1) to a graph

automorphism ψσ : G → G, such that ψσGs = Gσs for a bijection σ : S → S.

DEFINITION 5.2. If an M-point model graph G is given then we define the cor-responding self-similar symmetric sequence of finite graphs {Gn}∞

n=0 inductively asfollows:

(1) G0 is a complete graph of M vertices with ∂G0 = V (G0);(2) if ∂Gn ⊂ V (Gn) is an M point set, then Gn+1 is obtained by substituting each

Gs in G by a copy Gsn of Gn, so that ∂Gs = V (Gs) is substituted by ∂Gs

n;(3) ∂Gn+1 is defined as ∂G after this substitution.

For this self-similar sequence of finite graphs {Gn}∞n=0 there are bijections Bn: ∂G

→ ∂Gn ⊂ V (Gn) and graph monomorphisms Fns : Gn → Gn+1, s ∈ S, such that

for all n � 0


(1) each Fns is a graph isomorphism from Gn to Gs

n and Gn+1 = ⋃s∈S Gs

n;(2) Gs

n ∩ Gs ′n = ∂Gs

n ∩ ∂Gs ′n for all s, s′ ∈ S, s �= s′, where ∂Gs

n = Fns (∂Gn);

(3) for n � 1, we have Bn+1(x) = Fns (B

n(x)) if x ∈ ∂G ∩Gs ;(4) for all s, s′ ∈ S, s �= s′, we have Fn

s (x) = Fns ′(x

′) if and only if there arex, x′ ∈ ∂G such that x = Bn(x), x′ = Bn(x

′) and F 0s B

−10 (x) = F 0

s ′B−10 (x′).

Note that G1 can be naturally identified with G in such a way that ∂G1 is identifiedwith ∂G and B1(x) = x for all x ∈ ∂G.

LEMMA 5.3. Each Gn is symmetric with respect to ∂Gn.Proof. First note note that bijections σ of V0 = ∂G are in one-to-one correspon-

dence with bijections σn of ∂Gn via σn = BnσB−1n . Let σ , σ be as in Definition 5.1.

We have that G0 is always symmetric with respect to ∂G0. For n � 0 we defineψσn+1 : Gn+1 → Gn+1 by ψσn+1(x) = Fn

σs(Fns )

−1(x) if x ∈ Fns (Gn). Then ψσn are

the required well-defined graph automorphisms. ✷One can see that for a given M-point model graph there is a unique self-similar

symmetric sequence of finite graphs up to a natural isomorphism. This sequencecan be constructed inductively by Lemma 5.4. The proof of the lemma is elemen-tary.

LEMMA 5.4. For each n � 0 the graph Gn+1 is isomorphic to a graph Gn ×S/∼(see Lemma 4.7) where the relation ∼ on V (Gn) × S is defined as follows: if(x, s), (x′, s′) ∈ V (Gn) × S then (x, s) ∼ (x′, s′) if and only if there are x, x′ ∈V0 = ∂G such that x = Bn(x), x′ = Bn(x

′) and F 0s B

−10 (x) = F 0

s ′B−10 (x′).

Moreover, each maps F sn is the map x �→ (x, s) modulo ∼.

DEFINITION 5.5. Suppose anM-point model graph and a sequence K = {kn}∞n=0,

kn ∈ S, are fixed. If Gn ⊂ Gn+1 for each n � 0, and each Fnkn

is the identity (in-clusion) map then the corresponding self-similar infinite graph is G∞ = ⋃∞

n=0 Gn.We define ∂G∞ = ⋂∞

n=0 ∂Gn.

Clearly, for any given M-point model graph and a sequence K there exists aunique self-similar infinite graph (up to isomorphism). At the same time isomor-phic self-similar infinite graphs may correspond to different model graphs, and fordifferent sequences K even if the M-point model graph is the same. The graph G∞is always of polynomial growth.

DEFINITION 5.6. Expansion maps Kn: V (Gn) → V (Gn+1) are defined induc-tively by Kn(x) = Fn

s Kn−1(Fn−1s )−1(x) if x ∈ V (Gs

n−1) = Fn−1s (V (Gn−1)).

Although such s may not be unique, the expansion map does not depend on aparticular choice of s by Definition 5.2.

For a self-similar infinite graph an expansion map K∞: V (G∞) → V (G∞) isdefined by K∞|V (Gn) = Kn.


LEMMA 5.7. The expansion map Kn: V (Gn) → V (Gn+1) induces an isometryUn: H(Gn) → H(Gn+1) defined by Unf (x) = 0 if x /∈ ImKn and Unf (x) =f (K−1

n (x)) otherwise.Similarly, K∞ induces an isometry U∞: H(G∞) → yH(G∞).Proof. By the definition, Kn is an injection and so is Un. It is isometric since by

a simple induction we have degKn(x) = deg x for any x ∈ V (Gn). ✷By Lemmas 4.7 and 5.4, the graph Laplacian on Gn is spectrally similar to that

on G0. The next theorem gives a more useful spectral similarity result.

THEOREM 5.8. Let /n = /Gnand /∞ = /G∞ be the graph Laplacians on Gn

and G∞ respectively for a self-similar symmetric sequence of finite graphs. Then

(1) For any n � 0, the operator /n+1 is spectrally similar to /n with the isometryUn and rational functions ϕ0(z) and ϕ1(z) which do not depend on n. Theexceptional set (see Definition 3.5) E = E(/n+1,/n) = E(/1,/0) also doesnot depend on n.

(2) Let Dn = ⋃nm=0 R

−m(E ∪ σ (/0)), where R−m is the preimage of order m

under R(z) = ϕ1(z)/ϕ0(z). Then σ (/n) ⊆ Dn, where σ (·) is the spectrum ofan operator.

(3) The operator /∞ is spectrally self-similar with the isometry U∞, rationalfunctions ϕ0(z) and ϕ1(z) and the exceptional set E .

J(R) ⊆ σ (/∞) ⊆ J(R) ∪ D∞,

where D∞ = ⋃∞n=0 Dn and J(R) is the Julia set of the rational function R.

Remark 5.9. In particular, the Julia set of R is real. Moreover, one can showeasily that σ (/∞) ⊆ [−2, 0], and so J(R) ⊆ [−2, 0]. Note also that D∞ \ J(R)contains only isolated points, if any.

By this theorem many eigenvalues and eigenfunctions of /n+1 are ‘offsprings’of those of /n via maps defined in Theorem 3.6. (so-called ‘spectral decimation’).However a significant number of eigenfunctions might not fall into this category(see [4, 14, 19, 23]).

Proof. In order to prove (1) we apply Lemma 4.7 where we define G = G1 andA = Sn. The role of V0 will be played by ∂G1. Let the relation ∼ on ∂G1 × Abe as follows: if x, x′ ∈ ∂G1, α = sn · · · s1, α′ = s′

n · · · s′1 then (x, α) ∼ (x′, α′)

if and only if Fnsn

· · ·F 1s1(x) = Fn

s ′n · · ·F 1s ′1(x′). By induction we have that the graph

G = {G1 × A}/∼ is isomorphic to Gn+1. Similarly we define G0 = {G0 × A}/∼,where G0 is (temporarily, for this part of the proof only) identified with the com-plete graph over the set of vertices ∂G1. Note that ∂G1 and V0 are in one-to-onecorrespondence via B1, and so G0 is naturally isomorphic to the complete graphover ∂G1 via the map induced by B1. Then G0 is isomorphic to Gn. Moreover,


the expansion map Kn is a bijection from V (Gn) to V (G0) such that the degree ofdegGn−1

x = degG0Kn(x).

The proof of (2) follows from (1) and Theorem 3.6 by induction.Clearly, 0 ∈ σ (/∞) but 0 is not an eigenvalue. Hence, 0 is a point of spectrum

which is not isolated. Therefore, by Theorem 3.6, for any ε > 0 and n � 1 thereexists z ∈ σ (/∞) such that |z| < ε and R−n(z) ⊆ σ (/∞). Since σ (/∞) is aclosed set, this implies

Closure

(⋃n�1

R−n(0)

)⊆ σ (/∞).

We have by Lemma 4.9 that the point 0 is a repulsive fixed point of the rationalfunction R, and so 0 ∈ J(R) by Theorem 2.2 in [3]. Then

Closure

(⋃n�1

R−n(0)

)= J(R)

by Corollary 2.2 in [3].By Theorem 3.6, σ (/m) ⊆ D∞ for any m � 1. Therefore, σ (/∞) ⊆ Closure

(D∞) since /m converges strongly to /∞. We have

Closure(D∞) = J(R) ∪ D∞

by [3]. ✷

6. Examples

A large class of infinite graphs with spectral self-similarity is given by two-pointself-similar graphs in [15]. Any nested fractal [13] with two or three essential fixedpoints gives rise to a self-similar symmetric sequence of finite graphs and thusto spectral similarity. The same is true for a nested fractal which has its essentialfixed points in general position; that is, for the essential fixed points P0, . . . , Pk thevectors {[−−−→

P0, Pi]}ki=1 are linearly independent.Below we give a few concrete examples. All the examples here have a common

feature that the infinite self-similar graphs can be realized as infinite self-similarlattices in R2. Moreover, the corresponding fractals can also be realized as linear

Figure 1. Sierpinski gasket self-similar sequence of finite graphs.


Figure 2. Modified Koch self-similar sequence of finite graphs.

Figure 3. Vicsek set self-similar sequence of finite graphs.

self-similar fractals in R2. This means that these fractals are the limit sets of aniteration function system of contracting similitudes of R2. Then the maps Fn

s areexpanding similitudes that are inverse of the just mentioned contractions. Suchlattices are described in detail in [22]. Therefore we will avoid giving precisedefinitions since they are either obvious, or are given in the references provided.Note that the model graph G is the same as G1.

EXAMPLE 6.1. Sierpinski gasket self-similar sequence of finite graphs (Figure 1)[2, 4, 16, 17, 19, 20, 23]. Historically this is the first example of the spectral self-similarity we are interested in. Here R(z) = z(4z + 5).

EXAMPLE 6.2. Modified Koch self-similar sequence of finite graphs (Figure 2)[14]. In this example

R(z) = 2z(z − 1)(3z − 4)(3z − 5)

2z − 3.

EXAMPLE 6.3. Vicsek set self-similar sequence of finite graphs (Figure 3). Notethat these graphs are symmetric in the sense of Definition 4.1 although the givenR2-embedding does not have all the required symmetries (the essential fixed pointsare not in general position). By [7, 21] we have R(z) = z(6z + 3)(6z + 5).

EXAMPLE 6.4. Lindstrøm snowflake self-similar sequence of finite graphs (Fig-ure 4) [13]. This example is different from the previous ones in that the spectralsimilarity does not hold. One can see that the pairwise distances between theboundary points are not the same, and so the necessary condition of Lemma 4.6is not satisfied.


Figure 4. Lindstrøm snowflake self-similar sequence of finite graphs.

Acknowledgements

The authors thank M. Barlow, L. Bartholdi, M. Denker, P. Diaconis, R. Grigorchuk,J. Hubbard, J. Kigami, B. Krön, M. Lapidus, T. Lindstrøm, V. Metz, C. Sabot,T. Shima, B. Simon, R. Strichartz, E. Teufl, M. Zähle and A. Zuk for interestingand fruitful discussions related to this work.

References

1. Bartholdi, L. and Grigorchuk, R. I.: On the spectrum of Hecke type operators related to somefractal groups, Mat. Inst. Steklova, Din. Sist., Avtom. i Beskon. Gruppy 231 (2000), 5–45;translation in Proc. Steklov Inst. Math. 2000, No. 4 (231), 1–41.

2. Béllissard, J.: Renormalization group analysis and quasicrystals, In: Ideas and Methods inQuantum and Statistical Physics (Oslo, 1988), Cambridge Univ. Press, Cambridge, 1992,pp. 118–148.

3. Brolin, H.: Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144.4. Fukushima, M. and Shima, T.: On a spectral analysis for the Sierpinski gasket, Potential Anal.

1 (1992), 1–35.5. Grigorchuk, R. I., Linnell, P., Schick, T. and Zuk, A.: On a question of Atiyah, C.R. Acad. Sci.

Paris Sér. I Math. 331(9) (2000), 663–668.6. Grigorchuk, R. I. and Zuk, A.: The lamplighter group as a group generated by a 2-state

automaton, and its spectrum, Geom. Dedicata 100 (2003), 000–000.7. Grishin, S. A.: On a spectral analysis of the nested fractal sets, Diploma work (MS thesis),

Moscow State University (1991).8. Kigami, J.: Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335 (1993),

721–755.9. Kigami, J.: Analysis on Fractals, Cambridge Univ. Press, Cambridge, 2001.

10. Krön, B.: Green functions of self-similar graphs and bounds for the spectrum of the Laplacian,Ann. Inst. Fourier 52(6) (2002), 1875–1900.


11. Krön, B. and Teufl, E.: Asymptotics of the transition probabilities of the simple random walkon self-similar graphs, Preprint.

12. Kumagai, T.: Regularity, closedness and spectral dimensions of the Dirichlet forms on P.C.F.self-similar sets, J. Math. Kyoto Univ. 33 (1993), 765–786.

13. Lindstrøm, T.: Brownian motion on nested fractals, Mem. Amer. Math. Soc. 420 (1989).14. Malozemov, L.: Spectral theory of the differential Laplacian on the modified Koch curve,

Geometry of the Spectrum (Seattle, WA, 1993), Contemp. Math. 173, Amer. Math. Soc.,Providence, RI, 1994, pp. 193–224.

15. Malozemov, L. and Teplyaev, A.: Pure point spectrum of the Laplacians on fractal graphs, J.Funct. Anal. 129 (1994), 390–405.

16. Rammal, R.: Spectrum of harmonic excitations on fractals, J. Phys. 45 (1984), 191–206.17. Rammal, R. and Toulouse, G.: Random walks on fractal structures and percolation clusters, J.

Phys. Lett. 44 (1983), L13–L22.18. Sabot, C.: Spectral properties of hierarchical lattices and iteration of rational maps, Preprint,

2001.19. Shima, T.: The eigenvalue problem for the Laplacian on the Sierpinski gasket, In: Asymptotic

Problems in Probability Theory: Stochastic Models and Diffusions on Fractals (Sanda/Kyoto,1990), Pitman Res. Notes Math. Ser. 283, Longman, Harlow, 1993, pp. 279–288.

20. Shima, T.: On eigenvalue problems for the random walks on the Sierpinski pre-gaskets, JapanJ. Indust. Appl. Math. 8 (1991), 127–141.

21. Shima, T.: On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust.Appl. Math. 13 (1996), 1–23.

22. Strichartz, R. S.: Fractals in the large, Canad. J. Math. 50 (1998), 638–657.23. Teplyaev, A.: Spectral analysis on infinite Sierpinski gaskets, J. Funct. Anal. 159 (1998), 537–

567.


219

A Particle in a Magnetic Field of an InfiniteRectilinear Current

D. YAFAEVDepartment of Mathematics, University of Rennes, Campus Beaulieu, 35042, Rennes, France.e-mail: [email protected]

(Received: 30 September 2002)

Abstract. We consider the Schrödinger operator H = (i∇+A)2 in the spaceL2(R3)with a magnetic

potential A created by an infinite rectilinear current. We show that the operator H is absolutelycontinuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then wefind the large-time behavior of solutions exp(−iHt)f of the time dependent Schrödinger equation.Our main observation is that a quantum particle has always a preferable (depending on its charge)direction of propagation along the current. Similar result is true in classical mechanics.

Mathematics Subject Classifications (2000): Primary: 47A40; secondary: 81U05.

Key words: Biot–Savart–Laplace magnetic field, Schrödinger equation, explicit solutions, spectraland scattering theory.

1. Introduction

They are very few examples of explicit solutions of the Schrödinger equationwith a magnetic potential, the case of a constant magnetic field (see, e.g., [3])probably being the single one. Here we consider the magnetic field B(x, y, z)created by an infinite rectilinear current which we suppose to coincide with theaxis z. We assume that the axes (x, y, z) are positively oriented. According to theBiot–Savart–Laplace law (see, e.g., [6])

B(x, y, z) = αr−2(−y, x, 0), r = (x2 + y2)1/2, (1.1)

where |α| is proportional to the current strength and α > 0 (α < 0) if the currentstreams in the positive (negative) direction. The magnetic potential is defined bythe equation

B(x, y, z) = curlA(x, y, z)

and can be chosen as

A(x, y, z) = −α(0, 0, ln r). (1.2)

Thus, the corresponding Schrödinger operator in the space L2(R3) has the form

H = H(γ ) = −∂2x − ∂2

y + (i∂z − γ ln r)2, γ = ec−1α, (1.3)

220 D. YAFAEV

where e is the charge of a quantum particle of the mass 1/2 and c is the speed ofthe light.

Since the magnetic potential (1.2) grows at infinity, the Hamiltonian H doesnot fit to the well elaborated framework of spectral and scattering theory. Actually,we are aware of only one paper [4] (see also the book [1]) on this subject whereit was proven that the essential spectrum of the magnetic Schrödinger operatorcoincides with the positive half-line provided the field vanishes at infinity. Herewe obtain much more advanced information on the operator H and perform inSection 2 its spectral analysis almost explicitly. We show that the operator H isabsolutely continuous, its spectrum has infinite multiplicity and coincides with thepositive half-axis. Then we find in Section 3 the large-time behavior of solutionsexp(−iHt)f of the time dependent Schrödinger equation. Our main observationis that a positively (negatively) charged quantum particle always moves in thedirection of the current (in the opposite direction) and is localized in the orthogonalplane. Actually, somewhat similar results are true in classical mechanics. Since wewere unable to find a solution of the classical problem in the literature, it is givenin Section 4.

2. The Spectrum of the Operator H

Let us make the Fourier transform � = �z in the variable z. Then the operatorH = �H�∗ acts in the space L2(R

2 × R) as

H = −�+ (p + γ ln r)2,

where � is always the Laplacian in the variables (x, y) and p ∈ R is the variabledual to z. Thus,

(Hu)(x, y, p) = (h(p)u)(x, y, p),

where

h(p) = −�+ ln2(eprγ ) (2.1)

acts in the space L2(R2). Clearly, the spectrum of each operator h(p) is positive

and discrete. If we separate variables in the polar coordinates (r, ϕ) and denote byHm ⊂ L2(R

2) the subspace of functions f (r)eimϕ , where f ∈ L2(R+; r dr) andm = 0,±1,±2, . . . is the orbital quantum number, then

L2(R2) =

∞⊕m=−∞

Hm. (2.2)

Every subspace Hm is invariant with respect to the operator h(p). The spectrum ofits restriction

hm(p) = −r−1∂r(r∂r)+m2r−2 + ln2(eprγ ) (2.3)

A PARTICLE IN A MAGNETIC FIELD 221

on Hm consists of positive simple eigenvalues λm,1(p), λm,2(p), . . . which are an-alytic functions of p. We denote by ψm,1(r, p), ψm,2(r, p), . . . the correspondingeigenfunctions which are supposed to be normalized and real.

Quite similarly, if Hm ⊂ L2(R3) is the subspace of functions u(r, z)eimϕ where

u ∈ L2(R+ × R; r dr dz), then

L2(R3) =

∞⊕m=−∞

Hm. (2.4)

Every subspace Hm is invariant with respect to H. We denote by Hm the restric-tion of H on Hm. Actually, decompositions (2.2), (2.4) are needed only to avoidcrossings of different eigenvalues of the operators h(p). It allows us to use alwaysformulas of perturbation theory (see [5]) for simple eigenvalues.

Fixing γ , we often use the parameter a = ep/γ ∈ (0,∞) instead of p. Let usset

K(a) = −a2�+ γ 2 ln2 r, (2.5)

and let w(a),

(w(a)u)(x, y) = au(ax, ay),

be the unitary operator of dilations in the space L2(R2). Then the operator (2.1)

equals

h(p) = w(a)K(a)w∗(a), (2.6)

where, as always, a = ep/γ . We denote by µm,n(a) and φm,n(r, a) eigenvalues andeigenfunctions Km(a) of the restrictions of the operators K(a) on the subspacesHm. It follows from (2.6) that µm,n(a) = λm,n(p) and φm,n(a) = w∗(a)ψm,n(p).Below we usually fix m and omit it from the notation.

The following assertions are quite elementary.

LEMMA 2.1. For every n, we have that µ′n(a) > 0, for all a > 0.

Proof. Applying analytic perturbation theory to the family (2.5), we see that

µ′n(a) = 2a

∫R2

|∇φn(x, y, a)|2 dx dy. (2.7)

This expression is obviously positive since otherwise φn(x, y, a) = const. ✷The next lemma realizes an obvious idea that the spectrum of K(a) converges

as a → 0 to that of the multiplication operator by γ 2 ln2 r, which is continuous andstarts from zero.

LEMMA 2.2. For every n, we have that

lima→0

µn(a) = 0.

222 D. YAFAEV

Proof. Let ε > 0 be arbitrary and δ > 1 be such that γ 2 ln2 δ = ε. Suppose thatfunctions f1, f2, . . . , fn ∈ C∞

0 (δ−1, δ) are obtained from, say, f1, ‖f1‖ = 1, by

shifts and that they are disjointly supported. Set up(x, y) = r−1/2fp(r)eimϕ , then

(Km(a)up, up) = a2∫ ∞

0(|f ′

p(r)|2 + (m2 − 1/4)r−2|fp(r)|2) dr +

+ γ 2∫ ∞

0ln2 r|fp(r)|2 dr.

The first term here tends to zero as a → 0 and the second is bounded by ε. Thus,for sufficiently small a, the operator Km(a) has at least n eigenvalues below 2ε.This implies that µn(a) < 2ε. ✷

Let BR = {x2 + y2 � R2} and SR = {x2 + y2 = R2} be the disc and the circleof radius R. One of possible proofs of the next lemma relies on the Friedrichsinequality

R2∫

BR

|∇u(x, y)|2 dx dy + R

∫SR

|u(x, y)|2 dSR

� c1

∫BR

|u(x, y)|2 dx dy, (2.8)

where R is arbitrary and dSR = R dϕ. This inequality is usually verified first forR = 1 and then one makes the dilation transformation (x, y) �→ (Rx,Ry). Weneed also the standard Sobolev inequality where again the dilation transformationis taken into account:

R

∫SR

|u(x, y)|2 dSR

� R2∫

R2\BR

|∇u(x, y)|2 dx dy + c2

∫R2\BR

|u(x, y)|2 dx dy. (2.9)

LEMMA 2.3. For every n, we have that lima→∞µn(a) = ∞.Proof. It suffices to show, that the infimum of the operator K(a) tends to ∞ as

a → ∞. It follows from inequality (2.8) that

a2∫

BR

|∇u(x, y)|2 dx dy + a2R−1∫

SR

|u(x, y)|2 dSR +

+ γ 2∫

BR

ln2 r|u(x, y)|2 dx dy � c1a2R−2

∫BR

|u(x, y)|2 dx dy. (2.10)

Inequality (2.9) implies that

a2∫

R2\BR

|∇u(x, y)|2 dx dy − a2R−1∫

SR

|u(x, y)|2 dSR +


+ γ 2∫

R2\BR

ln2 r|u(x, y)|2 dx dy

� (γ 2 ln2 R − c2a2R−2)

∫R2\BR

|u(x, y)|2 dx dy. (2.11)

Let us now choose R such that γ 2R2 ln2 R = a2(c1 + c2). Combining estimates(2.10) and (2.11), we see that

(K(a)u, u) � γ 2c1(c1 + c2)−1 ln2 R‖u‖2

and the right-hand side tends to ∞ as a → ∞. ✷Of course, Lemma 2.3 and its proof remain valid for all dimensions and for

arbitrary potentials tending to ∞ at infinity.Let us again fix the orbital quantum numberm. In terms of eigenvalues λn(p) =

λm,n(p) of the operators h(p) = hm(p) Lemmas 2.1–2.3 mean that λ′n(p) > 0 for

all p ∈ R and

limp→−∞ λn(p) = 0, lim

p→∞λn(p) = ∞ (2.12)

if γ > 0. Note that eigenvalues of the operators (2.1) satisfy the identity

λn(p, γ ) = λn(−p,−γ ),so we may assume without loss of generality that γ > 0.

Let ,n be multiplication operator by the function λn(p) in the space L2(R).It follows from the results on the eigenvalues λn(p) that the spectrum of ,n isabsolutely continuous, simple and coincides with the positive half axis. Let usintroduce a unitary mapping

-: L2(R+ × R; r dr dz) →∞⊕n=1

L2(R)

by the formula

(-f )n(p) =∫ ∞

0f (r, p)ψn(r, p)r

1/2 dr.

Then

-�H�∗-∗ =∞⊕n=1

,n (2.13)

(of course, H = Hm and ,n = ,m,n), and we obtain the following theorem:

THEOREM 2.4. The spectra of all operators Hm and H are absolutely continu-ous, have infinite multiplicity and coincide with the positive half axis.

224 D. YAFAEV

As a by-product of our considerations, we have constructed a complete set ofeigenfunctions of the operator H. They are parametrized by the orbital quantumnumber m, the momentum p in the direction of the z-axis and the number n ofan eigenvalue λm,n(p) of the operator hm(p) defined by (2.3) on the subspaceL2(R+; r dr). Thus, if we set

um,n,p(r, z, ϕ) = eipzeimϕψm,n(r, p),

then

Hum,n,p = λm,n(p)um,n,p.

3. Time Evolution

Explicit formulas obtained in the previous section allow us to find the asymptoticsfor large t of solutions u(t) = exp(−iHt)u0 of the time-dependent Schrödingerequation. It follows from (1.3) that

exp(−iH(γ )t)u0 = exp(iH(−γ )t)u0.

Therefore it suffices to consider the case γ > 0. Moreover, on every subspace Hm

with a fixed orbital quantum number m, the problem reduces to the asymptotics ofthe function u(t) = exp(−iHmt)u0.

Suppose that

(�u0)(r, p) = ψn(r, p)f (p), (3.1)

where f ∈ C∞0 (R). Then it follows from formula (2.13) that

u(r, z, t) = (2π)−1/2∫ ∞

−∞eipz−iλn(p)tψn(r, p)f (p) dp. (3.2)

The stationary points of this integral are determined by the equation

z = λ′n(p)t. (3.3)

Since, by Lemma 2.1, λ′n(p) > 0 for γ > 0, Equation (3.3) has a solution only

if zt > 0. We need the following information on the eigenvalues µn(a) of theoperator (2.5).

LEMMA 3.1. For every n, we have that lima→0 aµ′n(a) = 0.

Proof. It follows from Equation (2.7) that aµ′n(a) � 2µn(a). Therefore, it

remains to use Lemma 2.2. ✷The following conjecture is physically quite plausible and is used mainly to

formulate Theorem 3.3 below in a simpler form.


CONJECTURE 3.2. For every n, we have that (aµ′n(a))

′ > 0 for all a > 0 and

lima→∞ aµ

′n(a) = ∞. (3.4)

In terms of eigenvalues λn(p) of the operators h(p), Lemma 3.1 and Conjec-ture 3.2 mean that λ′′

n(p) > 0 for all p ∈ R and

limp→−∞ λ

′n(p) = 0, lim

p→∞λ′n(p) = ∞. (3.5)

Therefore equation λ′n(p) = α has a unique solution pn = νn(α) for every α > 0.

Clearly,

λ′′n(νn(α))ν

′n(α) = 1. (3.6)

Let

�n(α) = νn(α)α − λn(νn(α)), (3.7)

θ(α) = 1 for α > 0, θ(α) = 0 for α < 0 and ±i = e±πi/2. Applying to the integral(3.2) the stationary phase method and taking into account identity (3.6), we findthat

u(r, z, t) = ei�n(z/t)tψn

(r, νn

(z

t

))ν′n

(z

t

)1/2

f

(νn

(z

t

))(it)−1/2θ

(z

t

)+

+ u∞(r, z, t), (3.8)

where

limt→±∞ ‖u∞(·, t)‖ = 0. (3.9)

Note that the norm in the space L2(R+ × R) of the first term in the right-hand sideof (3.8) equals ‖u0‖. The asymptotics (3.8) extends of course to all f ∈ L2(R) andto linear combinations of functions (3.1) over different n. Thus, we have proven

THEOREM 3.3. Assume that Conjecture 3.2 is verified. Suppose γ > 0. Letu(t) = exp(−iHmt)u0 where u0 satisfies (3.1) with f ∈ L2(R). Then the as-ymptotics as t → ±∞ of this function is given by relations (3.8), (3.9) where �n

is the phase function (3.7). Moreover, if f ∈ C∞0 (R) and ∓z > 0, then the function

u(r, z, t) tends to zero faster than any power of |t|−1 as t → ±∞.Conversely, for any g ∈ L2(R+) there exists the function u0, namely

(�u0)(r, p) = ψn(r, λ′n(p))λ

′′n(p)

1/2g(λ′n(p)),

such that u(t) = exp(−iHmt)u0 has the asymptotics as t → ±∞

u(r, z, t) = ei�n(z/t)tψn

(r,z

t

)g

(z

t

)(it)−1/2θ

(z

t

)+ u∞(r, z, t),

where u∞ satisfies (3.9).

226 D. YAFAEV

Formulas (3.8), (3.9) show that a positively (negatively) charged quantum par-ticle always moves in the magnetic field (1.1) in the direction of the current (in theopposite direction), and its motion is essentially free. Note however that for the freemotion the phase in (3.8) would be�(α) = α2/4, whereas for the Hamiltonian Hm

it is determined by the eigenvalues λn(p) (see formula (3.7)). On the contrary, aparticle remains localized in the plane orthogonal to the current.

As was already noted, Conjecture 3.2 is not really essential for Theorem 3.3.Remark first that λ′′

n(p) cannot vanish on an interval. Otherwise, λ′n(p) would be

a constant on the same interval, and hence by analyticity λ′n(p) = const for all

p ∈ R. This contradicts conditions (2.12). If λ′′n(p) < 0 on some interval, this

changes only the phase factor in (3.8). Finally, the condition (3.4), or equivalentlythe second condition (3.5), is required to guarantee that Equation (3.3) has solutionsfor arbitrary large z/t . We emphasize that the assertion that u(r, z, t) ‘lives’ in thehalf-space ±z > 0 for ±t > 0 is true without Conjecture 3.2.

4. Classical Mechanics

Let us consider the motion of a classical particle of mass m = 1/2 and charge ein a magnetic field B(x, y, z). It is natural to study somewhat more general casewhere

A(x, y, z) = (0, 0,A(r)), r = (x2 + y2)1/2,

and A(r) is an arbitrary C2(R+) function such that A(r) = o(r−1) as r → 0 and|A(r)| → ∞ as r → ∞. For such magnetic potentials

B(x, y, z) = A′(r)r−1(y,−x, 0). (4.1)

The force exercised by a magnetic field on a particle with a velocity v at a pointr = (x, y, z) equals ec−1v × B(r) (see [6]). Therefore, the Newton equation readsas

r′′(t) = e0r′(t)× B(r(t)), (4.2)

where e0 = 2ec−1. Clearly, the expression

d|r′(t)|2dt

= 2〈r′(t), r′′(t)〉 = 2e0〈r′(t), r′(t)× B(r(t))〉 = 0

since the vectors r′(t) and r′(t) × B(r(t)) are orthogonal. Therefore, as is wellknown, the kinetic energy

|r′(t)|2 = x′(t)2 + y′(t)2 + z′(t)2 = K2 (4.3)

of a particle in a magnetic field does not depend on time.


For the magnetic field (4.1), Equation (4.2) is equivalent to the equations

x′′(t) = e0z′(t)x(t)A′(r(t))r−1(t), (4.4)

y′′(t) = e0z′(t)y(t)A′(r(t))r−1(t), (4.5)

z′′(t) = −e0(x′(t)x(t) + y′(t)y(t))A′(r(t))r−1(t). (4.6)

It is convenient to rewrite these equations in cylindrical coordinates using theobvious identification (x, y) ↔ x+ iy = reiϕ . Differentiating this identity, we findthat

x′′(t)+ iy′′(t)= (r ′′(t)− ϕ′(t)2r(t) + iϕ′′(t)r(t)+ 2iϕ′(t)r ′(t))eiϕ(t). (4.7)

Multiplying Equation (4.5) by i and taking its sum with Equation (4.4), we see that

x′′(t)+ iy′′(t) = e0z′(t)A′(r(t))eiϕ(t). (4.8)

Comparing the right-hand sides of (4.7) and (4.8), we obtain that

r ′′(t)− ϕ′(t)2r(t) = e0z′(t)A′(r(t)), (4.9)

ϕ′′(t)r(t)+ 2ϕ′(t)r ′(t) = 0. (4.10)

Since, moreover,

x′(t)x(t) + y′(t)y(t) = r ′(t)r(t),

Equation (4.6) is equivalent to

z′′(t) = −e0r′(t)A′(r(t)). (4.11)

Similarly, the conservation law (4.3) means that

r ′(t)2 + r(t)2ϕ′(t)2 + z′(t)2 = K2. (4.12)

Integrating Equations (4.10), (4.11), we find that

ϕ′(t)r(t)2 = σ, σ = ϕ′(0)r(0)2 �= 0, (4.13)

z′(t) = −e0(A(r(t))+ C), C = −e−10 z′(0)− A(r(0)). (4.14)

Plugging these expressions into (4.12), we see that

r ′(t)2 + V (r(t)) = K2, (4.15)

where

V (r) = σ 2r−2 + e20 (A(r)+ C)2. (4.16)

228 D. YAFAEV

Clearly, (4.15) is the equation of one-dimensional motion (see [2]) with the ef-fective potential energy V (r) and the total energy K2. It admits the separation ofvariables and can be integrated by the formula

t = ±∫(K2 − V (r))−1/2 dr. (4.17)

Note that V (r) → ∞ as r → 0 or r → ∞. Let rmin and rmax be the roots of theequation V (r) = K2 (it has exactly two roots for given initial data). It follows from(4.17) that the function r(t) is periodic with period

T = 2∫ rmax

rmin

(K2 − V (r))−1/2 dr. (4.18)

One can imagine, for example, that on the period [0, T ] the function r(t) increasesmonotonically from rmin to rmax and then decreases from rmax to rmin. Having foundr(t), we determine ϕ(t) from Equation (4.13):

ϕ(t) = ϕ(0)+ σ

∫ t

0r(s)−2 ds. (4.19)

To find a motion in the variable z, we use Equation (4.14) which gives

z(t)− z(0) = −e0

∫ t

0(A(r(s))+ C) ds. (4.20)

Thus, we have integrated the system (4.4)–(4.6).

THEOREM 4.1. In the variable r a classical particle moves periodically accord-ing to Equation (4.17) with period (4.18). The angular variable is determined byEquation (4.19) so that ϕ(t) = ϕ0t + O(1), where

ϕ0 = σT −1∫ T

0r(s)−2 ds,

as |t| → ∞. The variable z is determined by Equation (4.20) where C is the sameconstant as in (4.14).

It follows from equation (4.14) that z′(t) � 0 or z′(t) � 0 for all t if and only if

z′(0) � maxrmin�r�rmax

(e0A(r))− e0A(r(0))

or

z′(0) � minrmin�r�rmax

(e0A(r))− e0A(r(0)),

respectively. Otherwise, ±z′(t) � 0 if and only if

±e0(A(r(t))− A(r(0))) � ±z′(0),


so that a particle can move both in positive and negative directions in the variable z.Nevertheless one gives simple sufficient conditions for the inequality

±(z(T )− z(0)) > 0. (4.21)

Indeed, it follows from Equations (4.9) and (4.13) that

e0z′(t) = r ′′(t)A′(r(t))−1 − σ 2r(t)−3A′(r(t))−1.

Integrating this equation and taking into account the periodicity of the functionr(t), we see that

e0(z(T )− z(0)) =∫ T

0r ′′(t)A′(r(t))−1 dt − σ 2

∫ T

0r(t)−3A′(r(t))−1 dt

=∫ T

0r ′(t)2A′(r(t))−2A′′(r(t)) dt −

− σ 2∫ T

0r(t)−3A′(r(t))−1 dt. (4.22)

In particular,

z(nT )− z(0) = n(z(T )− z(0)).

Let us formulate the results obtained.

THEOREM 4.2. The increment of the variable z at every period is determined byEquation (4.22). In particular, if ±e0A

′(r) < 0 and ±e0A′′(r) � 0 for all r, then

inequality (4.21) holds. In this case z(t) = z0t+O(1) with z0 = T −1(z(T )−z(0)),±z0 > 0, as |t| → ∞.

Let us finally discuss the magnetic potential A(r) = −α ln r of an infiniterectilinear current. Such potentials satisfy all the assumptions of this section. Nowthe Equation (4.14) reduces to

z′(t) = 2γ ln br(t), b = r(0)−1e(2γ )−1z′(0) > 0,

and effective potential (4.16) is given by

V (r) = σ 2r−2 + 4γ 2 ln2 br,

where γ = αec−1. Then ±z′(t) � 0 for all t if and only if

±z′(0) � ±2γ ln(r(0)/rmax)

for ∓γ > 0 and if and only if

±z′(0) � ±2γ ln(r(0)/rmin)

230 D. YAFAEV

for ±γ > 0. Otherwise, ±z′(t) � 0 for ±γ > 0 if r(t) � b−1 and ±z′(t) � 0 for∓γ > 0 if r(t) � b−1. Equation (4.22) now takes the form

z(T )− z(0) = (2γ )−1∫ T

0(r ′(t)2 + σ 2r(t)−2) dt.

This expression is strictly positive (negative) if αe > 0 (if αe < 0).Thus, positively charged classical and quantum particles always move asymp-

totically in the direction of the current and never in the opposite direction. Simi-larly, negatively charged particles always move asymptotically against direction ofthe current and never in the same direction. In the plane orthogonal to the directionof the current classical and quantum particles are essentially localized.

Acknowledgement

I thank A. Its for a useful discussion of the classical problem.

References

1. Cycon, H., Froese, R., Kirsch, W. and Simon, B.: Schrödinger Operators, Texts Monogr. Phys.,Springer, Berlin, 1987.

2. Landau, L. D. and Lifshitz, E. M.: Classical Mechanics, Pergamon Press, Oxford, 1960.3. Landau, L. D. and Lifshitz, E. M.: Quantum Mechanics, Pergamon Press, Oxford, 1965.4. Miller, K. and Simon, B.: Quantum magnetic Hamiltonians with remarkable spectral properties,

Phys. Rev. Lett. 44 (1980), 1706–1707.5. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics IV, Academic Press, New

York, 1978.6. Rocard, Y.: Électricité, Masson, Paris, 1956.


231

Large Deviations for the Boundary DrivenSymmetric Simple Exclusion Process �

L. BERTINI1, A. DE SOLE2, D. GABRIELLI3, G. JONA-LASINIO4 andC. LANDIM5

1Dipartimento di Matematica, Università di Roma La Sapienza, P.le A. Moro 2, 00185 Rome, Italy.e-mail: [email protected] of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, U.S.A.e-mail: [email protected] di Matematica, Università dell’Aquila, 67100 Coppito, L’Aquila, Italy.e-mail: [email protected] di Fisica and INFN, Università di Roma La Sapienza, P.le A. Moro 2, 00185 Rome,Italy. e-mail: [email protected], Estrada Dona Castorina 110, J. Botanico, 22460 Rio de Janeiro, Brazil.CNRS UMR 6085, Université de Rouen, 76128 Mont-Saint-Aignan Cedex, France.e-mail: [email protected]

(Received: 7 November 2002; accepted in final form: 6 December 2002)

Abstract. The large deviation properties of equilibrium (reversible) lattice gases are mathematicallyreasonably well understood. Much less is known in nonequilibrium, namely for nonreversible sys-tems. In this paper we consider a simple example of a nonequilibrium situation, the symmetric simpleexclusion process in which we let the system exchange particles with the boundaries at two differentrates. We prove a dynamical large deviation principle for the empirical density which describes theprobability of fluctuations from the solutions of the hydrodynamic equation. The so-called quasipotential, which measures the cost of a fluctuation from the stationary state, is then defined by a vari-ational problem for the dynamical large deviation rate function. By characterizing the optimal path,we prove that the quasi potential can also be obtained from a static variational problem introducedby Derrida, Lebowitz, and Speer.

Mathematics Subject Classifications (2000): 82C22, 82C35, 60F10.

Key words: stationary nonreversible states, large deviations, boundary driven lattice gases.

1. Introduction

In previous papers [3, 4] we began the study of the macroscopic properties ofstochastic nonequilibrium systems. Typical examples are stochastic lattice gaseswhich exchange particles with different reservoirs at the boundary. In these sys-tems, there is a flow of matter through the system and the dynamics is not re-versible. The main difference with respect to equilibrium (reversible) states is

� Partially supported by Cofinanziamento MURST 2000 and 2001.

232 L. BERTINI ET AL.

the following: in equilibrium the invariant measure, which determines the ther-modynamic properties, is given for free by the Gibbs distribution specified bythe Hamiltonian. On the contrary, in nonequilibrium states the construction ofthe appropriate ensemble, that is the invariant measure, requires the solution ofa dynamical problem.

For equilibrium states, the thermodynamic entropy S is identified [6, 20, 22]with the large deviation rate function for the invariant measure. The rigorous studyof large deviations has been extended to hydrodynamic evolutions of stochastic in-teracting particle systems [10, 17]. Developing the methods of [17], this theory hasbeen extended to nonlinear hydrodynamic regimes [15]. In a dynamical setting onemay ask new questions, for example what is the most probable trajectory followedby the system in the spontaneous emergence of a fluctuation or in its relaxationto equilibrium. In the physics literature, the Onsager–Machlup theory [23] givesthe following answer under the assumption of time reversibility. In the situationof a linear macroscopic equation, that is, close to equilibrium, the most probableemergence and relaxation trajectories are one the time reversal of the other.

In [3, 4] we have heuristically shown how this theory has to be modified fornonequilibrium systems. At thermodynamic level, we do not need all the infor-mation carried by the invariant measure, but only its rate function S. This canbe obtained, by solving a variational problem, from the dynamical rate functionwhich describes the probability of fluctuations from the hydrodynamic behavior.The physical content of the variational problem is the following. Let ρ be the rele-vant thermodynamic variable, for instance the local density, whose stationary valueis given by some function ρ(u). The entropy S(ρ) associated to some profile ρ(u)

is then obtained by minimizing the dynamical rate function over all possible pathsπ(t) = π(t, u) connecting ρ to ρ. We have shown that the optimal path π∗(t) issuch that π∗(−t) is a solution of the hydrodynamic equation associated to the timereversed microscopic dynamics, which we call adjoint hydrodynamics. This rela-tionship is the extension of the Onsager–Machlup theory to nonreversible systems.Moreover, we have also shown that S solves an infinite-dimensional Hamilton–Jacobi equation and how the adjoint hydrodynamics can be obtained once S isknown.

In the present paper we study rigorously the symmetric one-dimensional exclu-sion process. In this model there is at most one particle for each site of the lattice{−N, . . . , N} which can move to a neighboring site only if this is empty, withrate 1/2 for each side. Moreover, a particle at the boundary may leave the systemat rate 1/2 or enter at rate γ−/2, respectively γ+/2, at the site −N , respectively+N . In this situation there is a unique invariant measure µN which reduces toa Bernoulli measure if γ− = γ+. On the other hand, if γ− �= γ+, the measureµN exhibits long range correlations [7, 24] and it is not explicitly known. By us-ing a matrix representation and combinatorial techniques, Derrida, Lebowitz, andSpeer [8, 9] have recently shown that the rate function for µN can be obtained solv-ing a nonlinear boundary value problem on the interval [−1, 1]. We here analyze

THE BOUNDARY DRIVEN SYMMETRIC SIMPLE EXCLUSION PROCESS 233

the macroscopic dynamical behavior of this system. The hydrodynamic limit forthe empirical density has been proven in [12, 13]. We prove the associated dynam-ical large deviation principle which describes the probability of fluctuations fromthe solutions of the hydrodynamic equation. We then define the quasi-potential viathe variational problem mentioned above. By characterizing the optimal path weprove that the quasi-potential can also be obtained from a static variational problemintroduced in [8, 9]. Using the identification of the quasi-potential with the ratefunction for the invariant measure proven in [5], we finally obtain an independentderivation of the expression for the thermodynamic entropy found in [8, 9].

2. Notation and Results

For an integer N � 1, let �N := [−N,N] ∩ Z = {−N, . . . , N}. The sites of�N are denoted by x, y, and z while the macroscopic space variable (points inthe interval [−1, 1]) by u, v, and w. We introduce the microscopic state space as�N := {0, 1}�N which is endowed with the discrete topology; elements of �N ,called configurations, are denoted by η. In this way η(x) ∈ {0, 1} stands for thenumber of particles at site x for the configuration η.

The one-dimensional boundary driven simple exclusion process is the Markovprocess on the state space �N with infinitesimal generator

LN := L−,N + L0,N + L+,N

defined by

(L0,Nf )(η) := N2

2

N−1∑x=−N

[f (σ x,x+1η)− f (η)

],

(L±,Nf )(η) := N2

2[γ± + (1 − γ±)η(±N)][f (σ±Nη)− f (η)

]for every function f : �N → R. In this formula σ x,yη is the configuration obtainedfrom η by exchanging the occupation variables η(x) and η(y):

(σ x,yη)(z) :=

η(y) if z = x,

η(x) if z = y,

η(z) if z �= x, y

and σ xη is the configuration obtained from η by flipping the configuration at x:

(σ xη) (z) := η(z)[1− δx,z] + δx,z[1 − η(z)],where δx,y is the Kronecker delta. Finally, γ± ∈ (0,∞) are the activities of thereservoirs at the boundary of �N .

Notice that the generators are speeded up by N2; this corresponds to the diffu-sive scaling. We denote by ηt the Markov process on �N with generator LN and


by Pη its distribution if the initial configuration is η. Note that Pη is a probabilitymeasure on the path space D(R+, �N), which we consider endowed with the Sko-rokhod topology and the corresponding Borel σ -algebra. Expectation with respectto Pη is denoted by Eη.

Our first main result is the dynamical large deviation principle for the measurePη. We denote by 〈·, ·〉 the inner product in L2([−1, 1], du) and let

M := {ρ ∈ L∞

([−1, 1], du) : 0 � ρ(u) � 1 a.e.

}(2.1)

which we equip with the topology induced by weak convergence, namely ρn → ρ

in M if and only if 〈ρn,G〉 → 〈ρ,G〉 for each continuous function G: [−1, 1] →R; we consider M also endowed with the corresponding Borel σ -algebra. Let usdefine the map πN : �N → M as

πN(η) :=N∑

x=−Nη(x)1

{[x

N− 1

2N,x

N+ 1

2N

)}, (2.2)

where 1{A} stands for the indicator function of the set A; namely πN = πN(η) isthe empirical density obtained from the configuration η. Notice that πN(η) ∈ M,i.e. 0 � πN(η) � 1, because η(x) ∈ {0, 1}.

Let ηN be a sequence of configurations for which the empirical density πN(ηN)

converges in M, as N ↑ ∞, to some function ρ, namely for each G ∈ C([−1, 1])

limN→∞〈π

N(ηN),G〉 = limN→∞

N∑x=−N

ηN(x)

∫ 1∧( xN+ 1

2N )

(−1)∨( xN − 12N )

duG(u)

=∫ 1

−1duρ(u)G(u), (2.3)

where we used the notation a ∧ b := min{a, b} and a ∨ b := max{a, b}. If (2.3)holds we say that the sequence {ηN : N � 1} is associated to the profile ρ ∈ M.

For T > 0 and positive integers m, n we denote by Cm,n0 ([0, T ] × [−1, 1]) the

space of functions G: [0, T ]×[−1, 1] → R with m continuous derivatives in time,n continuous derivatives in space and which vanish at the boundary: G(·,±1) = 0.Let also D([0, T ],M) be the Skorokhod space of paths from [0, T ] to M equippedwith its Borel σ -algebra. Elements of D([0, T ],M) will be denoted by π(t) =π(t, u).

Let ρ± := γ±/[1+γ±] ∈ (0, 1) be the density at the boundary of [−1, 1] and fixa function ρ ∈ M which corresponds to the initial profile. For H ∈ C

1,20 ([0, T ] ×

[−1, 1]), let JT,H,ρ = JH : D([0, T ],M) → R be the functional given by

JH(π) := 〈π(T ),H(T )〉 − 〈ρ,H(0)〉 −∫ T

0dt

⟨π(t), ∂tH(t)+ 1

2(H(t)⟩+

+ ρ+2

∫ T

0dt∇H(t, 1)− ρ−

2

∫ T

0dt∇H(t,−1)−

− 1

2

∫ T

0dt

⟨χ(π(t)),

(∇H(t))2⟩

, (2.4)


where ∇ denotes the derivative with respect to the macroscopic space variable u,( is the Laplacian on (−1, 1), and we have set χ(a) := a(1 − a). Let finallyIT ( · |ρ): D([0, T ],M)→ [0,+∞] be the functional defined by

IT (π |ρ) := supH∈C1,2

0 ([0,T ]×[−1,1])JH (π). (2.5)

Notice that, if π(t) solves the heat equation with boundary condition π(t,±1) =ρ± and initial datum π(0) = ρ, then IT (π |ρ) = 0.

THEOREM 2.1. Fix T > 0 and a profile ρ ∈ M bounded away from 0 and1, namely such that there exists δ > 0 with δ � ρ � 1 − δ a.e. Consider asequence ηN of configurations associated to ρ. Then the measure PηN ◦ (πN)−1

on D([0, T ],M) satisfies a large deviation principle with speed N and convexlower semi-continuous rate function IT (·|ρ). Namely, for each closed set C ⊂D([0, T ],M) and each open set O ⊂ D([0, T ],M),

lim supN→∞

1

Nlog PηN [πN ∈ C] � − inf

π∈CIT (π |ρ),

lim infN→∞

1

Nlog PηN [πN ∈ O] � − inf

π∈OIT (π |ρ).

It is possible to obtain a more explicit representation of the functional IT (·|ρ),see Lemma 3.6 below. If the particle system is considered with periodic boundaryconditions, i.e. �N is replaced by the discrete torus of length N , this theorem hasbeen proven in [17]. As we shall see later, the main difference with respect to thecase with periodic boundary condition is the lack of translation invariance and thefact that the path π(t, ·) is fixed at the boundary.

We now define precisely the variational problem mentioned in the Introduction.Let ρ ∈ M be the linear profile ρ(u) := [ρ−(1 − u)+ ρ+(1+ u)]/2, u ∈ [−1, 1],which is the density profile associate to the invariant measure µN , see Section 3below. We then define V :M → [0,+∞] as the quasi potential for the rate functionIT ( · |ρ):

V (ρ) := infT>0

infπ(·) : π(T )=ρ

IT (π |ρ) (2.6)

which measures the minimal cost to produce the profile ρ starting from ρ.Let us first describe how the variational problem (2.6) is solved when γ− =

γ+ = γ . In this case ρ = γ /(1 + γ ) is constant and the process is reversible withrespect to the Bernoulli measure with density ρ. We have that IT (π |ρ0) = 0 ifπ(t) solves the hydrodynamic equation which for this system is given by the heatequation:

∂tρ(t) = (1/2)(ρ(t),

ρ(t,±1) = ρ±, (2.7)

ρ(0, ·) = ρ0(·).Note that ρ(t) → ρ as t →∞.


It can be easily shown that the minimizer for (2.6), defined on the time interval(−∞, 0] instead of [0,+∞) as in (2.6), is given by π∗(t) = ρ(−t), where ρ(t) isthe solution of (2.7) with initial condition ρ0 = ρ. This symmetry of the relaxationand fluctuation trajectories is the Onsager–Machlup principle mentioned before.

Moreover the quasi-potential V (ρ) coincides with the entropy of the Bernoullimeasure with density ρ, that is, understanding 0 log 0 = 0,

V (ρ) = S0(ρ) :=∫ 1

−1du

[ρ(u) log

ρ(u)

ρ+ [1 − ρ(u)] log

1 − ρ(u)

1 − ρ

]. (2.8)

In the context of Freidlin–Wentzell theory [14] for diffusions in Rn, the situa-

tion just described is analogous to the so called gradient case in which the quasipotential coincides with the potential. This structure reflects the reversibility ofthe underlying process. In general for nongradient systems, the solution of thedynamical variational problem, or of the associated Hamilton–Jacobi equation,cannot be explicitly calculated. The case γ+ �= γ− is analogous to a nongradientsystem, but for this particular model we shall prove that the quasi potential V (ρ), asdefined in (2.6), coincides with the functional S(ρ) defined by a time independentvariational problem introduced in [8, 9] which is stated below. This is the secondmain result of this paper.

Denote by C1([−1, 1]) the space of once continuously differentiable functionsf : [−1, 1] → R endowed with the norm ‖f ‖C1 := supu∈[−1,1]{|f (u)| + |f ′(u)|}.Let

F := {f ∈ C1([−1, 1]) : f (±1) = ρ±, [ρ+ − ρ−]f ′(u) > 0,

u ∈ [−1, 1]}, (2.9)

where f ′ denotes the derivative of f . Note that if f ∈ F , then

0 < ρ− ∧ ρ+ � f (u) � ρ− ∨ ρ+ < 1 for all − 1 � u � 1.

For ρ ∈ M and f ∈ F we set

G(ρ, f ) :=∫ 1

−1du

[ρ(u) log

ρ(u)

f (u)+ [

1 − ρ(u)]

log1− ρ(u)

1 − f (u)+

+ logf ′(u)

[ρ+ − ρ−]/2

](2.10)

and

S(ρ) := supf∈F

G(ρ, f ). (2.11)

Theorem 4.5 below, which formalizes the arguments in [9], states that the supre-mum in (2.11) is uniquely attained for a function f which solves a nonlinear bound-ary value problem. We shall denote it by F = F(ρ) to emphasize its dependenceon ρ; therefore S(ρ) = G(ρ, F (ρ)).


THEOREM 2.2. Let V and S as defined in (2.6) and (2.11). Then for each ρ ∈ Mwe have V (ρ) = S(ρ).

In the proof of the above theorem we shall construct a particular path π∗(t) inwhich the infimum in (2.6) is almost attained. As recalled in the introduction, bythe heuristic arguments in [4], π∗(−t) is the solution of the hydrodynamic equationcorresponding to the process with generator L∗

N , the adjoint of LN in L2(�N, dµN)

and initial condition ρ. In analogy to the Freidlin–Wentzell theory [14], we expectthat the exit path from a neighborhood ρ to a neighborhood of ρ should, withprobability converging to one as N ↑ ∞, take place in a small tube around the pathπ∗(t).

The optimal path can be described in a rather simple fashion. Recalling thatwe denoted by F = F(ρ) the maximizer for (2.11), consider the heat equation in[−1, 1] with boundary conditions ρ± and initial datum F :

∂t.(t) = (1/2)(.(t),

.(t,±1) = ρ±, (2.12)

.(0, ·) = F(ρ).

We next define ρ∗(t) = ρ∗(t, u) by

ρ∗(t) := .(t)+.(t)[1 −.(t)] (.(t)

(∇.(t))2· (2.13)

In view of (4.3) below, ρ∗(0) = ρ and, by Lemma 5.6, limt→∞ ρ∗(t) = ρ. Theoptimal path π∗(t), defined on the time interval (−∞, 0] instead of [0,+∞) asin (2.6), is then given by π∗(t) = ρ∗(−t).

From the dynamical large deviation principle we can obtain, by means of thequasi potential, the large deviation principle for the empirical density when theparticles are distributed according to the invariant measure of the process ηt . Notethat the finite state Markov process ηt with generator LN is irreducible, therefore ithas a unique invariant measure µN .

Let us introduce PN := µN ◦(πN)−1 which is a probability on M and describesthe behavior of the empirical density under the invariant measure. In [7, 12, 13, 24]it is shown, see also Section 3 below, that PN satisfies the law of large numbersPN ⇒ δρ in which ⇒ stands for weak convergence of measures on M and ρ is thelinear profile already introduced.

Since ρ is globally attractive for (2.7), the quasi-potential with respect to ρ

defined in (2.6) gives the rate function for the family PN . In [3, 4] we have heuristi-cally derived this identification via a time reversal argument. For the present modela rigorous proof, in the same spirit of the Freidlin–Wentzell theory, is given in [5];that is we have the following theorem.

THEOREM 2.3. Let V as defined in (2.6). Then the measure PN satisfies a largedeviation principle with speed N and rate function V .


The identification of the rate function for PN with the functional S now followsfrom Theorems 2.1, 2.2 and 2.3.

COROLLARY 2.4. Let S as defined in (2.11). The measure PN satisfies a largedeviation principle on M with speed N and convex lower semi-continuous ratefunction S. Namely for each closed set C ⊂ M and each open set O ⊂ M,

lim supN→∞

1

Nlog µN [πN ∈ C] � − inf

ρ∈CS(ρ),

lim infN→∞

1

Nlog µN [πN ∈ O] � − inf

ρ∈OS(ρ).

As already mentioned, the rate function S has been first obtained in [8, 9]by using a matrix representation of the invariant measure µN and combinatorialtechniques. By means of Theorems 2.1, 2.2, and 2.3 we prove here, indepen-dently of [8, 9], the large deviation principle by following the dynamical/variationalroute explained in [4] which is analogous to the Freidlin–Wentzell theory [14] fordiffusions on R

n.We remark that it should be possible, modulo technical problems, to extend

Theorems 2.1 and 2.3 to other boundary driven diffusive lattice gases, see [4] fora heuristic discussion. The characterization of the rate function for the invariantmeasure as the quasi potential allows to obtain some information on it directly fromthe variational problem (2.6). In particular, in Appendix, we discuss the symmetricsimple exclusion in any dimension and get a lower bound on V in terms of theentropy S0 of the equilibrium system. In the one-dimensional case, this bound hasbeen proven in [8, 9] by using instead the variational problem (2.11).

Outline. In Section 3 we recall the hydrodynamic behavior of the boundarydriven exclusion process and prove the associated large deviation principle de-scribed by Theorem 2.1. In Sections 4 and 5, which are more technical, we stateand prove some properties of the functional S which is then shown to coincidewith the quasi potential V . Finally, in Appendix, we consider the symmetric simpleexclusion in any dimension and prove a lower bound on V .

3. Dynamical Behavior

In this section we study the dynamical properties of the empirical density for theboundary driven simple exclusion process in a fixed (macroscopic) time interval[0, T ]. In particular, we review the hydrodynamic limit (law of large numbers) andprove the corresponding large deviation principle. This problem was consideredbefore by Kipnis, Olla and Varadhan in [17] for the exclusion process with periodicboundary condition. For this reason, we present only the modifications needed inthe argument and refer to [2, 16, 17] for the missing arguments.


As already stated, the invariant measure µN is not known explicitly but someof its properties have been derived. For example, the one-site marginals or thecorrelations can be computed explicitly. To compute the one-site marginals, whichwill be used later, let ρN(x) = EµN [η(x)] for −N � x � N . Since µN is invariant,EµN [LNη(x)] = 0. Computing LNη(x), we obtain a closed difference equation forρN(x):

((NρN)(x) = 0 for −N + 1 � x � N − 1,

ρN(N − 1)− ρN(N)+ γ+[1 − ρN(N)] − ρN(N) = 0,

ρN(−N + 1)− ρN(−N)+ γ−[1 − ρN(−N)] − ρN(−N) = 0.

In this formula, (N stands for the discrete Laplacian so that ((Nf )(x) = f (x +1)+ f (x − 1)− 2f (x). The unique solution of this discrete elliptic equation givesthe one-site marginals of µN .

Denote by νN = νNγ−,γ+ the product measure on �N with marginals given by

νN {η : η(x) = 1} = ρN(x)

and observe that the generators L−,N , L+,N are reversible with respect to νN .Denote by {τx : x ∈ Z} the group of translations in {0, 1}Z so that (τxζ )(z) =

ζ(x + z) for all x, z in Z and configuration ζ in {0, 1}Z. Translations are extendedto functions and measures in a natural way. Eyink et al. [12] and De Masi et al. [7]proved that

limN→∞

EµN [τ[uN]f ] = Eνρ(u)[f ]

for every local function f and u in (−1, 1). Here ρ is the unique solution of

(1/2)(ρ = 0, ρ(±1) = ρ±,

namely ρ is the linear interpolation between ρ− and ρ+ and {να : 0 � α � 1}stands for the Bernoulli product measure in {0, 1}Z with density α and ρ± =γ±/[1 + γ±] is the density at the boundary of [−1, 1].

3.1. HYDRODYNAMIC LIMIT

Recall that, for each configuration η ∈ �N , we denote by πN = πN(η) ∈ Mthe empirical density obtained from η, see Equation (2.2). We say that a sequenceof configurations {ηN : N � 1} is associated to the profile γ if (2.3) holds forall continuous functions G: [−1, 1] → R. The following result is due to Eyink,Lebowitz and Spohn [13].

THEOREM 3.1. Consider a sequence ηN associated to some profile ρ0 ∈ M.Then, for all t > 0, πN(t) = πN(ηt ) converges (in the sense (2.3)) in probabilityto ρ(t, u), the unique weak solution of


∂tρ = (1/2)(ρ,

ρ(t,±1) = ρ±, (3.1)

ρ(0, ·) = ρ0(·).

By a weak solution of the Dirichlet problem (3.1) in the time interval [0, T ],we understand a bounded real function ρ which satisfies the following two condi-tions.

(a) There exists a function A(t, u) in L2([−1, 1] × [0, T ]) such that∫ t

0ds

∫ 1

−1duρ(s, u)(∇H)(u)

= {ρ+H(1)− ρ−H(−1)}t −∫ t

0ds

∫ 1

−1duA(s, u)H(u)

for every smooth function H : [−1, 1] → R and every 0 � t � T . A(t, u) willbe denoted by (∇ρ)(t, u).

(b) For every function H : [−1, 1] → R of class C1([−1, 1]) vanishing at theboundary and every 0 � t � T ,∫ 1

−1duρ(t, u)H(u) −

∫ 1

−1duρ0(u)H(u)

= −(1/2)∫ t

0ds

∫ 1

−1du (∇ρ)(s, u)(∇H)(u).

The classical H−1 estimates gives uniqueness of weak solutions of Equation(3.1). Note that here the weak solution coincides with the semi-group solutionρ(t) = ρ+et(

0/2(ρ0− ρ), where ρ is the stationary profile and (0 is the Laplacianwith zero boundary condition.

3.2. SUPER-EXPONENTIAL ESTIMATE

We now turn to the problem of large deviations from the hydrodynamic limit. Itis well known that one of the main steps in the derivation of a large deviationprinciple for the empirical density is a super-exponential estimate which allowsthe replacement of local functions by functionals of the empirical density in thelarge deviations regime. Essentially, the problem consists in bounding expressionssuch as 〈V, f 2〉µN in terms of the Dirichlet form 〈−LNf, f 〉µN . Here V is a lo-cal function and 〈·, ·〉µN indicates the inner product with respect to the invariantstate µN .

In the context of boundary driven simple exclusion processes, the fact that theinvariant state is not known explicitly introduces a technical difficulty. Follow-ing [19] we fix νN , the product measure defined in the beginning of this section, asreference measure and estimate everything with respect to νN . However, since νN

is not an invariant state, there are no reasons for 〈−LNf, f 〉νN to be positive. Thefirst statement shows that this expression is almost positive.


For a function f : �N → R, let

DN(f ) =N−1∑x=−N

∫[f (σ x,x+1η)− f (η)]2 dνN(η).

LEMMA 3.2. There exists a finite constant C0 depending only on γ± such that

〈L0,Nf, f 〉νN � −N2

4DN(f )+ C0N〈f, f 〉νN

for all functions f : �N → R.

The proof of this lemma is elementary and left to the reader. Notice, on theother hand, that both 〈−L+,Nf, f 〉νN , 〈−L−,Nf, f 〉νN are positive because νN is areversible state by our choice of the profile ρN .

This lemma together with the computation presented in [2, p. 78] for non-reversible processes, permits to prove the super-exponential estimate. The state-ment of this result requires some notation. For a cylinder function 5, denote theexpectation of 5 with respect to the Bernoulli product measure να by 5(α):

5(α) := Eνα [5].For a positive integer 6 and −N � x � N , denote the empirical mean density on abox of size 26+ 1 centered at x by η6(x), namely

η6(x) = 1

|�6(x)|∑

y∈�6(x)

η(y),

where

�6(x) = �N,6(x) = {y ∈ �N : |y − x| � 6}.Let H ∈ C([0, T ] × [−1, 1]) and 5 a cylinder function. For ε > 0, define also

VH,5N,ε (t, η) = 1

N

∑x

H(t, x/N){τx5(η)− 5

(ηNε(x)

)},

where the summation is carried over all x such that the support of τx5 belongs to�N . For a continuous function G: [0, T ] → R, let

W±G =

∫ T

0ds G(s)[ηs(±N)− ρ±].

THEOREM 3.3. Fix H in C([0, T ]×[−1, 1]), G ∈ C([0, T ]), a cylinder function5, and a sequence {ηN ∈ �N : N � 1} of configurations. For any δ > 0 we have

lim supε→0

lim supN→∞

1

Nlog PηN

[∣∣∣∣∫ T

0V

H,5N,ε (t, ηt ) dt

∣∣∣∣ > δ

]= −∞,

lim supN→∞

1

Nlog PηN

[|W±G | > δ

] = −∞.


3.3. UPPER BOUND

The proof of the upper bound of the large deviation principle is essentially the sameas in [17]. There is just a slight difference in the definition of the functionals JHdue to the boundary conditions.

For H in C1,20 ([0, T ]×[−1, 1]) consider the exponential martingale MH

t definedby

MHt = exp

{N

[〈πN(t),H(t)〉 − 〈πN(0),H(0)〉 −

− 1

N

∫ t

0e−N〈π

N (s),H(s)〉(∂s +N2LN)eN〈πN (s),H(s)〉 ds

]}.

An elementary computation shows that

MHT = expN

{JH(π

N ∗ ιε)+ VHN,ε + CH(ε)

},

where limε→0 CH(ε) = 0, ιε stands for the approximation of the identity ιε(u) =(2ε)−11{u ∈ [−ε, ε]}, ∗ stands for convolution,

VHN,ε =

∫ T

0V

H,50N,ε (t, ηt ) dt +W+

∇H(·,1) −W−∇H(·,−1)

and 50(η) = η(0)[1 − η(1)].Fix a subset A of D([0, T ],M) and write

1

Nlog PηN [πN ∈ A] = 1

Nlog EηN

[MH

T (MHT )−11{πN ∈ A}].

Maximizing over πN in A, we get from previous computation that the last term isbounded above by

− infπ∈A JH (π ∗ ιε)+ 1

Nlog EηN

[MH

T e−NVHN,ε

]− CH(ε).

Denote by PHηN

the measure PηNMHT . Since the martingale is bounded by exp{CN}

for some finite constant depending only on H and T , Theorem 3.3 holds for PHηN

in place of PηN . In particular, the second term of the previous formula is boundedabove by CH(ε,N) such that limε→0 lim supN→∞ CH(ε,N) = 0. Hence, for everyε > 0, and every H in C

1,20 ([0, T ] × [−1, 1]),

lim supN→∞

1

Nlog PηN [πN ∈ A] � − inf

π∈A JH(π ∗ ιε)+ C ′H(ε),

where limε→0 C′H(ε) = 0.

Assume now that the set A is a compact set K. Since JH(·∗ ιε) is continuous forevery H and ε > 0, we may apply the arguments presented in Lemma 11.3 of [25]


to exchange the supremum with the infimum. In this way we obtain that the lastexpression is bounded above by

lim supN→∞

1

Nlog PηN [πN ∈ K] � − inf

π∈K supH,ε

{JH(π ∗ ιε)+ C ′

H(ε)}.

First, letting ε ↓ 0, since JH (π∗ιε) converges to JH(π) for every H in C1,20 ([0, T ]×

[−1, 1]), in view of the definition (2.5) of IT (π |γ ), we deduce that

lim supN→∞

1

Nlog PηN [πN ∈ K] � − inf

π∈KIT (π |γ ),

which proves the upper bound for compact subsets.To pass from compact sets to closed sets, we have to obtain ‘exponential tight-

ness’ for the sequence PηN [πN ∈ ·]. The proof presented in [1] for the noninteract-ing zero-range process is easily adapted to our context.

3.4. HYDRODYNAMIC LIMIT OF WEAKLY ASYMMETRIC EXCLUSIONS

Fix a function H in C1,20 ([0, T ] × [−1, 1]) and recall the definition of the mar-

tingale MHT . Denote by P

HηN

the probability measure on D([0, T ], �N) defined by

PHηN[A] = EηN [MH

T 1{A}]. Under PHηN

, the coordinates {ηt : 0 � t � T } form a

Markov process with generator LHN = L+,N + LH

0,N + L−,N , where

(LH0,Nf )(η)

= N2

2

N−1∑x=−N

e−{H(t,[x+1]/N)−H(t,x/N)}{η(x+1)−η(x)}[f (σ x,x+1η)− f (η)].

The next result is due to Eyink et al. [13]. Recall χ(ρ) = ρ(1− ρ).

LEMMA 3.4. Consider a sequence ηN associated to some profile γ ∈ M and fixH in C

1,20 ([0, T ] × [−1, 1]). Then, for all t > 0, πN(t) = πN(ηt ) converges in

probability (in the sense (2.3)) to ρ(t, u), the unique weak solution of

∂tρ = (1/2)(ρ −∇{χ(ρ)∇H },ρ(t,±1) = ρ±, (3.2)ρ(0, ·) = γ (·).

As in Subsection 3.1, by a weak solution of the Dirichlet problem (3.2) in thetime interval [0, T ], we understand a bounded real function ρ which satisfies thefollowing two conditions.

(a) There exists a function A(t, u) in L2([−1, 1] × [0, T ]) such that

244 L. BERTINI ET AL.∫ t

0ds

∫ 1

−1duρ(s, u)(∇G)(u)

= {ρ+G(1)− ρ−G(−1)}t −∫ t

0ds

∫ 1

−1duA(s, u)G(u)

for every smooth function G: [−1, 1] → R and every 0 � t � T . A(t, u) willbe denoted by (∇ρ)(t, u).

(b) For every function G ∈ C1([−1, 1]) vanishing at the boundary and everyt � 0,∫ 1

−1duρ(t, u)G(u) −

∫ 1

−1du γ (u)G(u)

=∫ t

0ds

∫ 1

−1du (∇G)(u)

{− (1/2)(∇ρ)(s, u)+ χ(ρ(s, u))(∇H)(s, u)}.

The classical H−1 estimates gives uniqueness of weak solutions of Equation (3.2).

3.5. THE RATE FUNCTION

We prove in this subsection some properties of the rate function IT ( · |γ ). Wefirst claim that this rate function is convex and lower semi-continuous. In viewof the definition of IT ( · |γ ), to prove this assertion, it is enough to show that JHis convex and lower semi-continuous for each H in C

1,20 ([0, T ] × [−1, 1]). It is

convex because χ(a) = a(1−a) is a concave function. It is lower semi-continuousbecause for any positive, continuous function G: [0, T ]×[−1, 1] → R and for anysequence πn converging to π in D([0, T ],M),∫ T

0dt 〈χ(π(t)),G(t)〉 = lim

ε→0

∫ T

0dt 〈χ(π(t) ∗ ιε),G(t)〉

= limε→0

limn→∞

∫ T

0dt 〈χ(πn(t) ∗ ιε),G(t)〉.

Since χ is concave and G positive, a change of variables shows that this expressionis bounded below by

limε→0

lim supn→∞

∫ T

0dt〈χ(πn(t)),G(t) ∗ ιε〉 = lim sup

n→∞

∫ T

0dt〈χ(πn(t)),G(t)〉

because G is continuous and χ is bounded. This proves that JH is lower semi-continuous for every H in C

1,20 ([0, T ] × [−1, 1]).

Denote by Dγ the subset of D([0, T ],M) of all paths π(t, u) which satisfy theboundary conditions π(0, ·) = γ (·), π(·,±1) = ρ±, in the sense that for every0 � t0 < t1 � T ,

limδ→0

∫ t1

t0

dt1

δ

∫ −1+δ

−1π(t, u) du = (t1 − t0)ρ−

and a similar identity at the other boundary.


LEMMA 3.5. IT (π |γ ) = ∞ if π does not belong to Dγ .Proof. Fix π in D([0, T ],M) such that IT (π |γ ) < ∞. We first show that

π(0, ·) = γ (·). For δ > 0, consider the function Hδ(t, u) = hδ(t)g(u), hδ(t) =(1−δ−1t)+, g(·) vanishing at the boundary ±1. Here a+ stands for the positive partof a. Of course, Hδ can be approximated by smooth functions. Since π is boundedand since t → π(t, ·) is right-continuous for the weak topology,

limδ↓0

JHδ(π) = 〈π(0), g〉 − 〈γ, g〉,

which proves that π(0) = γ a.s. because IT (π |γ ) <∞.A similar argument shows that π(t,±1) = ρ±; to prove this statement we may

consider the sequence of functions Hδ(t, u) = h(t)gδ(u), where h(t) approximatesthe indicator of some time interval [t0, t1] and where

g′δ(u) ={A− (A+ b)(1 + u)/δ, if − 1 � u � −1 + δ,

−b, if − 1+ δ � u � 1.

Here A > 0 is large and fixed and b = b(A, δ) > 0 is chosen for the integral over[−1, 1] of g′δ to vanish. ✷

Fix π in Dγ and denote by H1(π) the Hilbert space induced by C1,20 ([0, T ] ×

[−1, 1]) endowed with the inner product 〈·, ·〉π defined by

〈H,G〉π =∫ T

0dt

∫ 1

−1duχ(π)(∇G)(∇H).

LEMMA 3.6. Fix a trajectory π in Dγ and assume that IT (π |γ ) is finite. Thereexists a function H in H1(π) such that π is the unique weak solution of

∂tπ = (1/2)(π −∇{χ(π)π(1− π)∇H },π(t,±1) = ρ±, (3.3)

π(0, ·) = γ (·).Moreover,

IT (π |γ ) = (1/2)∫ T

0dt

∫ 1

−1duχ(π)(∇H)2. (3.4)

We refer the reader to [16, 17] for a proof. One of the consequences of thislemma is that every trajectory t !→ π(t) with finite rate function is continuous inthe weak topology, π ∈ C([0, T ];M). Indeed, by the previous lemma, for π suchthat IT (π |γ ) <∞, and every G in C2

0([−1, 1]),〈π(t),G〉 − 〈π(s),G〉= (1/2)

∫ t

s

dr〈π(r),(G〉 +∫ t

s

dr〈χ(π(r)),∇G∇H 〉 −− (1/2){(∇G)(1) ρ+ − (∇G)(−1) ρ−}(t − s)


for some H in H1(π). Since G is smooth and H belongs to H1(π), the right-handside vanishes as |t − s| → 0.

3.6. LOWER BOUND

Denote by D0γ the set of trajectories π in D([0, T ],M) for which there exists H in

C1,20 ([0, T ] × [−1, 1]) such that π is the solution of (3.3). For each π in D0

γ , andfor each neighborhood Nπ of π

lim infN→∞

1

Nlog PηN [πN ∈ Nπ ] � −IT (π |γ ).

This statement is proved as in the periodic boundary case, see [16]. To completethe proof of the lower bound, it remains to show that for every trajectory π suchthat IT (π |γ ) < ∞, there exists a sequence πk in D0

γ such that limk πk = π ,limk IT (πk|γ ) = IT (π |γ ).

This is not too difficult in our context because the rate function is convexand lower semi continuous. We first show that any path π with finite rate func-tion can be approximated by a path which is bounded away from 0 and 1. Fix apath π such that IT (π |γ ) < ∞. Fix δ > 0 and denote by ρ(t, u) the solutionof the hydrodynamic equation (3.1) with initial condition γ instead of ρ0. Letπδ = δρ + (1 − δ)π . Of course, πδ converges to π as δ ↓ 0. By lower semi-continuity, IT (π |γ ) � lim infδ→0 IT (πδ|γ ). On the other hand, since IT ( · |γ ) isconvex, IT (πδ|γ ) � (1− δ)IT (π |γ ) because ρ is the solution of the hydrodynamicequation and IT (ρ|γ ) = 0. This shows that limδ→0 πδ = π , limδ→0 IT (πδ|γ ) =I (π). Since 0 < γ < 1, 0 < ρ± < 1, πδ is bounded away from 0 and 1, provingthe claim.

Fix now a path π with finite rate function and bounded away from 0 and 1. Weclaim that this trajectory may be approximated by a path in D0

γ . Since IT (π |γ ) <∞, by Lemma 3.6, there exists H in H1(π) satisfying (3.3). Since π is boundedaway from 0 and 1, H1(π) coincides with the usual Sobolev space H1 associatedto the Lebesgue measure. Consider a sequence of smooth functions Hn: [0, T ] ×[−1, 1] → R vanishing at the boundary and such that∇Hn converges in L2([0, T ]×[−1, 1]) to ∇H . Denote by πn the solution of (3.2) with Hn instead of H . We claimthat limn→∞ πn = π , limn→∞ IT (π

n|γ ) = IT (π |γ ).The proof that πn converges to π is divided in two pieces. We first show that

the sequence is tight in C([0, T ],M) and then we prove that all limit points are so-lution of Equation (3.2). We start with a preliminary estimate which will be neededrepeatedly. Recall that ρ is the stationary profile. Computing the time derivative of∫ 1−1 du (πn(t)− ρ)2, we obtain that∫ T

0dt

∫ 1

−1du (∇πn(t))2 � C (3.5)

for some finite constant independent of n.


From the previous bound and since πn(t, u) belongs to [0, 1], it is not difficultto show that the sequence πn is tight in C([0, T ],M). To check uniqueness oflimit points, consider any limit point β in C([0, T ],M). We claim that β is a weaksolution of Equation (3.2). Of course β is positive and bounded above by 1. Theexistence of a function A(s, u) in L2([−1, 1] × [0, T ]) for which (a) holds followsfrom (3.5), which guarantees the existence of weak converging subsequences. Theunique difficulty in the proof of identity (b) is to show that for any 0 � t � T , Gin L2([0, T ] × [−1, 1]),

limn→∞

∫ t

0ds〈χ(πn(s)),G(s)〉 =

∫ t

0ds〈χ(β(s)),G(s)〉 (3.6)

for any sequence πn converging to β in C([0, T ],M) and satisfying (3.5). Thisidentity holds because for any δ > 0

limn→∞

∫ t

0ds〈χ(πn(s) ∗ ιδ),G(s)〉 =

∫ t

0ds〈χ(β(s) ∗ ιδ),G(s)〉

and because, by Schwartz inequality and |χ(a)− χ(b)| � |a − b|,( ∫ t

0ds〈χ(πn(s) ∗ ιδ)− χ(πn(s)),G(s)〉

)2

�∫ t

0ds〈G(s)2〉

∫ t

0ds〈[πn(s) ∗ ιδ − πn(s)]2〉.

It is not difficult to show, using estimate (3.5), that this term vanishes as δ ↓ 0,uniformly in n, proving (3.6). In conclusion, we proved that the sequence πn is tightin C([0, T ],M) and that all its limit points are weak solutions of Equation (3.2).By uniqueness of weak solutions, πn converges in C([0, T ],M) to π .

It remains to see that IT (πn|γ ) converges to IT (π |γ ). Since πn → π and

IT ( · |γ ) is lower semi-continuous, we just need to check that lim supn IT (πn|γ ) �

IT (π |γ ). Here again the concavity and the boundness of χ help. Since ∇Hn con-verges in L2 to ∇H and χ is bounded, the main problem is to show that

lim supn→∞

∫ T

0dt〈χ(πn(t)), (∇H(t))2〉 �

∫ T

0dt〈χ(π(t)), (∇H(t))2〉.

Since π ∗ ιδ converges almost surely to π as δ ↓ 0,∫ T

0dt〈χ(π(t)), (∇H(t))2〉

= limδ→0

∫ T

0dt〈χ(π(t) ∗ ιδ), (∇H(t))2〉

= limδ→0

limn→∞

∫ T

0dt〈χ(πn(t) ∗ ιδ), (∇H(t))2〉.


Since χ is concave, the previous expression is bounded below by

limδ→0

lim supn→∞

∫ T

0dt〈χ(πn(t)) ∗ ιδ, (∇H(t))2〉.

Since χ is bounded and (∇H)2 integrable, a change of variables shows that theprevious expression is equal to

lim supn→∞

∫ T

0dt〈χ(πn(t)), (∇H(t))2〉,

concluding the proof of the lower bound.

4. The Rate Function for the Invariant Measure

In this section we discuss some properties of the functional S(ρ) which are neededlater. The results stated here are essentially contained in [9], but, for the sake ofcompleteness, we review them and give more detailed proofs. Without any lossof generality, from now on we shall assume that 0 < ρ− < ρ+ < 1. Recall thedefinitions of the set F , (2.9), and of the functional G(ρ, f ), (2.10).

The Euler–Lagrange equation associated to the variational problem (2.11) isgiven by the nonlinear boundary value problem

F ′′ = (ρ − F)(F ′)2

F(1 − F)in (−1, 1),

F (±1) = ρ±.(4.1)

We introduce the notation, which we will use throughout this section,

R(u) = R(ρ, F ;u) = (ρ(u)− F(u))F ′(u)

F (u)(1 − F(u)). (4.2)

Using this notation, Equation (4.1) takes the form

F ′′ = F ′R in (−1, 1),F (±1) = ρ±.

(4.3)

In order to state and prove an existence and uniqueness result for F ∈ F weformulate (4.3) as the integro-differential equation

F(u) = ρ− + (ρ+ − ρ−)∫ u

−1 dv exp{∫ v

−1 dwR(ρ, F ;w)}

∫ 1−1 dv exp

{∫ v

−1 dwR(ρ, F ;w)} . (4.4)

We will denote its solution by F = F(ρ) to emphasize its dependence on ρ. Weobserve that if ρ = ρ then F = F(ρ) = ρ solves (4.4) and (4.3).

Notice that if F ∈ C2([−1, 1]) is a solution of the boundary value problem (4.3)such that F ′(u) > 0 for u ∈ [−1, 1], then F is also a solution of the integro-differential equation (4.4). Conversely, if F ∈ C1([−1, 1]) is a solution of (4.4),then F ′(u) > 0, F ′′(u) exists for almost every u and (4.3) holds almost everywhere.Moreover, if ρ ∈ C([−1, 1]), then F ∈ C2([−1, 1]) and (4.3) holds everywhere.


Remark 4.1. There are nonmonotone solutions of Equation (4.3). For example,for the constant profile ρ = 1/2, it is easy to check that the functions

F(u) = 12 [1 + sin(λu+ ϕ)];

satisfy Equation (4.3) for countably many choices of the parameters λ and ϕ (fixedin order to satisfy the boundary conditions in (4.3)). However only one such func-tion is monotone. In fact, under the monotonicity assumption on F , we will proveuniqueness (and existence) of the solution of the boundary value problem (4.3).

The following theorem gives us the existence and uniqueness result for (4.4)together with a continuous dependence of the solution on ρ. Recall that we de-note by C1([−1, 1]) the Banach space of continuously differentiable functionsf : [−1, 1] → R endowed with the norm ‖f ‖C1 := supu∈[−1,1]{|f (u)| + |f ′(u)|}.

THEOREM 4.2. For each ρ ∈ M there exists in F a unique solution F = F(ρ)

of (4.4). Moreover:

(i) if ρ ∈ C([−1, 1]), then F = F(ρ) ∈ C2([−1, 1]) and it is the unique solutionin F ∩ C2([−1, 1]) of (4.3);

(ii) if ρn converges to ρ in M as n → ∞, then Fn = F(ρn) converges to F =F(ρ) in C1([−1, 1]);

(iii) fix T > 0 and consider a function ρ = ρ(t, u) ∈ C1,0([0, T ]×[−1, 1]). ThenF = F(t, u) = F(ρ(t, ·))(u) ∈ C1,2([0, T ] × [−1, 1]).

The existence result in Theorem 4.2 will be proven by applying Schauder’s fixedpoint theorem. For each ρ ∈ M consider the map Kρ : F → C1([−1, 1]) given by

Kρ(f )(u) := ρ− + (ρ+ − ρ−)

∫ u

−1 dv exp{∫ v

−1 dwR(ρ, f ;w)}

∫ 1−1 dv exp

{∫ v

−1 dwR(ρ, f ;w)} . (4.5)

Let us also define the following closed, convex subset of C1([−1, 1]):B := {

f ∈ C1([−1, 1]) : f (±1) = ρ±, b � f ′(u) � B} ⊂ F , (4.6)

where, recalling we are assuming γ− < γ+,

b := ρ+ − ρ−2

γ−γ+

, B := ρ+ − ρ−2

γ+γ−

.

LEMMA 4.3. For each ρ ∈ M, Kρ is a continuous map on F and Kρ(F ) ⊂ B.Furthermore Kρ(B) has compact closure in C1([−1, 1]). Hence, by Schauder’sfixed point theorem, for each ρ ∈ M Equation (4.4) has a solution F = Kρ(F ) ∈B. Moreover, there exist a constant C ∈ (0,∞) depending on ρ± such that for anyρ ∈ M and any u, v ∈ [−1, 1] we have |F ′(u)− F ′(v)| � C |u− v|.


Proof. It is easy to check that Kρ is continuous and Kρ(f )(±1) = ρ±. Let usdefine gρ := Kρ(f ), we have

g′ρ(u) = (ρ+ − ρ−)exp

{∫ u

−1 dwR(ρ, f ;w)}

∫ 1−1 dv exp

{∫ v

−1 dw R(ρ, f ;w)} · (4.7)

Since ρ(w)− f (w) � 1− f (w), ρ(w)− f (w) � −f (w), and f ′(w) � 0, we get

(1 − f )′

1− f� R � f ′

f

which implies b � g′ρ(u) � B for all u ∈ [−1, 1]. In particular Kρ(F ) ⊂ B.To show that Kρ(B) has a compact closure, by Ascoli–Arzela theorem, it is

enough to prove that g′ρ is Lipschitz uniformly for f ∈ B. Indeed, by using (4.7), itis easy to check that there exists a constant C = C(ρ−, ρ+) <∞ such that for anyu, v ∈ [−1, 1], any f ∈ B, and any ρ ∈ M we have |g′ρ(u)−g′ρ(v)| � C|u−v|. ✷

Proof of Theorem 4.2. The existence of solutions for (4.4) has been proven inLemma 4.3; to prove uniqueness we follow closely the argument in [9]. Considera solution F ∈ F of (4.4). Since it solves (4.3) almost everywhere, we get

F ′(u) = F ′(−1)+∫ u

−1dwF ′(w)R(ρ, F ;w) (4.8)

for all u in [−1, 1]. Moreover, taking into account that F is strictly increasing, weget from (4.3) that(

F(1 − F)

F ′

)′= 1 − F − ρ

holds a.e., so that

F(u)[1 − F(u)]F ′(u)

= ρ−[1 − ρ−]F ′(−1)

+∫ u

−1dv [1 − F(v)− ρ(v)] (4.9)

for all u in [−1, 1].Let F1, F2 ∈ F be two solutions of (4.4). If F ′

1(−1) = F ′2(−1) an application

of Gronwall inequality in (4.8) yields F1 = F2. We next assume F ′1(−1) < F ′

2(−1)and deduce a contradiction. Keep in mind that F ′

i > 0 because Fi belongs to Fand recall (4.9). Let u := inf{v ∈ (−1, 1] : F1(v) = F2(v)} which belongs to(−1, 1] because F1(±1) = F2(±1) and F ′

1(−1) < F ′2(−1). By definition of u,

F1(u) < F2(u) for any u ∈ (−1, u), F1(u) = F2(u) and F ′1(u) � F ′

2(u). By (4.9),we also obtain

F1(u)[1 − F1(u)]F ′

1(u)>

F2(u)[1− F2(u)]F ′

2(u)


or, equivalently, F ′1(u) < F ′

2(u), which is a contradiction and concludes the proofof the first statement of Theorem 4.2.

We turn now to statement (i). Existence follows from identity (4.3), which nowholds for all points u in [−1, 1] because ρ is continuous. Uniqueness follows fromthe uniqueness for the integro-differential formulation (4.4).

To prove (ii), let ρn be a sequence converging to ρ in M and denote by Fn =F(ρn) the corresponding solution of (4.4). By Lemma 4.3 and Ascoli–Arzela the-orem, the sequence Fn is relatively compact in C1([−1, 1]). It remains to showuniqueness of its limit points. Consider a subsequence nj and assume that Fnj

converges to G in C1([−1, 1]). Since ρnj converges to ρ in M and Fnj con-verges to G in C1([−1, 1]), by (4.5) Kρnj

(Fnj ) converges to Kρ(G). In particular,G = limj Fnj = limj Kρnj

(Fnj ) = Kρ(G) so that, by the uniqueness result,G = F(ρ). This shows that F(ρ) is the unique possible limit point of the sequenceFn, and concludes the proof of (ii).

We are left to prove (iii). If ρ(t, u) ∈ C1,0([0, T ] × [−1, 1]), we have from (i)and (ii) that F(t, u) = F(ρ(t, ·))(u) ∈ C0,2([0, T ] × [−1, 1]). We then just needto prove that F(t, u), as a function of t , is continuously differentiable. This will beaccomplished by Lemma 4.4 below. ✷

In order to prove the differentiability of t !→ F(t, u) := F(ρ(t, ·))(u) it isconvenient to introduce the new variable

ϕ(t, u) := logF(t, u)

1 − F(t, u), (t, u) ∈ [0, T ] × [−1, 1]. (4.10)

Note that ϕ ∈ [ϕ−, ϕ+] where ϕ± := log[ρ±/(1− ρ±)] = log γ± and u !→ ϕ(t, u)

is strictly increasing. We remark that, as discussed in [4], while the function F isanalogous to a density, the variable ϕ can be interpreted as a thermodynamic force.The advantage of using ϕ instead of F lies in the fact that, as a function of ϕ, thefunctional G is concave. This property plays a crucial role in the sequel.

Let us fix a density profile ρ ∈ C1,0([0, T ]×[−1, 1]). By (i)–(ii) in Theorem 4.2and elementary computations, we have that ϕ ∈ C0,2([0, T ]× [−1, 1]) and it is theunique strictly increasing (w.r.t. u) solution of the problem

(ϕ(t, u)

(∇ϕ(t, u))2 +1

1 + eϕ(t,u)= ρ(t, u), (t, u) ∈ [0, T ] × (−1, 1),

ϕ(t,±1) = ϕ±, t ∈ [0, T ].(4.11)

Note also that, by Lemma 4.3, there exists a constant C1 = C1(ρ−, ρ+) ∈ (0,∞)

such that1

C1� ∇ϕ(t, u) � C1, ∀(t, u) ∈ [0, T ] × [−1, 1]. (4.12)

LEMMA 4.4. Let ρ ∈ C1,0 ([0, T ] × [−1, 1]) and ϕ = ϕ(t, u) be the correspond-ing solution of (4.11). Then ϕ ∈ C1,2 ([0, T ] × [−1, 1]) and ψ(t, u) := ∂tϕ(t, u)

is the unique classical solution of the linear boundary value problem


∇[ ∇ψ(t, u)

(∇ϕ(t, u))2

]− eϕ(t,u)

(1+ eϕ(t,u))2ψ(t, u) = ∂tρ(t, u) (4.13)

for (t, u) ∈ [0, T ]×(−1, 1) with the boundary condition ψ(t,±1) = 0, t ∈ [0, T ].Proof. Fix t ∈ [0, T ], for h �= 0 such that t + h ∈ [0, T ] let us introduce

ψh(t, u) := [ϕ(t + h, u) − ϕ(t, u)]/h. Note that, by (i)–(ii) in Theorem 4.2,ψh(t, ·) ∈ C2 ([−1, 1]). By using (4.11), we get that ψh solves

∇[ ∇ψh(t, u)

∇ϕ(t, u)∇ϕ(t + h, u)

]− eϕ(t,u)

(1+ eϕ(t,u))(1+ eϕ(t+h,u))ehψh(t,u) − 1

h

= ρ(t + h, u)− ρ(t, u)

h(4.14)

for (t, u) ∈ [0, T ] × (−1, 1) with the boundary condition ψh(t,±1) = 0, t ∈[0, T ].

Multiplying the above equation by ψh(t, u) and integrating in du, after usingthe inequality x(ex − 1) � 0 and an integration by parts, we get∫ 1

−1du

(∇ψh(t, u))2

∇ϕ(t, u)∇ϕ(t + h, u)

�∣∣∣∣∫ 1

−1duψh(t, u)

ρ(t + h, u)− ρ(t, u)

h

∣∣∣∣� ε

∫ 1

−1duψh(t, u)

2 + 1

4ε

∫ 1

−1du

(ρ(t + h, u)− ρ(t, u)

h

)2

,

where we used Schwartz inequality with ε > 0. Recalling the Poincaré inequality(with f (±1) = 0)∫ 1

−1duf (u)2 � 4

π2

∫ 1

−1duf ′(u)2

using (4.12) and choosing ε small enough we finally find

lim suph→0

∫ 1

−1du

[∇ψh(t, u)]2 � C2

∫ 1

−1du

[∂tρ(t, u)

]2

for some constant C2 depending only on ρ+, ρ−.Hence, by Sobolev embedding, the sequence ψh(t, ·) is relatively compact in

C([−1, 1]). By taking the limit h → 0 in (4.14) it is now easy to show any limitpoint is a weak solution of (4.13). By classical theory on the one-dimensionalelliptic problems, see, e.g., [21, IV, §2.1], there exists a unique weak solutionof (4.13) which is in fact the classical solution since ∂tρ(t, ·) ∈ C([−1, 1]). Thisimplies there exists a unique limit point ψ(t, u) which is twice differentiable w.r.t.u. The continuity of t !→ ψ(t, ·) follows from the continuos dependence (in theC2([−1, 1]) topology) of the solution of (4.13) w.r.t. ∂tρ(t, ·) (in the C([−1, 1])topology). ✷


The link between the boundary value problem (4.3) and the variational prob-lem (2.11) is established by the following theorem.

THEOREM 4.5. Let S be the functional on M defined in (2.11). Then S is bounded,convex and lower semi-continuous on M. Moreover, for each ρ ∈ M, we have thatS(ρ) = G(ρ, F (ρ)) where F(ρ) is the solution of (4.4).

Proof. For each f ∈ F we have that G(·, f ) is a convex lower semi-continuousfunctional on M. Hence the functional S(·) defined in (2.11), being the supremumof convex lower semi-continuous functionals, is a convex lower semi-continuousfunctional on M. Furthermore, by choosing f = ρ in (2.11) we obtain that 0 �S0(ρ) � S(ρ). Finally, by using the concavity of x !→ log x, Jensen’s inequality,and f (±1) = ρ±, we get that G(ρ, f ) is bounded by some constant dependingonly on ρ− and ρ+.

In order to show the supremum in (2.11) is uniquely attained when f = F(ρ)

solves (4.4), it is convenient to make, as in Lemma 4.4, the change of variables ϕ =φ(f ) defined by ϕ(u) := log{f (u)/[1−f (u)]}. Note that f (u) = eϕ(u)/[1+eϕ(u)].We then need to show that the supremum of the functional

G(ρ, ϕ) := G(ρ, φ−1(ϕ))

=∫ 1

−1du

{ρ(u) log ρ(u)+ [1 − ρ(u)] log[1 − ρ(u)] +

+ [1 − ρ(u)]ϕ(u)− log[1+ eϕ(u)

]+ logϕ′(u)

[ρ+ − ρ−]/2

}(4.15)

for

ϕ ∈ F := φ(F ) = {ϕ ∈ C1([−1, 1]) : ϕ(±1) = ϕ±, ϕ′(u) > 0

}is uniquely attained when ϕ = φ(F(ρ)). We recall that F(ρ) denotes the solutionof (4.4).

Since the real functions x !→ log x and x !→ − log(1+ ex) are strictly concave,for each ρ ∈ M the functional G(ρ, ·) is strictly concave on F . Moreover, it iseasy to show that G(ρ, ·) is Gateaux differentiable on F with derivative given by⟨

δG(ρ, ϕ)

δϕ, g

⟩=

∫ 1

−1du

{g′(u)ϕ′(u)

+[

1

1+ eϕ(u)− ρ(u)

]g(u)

}.

By standard convex analysis, see, e.g., [11, I, Prop. 5.4], for any ϕ �= ψ ∈ F wehave

G(ρ,ψ) < G(ρ, ϕ)+⟨δG(ρ, ϕ)

δϕ,ψ − ϕ

⟩.

By noticing that δG(ρ, ϕ)/δϕ = 0 if ϕ solves (4.11) a.e. we conclude the proofthat the supremum on F of G(ρ, ·) is uniquely attained when ϕ = φ(F(ρ)). ✷


Remark 4.6. Given ρ ∈ M, let us consider a sequence ρn ∈ C2([−1, 1]) ∩ Mwith ρn(±1) = ρ±, bounded away from 0 and 1, which converges to ρ a.e. Then, bydominated convergence and (ii) in Theorem 4.2, we have S(ρn) = G(ρn, F (ρn))→G(ρ, F (ρ)) = S(ρ).

5. The Quasi Potential

In this section we show that the quasi potential for the one-dimensional boundarydriven simple exclusion process, as defined by the variational problem (2.6), coin-cides with the functional S(ρ) defined in (2.11). In the proof we shall also constructan optimal path for the variational problem (2.6).

Let us first recall the heuristic argument given in [4]. Taking into account therepresentation of the functional IT (π |ρ) given in Lemma 3.6, to the variationalproblem (2.6) is associated the Hamilton–Jacobi equation

1

2

⟨∇ δV

δρ, ρ(1 − ρ)∇ δV

δρ

⟩+

⟨δV

δρ,

1

2(ρ

⟩= 0, (5.1)

where∇ denotes the derivative w.r.t. the macroscopic space coordinate u ∈ [−1, 1].We look for a solution in the form

δV

δρ= log

ρ

1 − ρ− log

f

1− f

and obtain a solution of (5.1) provided f solves the boundary value problem (4.3),namely f = F(ρ). On the other hand, by Theorem 4.5, we have

δS(ρ)

δρ= δG(ρ, f )

δρ

∣∣∣∣f=F(ρ)

+ δG(ρ, f )

δf

∣∣∣∣f=F(ρ)

δF (ρ)

δρ

= logρ

1− ρ− log

F(ρ)

1 − F(ρ)

since (4.3) is the Euler–Lagrange equation for the variational problem (2.11). Weget therefore V = S since we have V (ρ) = S(ρ) = 0.

Let π∗(t) = π∗(t, u) be the optimal path for the variational problem (2.6)and define ρ∗(t) := π∗(−t). By using a time reversal argument, in [4] it is alsoshown that ρ∗(t) solves the hydrodynamic equation associated to the adjoint proces(whose generator is the adjoint of LN in L2(dµN)) which takes the form

∂tρ∗(t) = −1

2(ρ∗(t)+∇

(ρ∗(t)[1 − ρ∗(t)]∇ δS(ρ)

δρ

∣∣∣ρ=ρ∗(t)

). (5.2)

We will not develop here a mathematical theory of the Hamilton–Jacobi equa-tion (5.1). We shall instead work directly with the variational problem (2.6), mak-ing explicit computations for smooth paths and using approximation arguments to


prove that we have indeed V = S. Of course, the description of the optimal pathwill also play a crucial role.

To identify the quasi potential V with the functional S we shall prove sepa-rately the lower bound V � S and the upper bound V � S. For this purpose westart with two lemmata, which connect S defined in (2.11) to the Hamilton–Jacobiequation (5.1), used for both inequalities. The bound V � S will then be provenby choosing the right test field H in (2.4). To prove V � S we shall exhibit a pathπ∗(t) = π∗(t, u) which connects the stationary profile ρ to ρ in some time interval[0, T ] and such that IT (π∗|ρ) � S(ρ). As outlined above, this path ought to be thetime reversal of the solution of the adjoint hydrodynamic equation (5.2) with initialcondition ρ. The adjoint hydrodynamic equation needs, however, infinite time torelax to the stationary profile ρ. We have therefore to follow the time reversedadjoint hydrodynamic equation in a time interval [0, T1] to arrive at some profileρ∗(T1), which is close to ρ if T1 is large, and then interpolate, in some interval[T1, T1 + T2], between ρ∗(T1) and ρ.

Recall that we are assuming ρ− < ρ+ and pick δ0 > 0 small enough for δ0 �ρ− < ρ+ � 1 − δ0. For δ ∈ (0, δ0] and T > 0, we introduce

Mδ :={ρ ∈ C2([−1, 1]) : ρ(±1) = ρ±, δ � ρ(u) � 1 − δ

}, (5.3)

DT,δ :={π ∈ C1,2([0, T ] × [−1, 1]) :π(t,±1) = ρ±, δ � π(t, u) � 1 − δ

}. (5.4)

LEMMA 5.1. Let π ∈ DT,δ and denote by F(t, u) = F(π(t, ·)) (u) the solutionof the boundary value problem (4.3) with ρ replaced by π(t). Set

H(t, u) = logπ(t, u)

1− π(t, u)− log

F(t, u)

1 − F(t, u). (5.5)

Then, for each T � 0,

S(π(T ))− S(π(0)) =∫ T

0dt〈∂tπ(t), H(t)〉. (5.6)

Proof. Note that F(t, ·) is strictly increasing for any t ∈ [0, T ] and F ∈C1,2([0, T ] × [−1, 1]) by (iii) in Theorem 4.2. Moreover, since F(t,±1) = ρ±,we have ∂tF (t,±1) = 0. By Theorem 4.5, dominated convergence, an explicitcomputation, and an integration by parts, we get

d

dtS(π(t)) = d

dtG(π(t), F (t))

= 〈∂tπ(t), H(t)〉 +⟨∂tF (t),

1− π(t)

1 − F(t)− π(t)

F (t)

⟩+

⟨1

∇F(t), ∂t∇F(t)

⟩

= 〈∂tπ(t), H(t)〉 +⟨∂tF (t),

F (t)− π(t)

F (t)[1 − F(t)] +(F(t)

(∇F(t))2

⟩.


The lemma follows by noticing that the last term above vanishes by (4.3). ✷LEMMA 5.2. Let ρ ∈ Mδ , denote by F(u) = F(ρ) (u) the solution of theboundary value problem (4.3), and set

H(u) = logρ(u)

1 − ρ(u)− log

F(u)

1 − F(u)·

Then,⟨ρ(1− ρ), (∇H)2⟩+ 〈(ρ,H〉 = 0. (5.7)

Proof. Note that F ∈ Mδ by Theorem 4.2. After an integration by parts andsimple algebraic manipulations (5.7) is equivalent to

−⟨∇ρ, ∇F

F(1 − F)

⟩+

⟨ρ(1 − ρ),

( ∇FF(1 − F)

)2⟩= 0. (5.8)

We rewrite the first term on the left-hand side as

−⟨∇F, ∇F

F(1− F)

⟩−

⟨∇(ρ − F),

∇FF(1 − F)

⟩

which, by an integration by parts, is equal to

−⟨∇F, ∇F

F(1− F)

⟩+

⟨ρ − F,

(F

F(1 − F)− (1− 2F)(∇F)2

[F(1 − F)]2⟩.

Hence, the left-hand side of (5.8) is given by⟨ρ − F,

(F

F(1 − F)

⟩−

⟨(∇F)2

[F(1 − F)]2 , (ρ − F)2

⟩

=⟨

ρ − F

F(1 − F),(F − (ρ − F)

(∇F)2

F(1 − F)

⟩= 0

thanks to (4.3). ✷Note that, for smooth paths, Lemma 5.1 identifies, in the sense given by Equa-

tion (5.6), H as the derivative of S. Lemma 5.2 then states that this derivativesatisfies the Hamilton–Jacobi equation (5.1).

5.1. LOWER BOUND

We can now prove the first relation between the quasi potential V and the func-tional S.

LEMMA 5.3. For each ρ ∈ M we have V (ρ) � S(ρ).


Proof. In view of the variational definition V , to prove the lemma we needto show that S(ρ) � IT (π |ρ) for any T > 0 and any path π ∈ D([0, T ];M)

which connects the stationary profile ρ to ρ in the time interval [0, T ]: π(0) = ρ,π(T ) = ρ.

Fix such a path π and let us assume first that π ∈ DT,δ. Denote by F(t) =F(π(t)) the solution of the elliptic problem (4.3) with π(t) in place of ρ. In viewof the variational definition of IT (π |ρ) given in (2.5), to prove that S(ρ) � IT (π |ρ)it is enough to exhibit some function H ∈ C

1,20 ([0, T ]×[−1, 1]) for which S(ρ) �

JT,H,ρ(π). We claim that H given in (5.5) fulfills these conditions.We have that H ∈ C

1,20 ([0, T ] × [−1, 1]) because: π ∈ DT,δ by hypothesis,

F ∈ C1,2([0, T ] × [−1, 1]) by (iii) in Theorem 4.2, H(t,±1) = 0 since π(t, ·)and F(t, ·) satisfy the same boundary conditions. Recalling (2.4) we get, afterintegration by parts,

JT,H,ρ(π) =∫ T

0dt

⟨∂tπ(t), H(t)

⟩−−1

2

∫ T

0dt

[ ⟨H(t),(π(t)

⟩+ ⟨π(t)[1− π(t)], [∇H(t)]2⟩ ] .

By Lemmata 5.1 and 5.2 we then have JT,H,ρ(π) = S(ρ).Up to this point we have shown that S(ρ) � IT (π |ρ) for smooth paths π

bounded away from 0 and 1. In order to obtain this result for general paths, wejust have to recall the approximations performed in the proof of the lower bound ofthe large deviation principle. Fix a path π with finite rate function: IT (π |ρ) < ∞.In Section 3.6 we proved that there exists a sequence {πn, n � 1} of smoothpaths such that πn converges to π and IT (πn|ρ) converges to IT (π |ρ). Let πn bedefined by (1 − n−1)πn + n−1ρ. Since πn converges to π , πn converges to π . Bylower semi-continuity of the rate function, IT (π |ρ) � lim infn→∞ IT (πn|ρ). Onthe other hand, by convexity, IT (πn|ρ) � (1 − n−1)IT (πn|ρ) + n−1IT (ρ|ρ) =(1 − n−1)IT (πn|ρ) so that lim supn→∞ IT (πn|ρ) � IT (π |ρ). Since πn belongs toDT,δ for some δ = δn > 0, each path π with finite rate function can be approx-imated by a sequence πn in DT,δn , for some set of strictly positive parameters δn,and such that IT (π |ρ) = limn IT (πn|ρ). Therefore, by the result on smooth pathsand the lower semi-continuity of S, we get

IT (π |ρ) = limn

IT (πn|ρ) � lim infn

S(πn(T )) � S(π(T )),

which concludes the proof of the lemma. ✷

5.2. UPPER BOUND

The following lemma explains which is the right candidate for the optimal path forthe variational problem (2.6).


LEMMA 5.4. Fix δ ∈ (0, δ0], a profile α ∈ Mδ , and a path π ∈ DT,δ with finiterate function, IT (π |α) < ∞. Denote by F(t, u) = F(π(t, ·)) (u) the solution ofthe boundary value problem (4.3) with ρ replaced by π(t). Then there exists afunction K ∈ H1(π) such that π is the weak solution of

∂tπ = − 12(π + ∇

(π(1− π)∇

[log F

1−F +K

]),

(t, u) ∈ [0, T ] × (−1, 1),π(t,±1) = ρ±, t ∈ [0, T ],π(0, u) = α(u), u ∈ [−1, 1].

(5.9)

Moreover,

IT (π |α) = S(π(T ))− S(α)+ 1

2

∫ T

0dt

⟨π(t)[1− π(t)], [∇K(t)]2⟩. (5.10)

The optimal path for the variational problem (2.6) will be obtained by taking apath π∗ for which the last term on the right-hand side of the identity (5.10) (whichis positive) vanishes, namely for a path π∗ which satisfies (5.9) with K = 0. Thenρ∗(t) = π∗(−t) will be a solution of (5.2).

Proof. Denote by H the function in H1(π) introduced in Lemma 3.6, let Has defined in (5.5), and set K := H − H . Note that K belongs to H1(π) be-cause π ∈ DT,δ by hypothesis, F ∈ C1,2([0, T ] × [−1, 1]) by Theorem 4.2, andH(t,±1) = 0. Then (5.9) follows easily from (3.3). To prove the identity (5.10),replace in (5.6) ∂tπ(t) by the right-hand side of the differential equation in (5.9).After an integration by parts we obtain

S(π(T ))− S(α) =∫ T

0dt

{1

2

⟨H(t),(π(t)

⟩+ ⟨π(t)[1− π(t)], [∇H(t)]2⟩−

− ⟨π(t)[1− π(t)],∇K(t)∇H(t)⟩}

=∫ T

0dt

⟨π(t)[1− π(t)], 1

2[∇H(t)]2 −∇H(t)∇K(t)

⟩,

where we used Lemma 5.2. Recalling K = H −H , we thus obtain

S(π(T ))− S(α)+ 1

2

∫ T

0dt

⟨π(t)[1− π(t)], [∇K(t)]2⟩

= 1

2

∫ T

0dt

⟨π(t)[1− π(t)], [∇H(t)]2⟩,

which concludes the proof of the lemma in view of (3.4). ✷We write more explicitly the adjoint hydrodynamic equation (5.2). In the present

paper, we shall use it only to describe a particular path which will be shown to bethe optimal one. For ρ ∈ M, consider the nonlocal differential equation


∂tρ∗ = 1

2(ρ∗ − ∇

(ρ∗(1 − ρ∗)∇ log

F

1− F

), (t, u) ∈ (0,∞)× [−1, 1],

F (t, u) = F(ρ∗(t, ·))(u), (t, u) ∈ (0,∞)× [−1, 1],ρ∗(t,±1) = ρ±, t ∈ (0,∞),

ρ∗(0, u) = ρ(u), u ∈ [−1, 1],

(5.11)

where we recall that F(t, u) = F(ρ∗(t, ·)) (u) means that F(t, u) has to be ob-tained from ρ∗(t, u) by solving (4.4) with ρ(u) replaced by ρ∗(t, u). Since∇ log[F/(1−F)] > 0, in (5.11) there is a positive drift to the right. Let us describehow it is possible to construct the solution of (5.11).

LEMMA 5.5. For ρ ∈ M let .(t) be the solution of the heat equation (2.12) anddefine ρ∗ = ρ∗(t, u) by (2.13). Then ρ∗ ∈ C1,2((0,∞)×[−1, 1])∩C([0,∞);M)

and solves (5.11). Moreover, if δ � ρ(u) � 1 − δ a.e. for some δ > 0, thereexists δ′ = δ′(ρ−, ρ+, δ) ∈ (0, 1), for which δ′ � ρ∗(t, u) � 1 − δ′ for any(t, u) ∈ (0,∞)× [−1, 1].

Proof. Let F(u) = F(ρ) (u), then, by Theorem 4.2, F ∈ C1([−1, 1]) and,by Lemma 4.3, there is a constant C ∈ (0,∞) depending only on ρ−, ρ+ suchthat C−1 � F ′(u) � C for any u ∈ [−1, 1]. Since .(t, u) solves (2.12), thereexists C1 = C1(ρ−, ρ+) ∈ (0,∞) such that C−1

1 � (∇.)(t, u) � C1 for any(t, u) ∈ [0,∞) × [−1, 1]. Moreover, .(t,±1) = ρ± so that (.(t,±1) =2∂t.(t,±1) = 0. Hence, ρ∗ defined by (2.13) satisfies the boundary conditionρ∗(t,±1) = .∗(t,±1) = ρ±. Furthermore, ρ∗ ∈ C1,2((0,∞)× [−1, 1]) .

For the reader’s convenience, we reproduce below from [4, Appendix B] theproof that ρ∗(t, u), as defined in (2.13), solves the differential equation in (5.11).From (2.13) we get that

ρ∗(1 − ρ∗).(1−.)

= 1 + (1− 2.)(.

(∇.)2−.(1−.)

((.)2

(∇.)4

recalling (2.12), by a somehow tedious computation of the partial derivatives whichwe omit, we get

(∂t − 1

2()[.(1−.)

(.

(∇.)2

]= −∇

(ρ∗(1 − ρ∗).(1−.)

∇.)

from which, by using again (2.13), we see that ρ∗ satisfies the differential equationin (5.11).

To conclude the proof of the lemma, notice that ρ∗ is the solution of

∂tρ∗ = 1

2(ρ∗ − ∇{ρ∗(1− ρ∗)∇H },

ρ∗(t,±1) = ρ±,ρ∗(0, ·) = ρ(·),

for some function H in C1,1([0,∞)×[−1, 1]) for which∇H is uniformly bounded.Though H does not vanish at the boundary, we may use a weakly asymmetric


boundary driven exclusion process to prove the existence of a weak solution λ(t, u),in the sense of Subsection 3.4, which takes values in the interval [0, 1]. Since∇H isbounded, the usual H−1 method gives uniqueness so that λ = ρ∗ and 0 � ρ∗ � 1.In particular ρ∗ ∈ C([0,∞);M).

Assume now that δ � ρ � 1 − δ for some δ > 0. Fix t > 0 and assumethat ρ∗(t, ·) has a local maximum at −1 < u0 < 1. Since ρ∗ is a smooth solutionof (5.11), a simple computation gives that at (t, u0)

(∂tρ∗) = 1

2(ρ∗ − ρ∗(1− ρ∗)(∇F)2

F 2(1− F)2(ρ∗ + F − 1)

because

(∇ρ∗)(t, u0) = 0 and ( log

{F

1 − F

}= (∇F)2 (ρ

∗ + F − 1)

F 2(1 − F)2.

Since u0 is a local maximum, (ρ∗ � 0. On the other hand, assume that ρ∗(t, u0) >

1 − ρ−, in this case, since ρ− � F , ρ∗ + F − 1 > 0 so that ∂tρ∗ < 0. In thesame way we can conclude that (∂tρ∗)(t, u1) > 0 if u1 is a minimum of ρ∗(t, ·)and ρ∗(t, u1) � 1 − ρ+. These two estimates show that min{δ, 1 − ρ+, ρ−} �ρ∗(t, u) � max{1− δ, 1 − ρ−, ρ+}, which concludes the proof of the lemma. ✷

We now prove that the solution of (5.11), as constructed in Lemma 5.5, con-verges, as t → ∞, to ρ uniformly with respect to the initial datum ρ. We usebelow the usual notation ‖f ‖∞ := supu∈[−1,1] |f (u)|.LEMMA 5.6. Given ρ ∈ M, let ρ∗(t) = ρ∗(t, u) be the solution (5.11). Then,

limt→∞ sup

ρ∈M

∥∥ρ∗(t)− ρ∥∥∞ = 0.

Proof. Let us represent the solution .(t) of (2.12) in the form .(t, u) = ρ(u)+5(t, u). Then 5(t) = P 0

t 5(0) where P 0t is the semigroup generated by (1/2)(0,

with (0 the Dirichlet Laplacian on [−1, 1]. Since 5(0) = F(ρ)− ρ and since thesolution F(ρ) of (4.4) as well as ρ are contained in the interval [ρ−, ρ+], we havethat ‖5(0)‖∞ � |ρ+ − ρ−| < 1. Therefore, by standard heat kernel estimates,

limt→∞ sup

ρ∈M

{‖5(t)‖∞ + ‖∇5(t)‖∞ + ‖(5(t)‖∞} = 0,

the lemma follows recalling that, by Lemma 5.5, ρ∗(t) is given by (2.13). ✷Lemma 5.6 shows that we may join a profile ρ in M to a neighborhood of the

stationary profile by using Equation (5.11) for a time interval [0, T1] which at thesame time regularizes the profile. On the other hand, from Lemma 5.4 we shalldeduce that this path pays S(ρ) − S(ρ∗(T1)). It thus remains to connect ρ∗(T1),which is a smooth profile close to the stationary profile ρ for large T1, to ρ. In thenext lemma we show this can be done by paying only a small price. We denote by‖ · ‖2 the norm in L2([−1, 1], du).


LEMMA 5.7. Let α ∈ Mδ0 be a smooth profile such that ‖α − ρ‖∞ � δ0/(16).Then there exists a smooth path π (t), t ∈ [0, 1] with δ0/2 � π � 1− δ0/2, namelyπ ∈ D1,δ0/2, with π(0) = ρ, π (1) = α and a constant C = C(δ0) ∈ (0,∞) suchthat

I1(π |ρ) � C‖α − ρ‖22.

In particular V (α) � C‖α − ρ‖22.

We remark that by using the ‘straight path’ π (t) = ρ (1 − t) + α t one wouldget a bound in terms of the H1 norm of α − ρ. Below, by choosing a more cleverpath, we get instead a bound only in term of the L2 norm.

Proof. Let (ek, λk), k � 1 be the spectral basis for −(1/2)(0, where (0 isthe Dirichlet Laplacian on [−1, 1], namely {ek}k�1 is an complete orthonormalsystem in L2([−1, 1], du) and −(1/2)(0ek = λkek. Explicitly we have ek(u) =cos(kπu/2) and λk = k2π2/8. We claim that the path π (t) = π(t, u), (t, u) ∈[0, 1] × [−1, 1] given by

π (t) = ρ +∞∑k=1

eλkt − 1

eλk − 1〈α − ρ, ek〉ek (5.12)

fulfills the conditions stated in the lemma.It is immediate to check that π (0) = ρ, π(1) = α and π(t,±1) = ρ±. Further-

more, by the smoothness assumption on α, we get that π ∈ C1,2([0, 1] × [−1, 1]).In order to show that δ0/2 � π � 1 − δ0/2, let us write π(t) = ρ + q(−t), thenq(t) = q(t, u), (t, u) ∈ [−1, 0] × [−1, 1] solves

∂tq(t) = 12(q(t) + g,

q(t,±1) = 0,

q(−1, u) = α(u)− ρ(u),

where g = g(u) is given by

g = −∞∑k=1

λk

eλk − 1〈α − ρ, ek〉ek.

Let us denote by ‖g‖H1 := ‖g′‖2 the H1 norm in [−1, 1]; a straightforwardcomputation shows

‖g‖2H1

=∞∑k=1

2λk

(λk

eλk − 1

)2

〈α − ρ, ek〉2 � 8

λ1

∞∑k=1

〈α − ρ, ek〉2

� 8

λ1‖α − ρ‖2

2 �(

8

π

)2

2

(δ0

16

)2

= 1

2π2δ2

0,

where we used that, for λ > 0, we have eλ − 1 � λ2/2.


Let P 0t = exp{t(0/2} be the heat semigroup on [−1, 1]; since ‖g‖∞ �√

2‖g‖H1 , we have

supt∈[−1,0]

‖q(t)‖∞

= supt∈[−1,0]

∥∥∥∥P 0t+1(α − ρ)+

∫ t

−1dsP 0

t−sg∥∥∥∥∞ � δ0

16+ 1

πδ0 � 7

16δ0,

so that π ∈ D1,δ0/2.We shall estimate I1(π |ρ) by using the representation given in Lemma 3.6. To

this end, let us define h = h(t, u) ∈ C([0, 1] × [−1, 1] by h := −∂t π + (1/2)(π

and let H = H(t, u) be the solution of

∇(π [1− π]∇H ) = h,

H(t,±1) = 0,

so that π solves (3.3) with H as above which belongs to H1(π).Let us denote by ‖ · ‖H−1 the usual negative Sobolev norm in [−1, 1], namely

‖h‖2H−1

:= supf �=0,f (±1)=0

〈f, h〉2〈∇f,∇f 〉 =

∞∑k=1

1

2λk〈h, ek〉2.

By using that π[1 − π ] � (δ0/2)2 a simple computations shows that∫ 1

0dt

⟨π(t)[1 − π(t)], (∇H(t))2

⟩� 4

δ20

∫ 1

0dt‖h(t)‖2

H−1.

By using the explicit expression for π we get

h(t) = −∞∑k=1

λk2eλkt − 1

eλk − 1〈α − ρ, ek〉ek,

hence, by a direct computation,

‖h(t)‖2H−1

=∞∑k=1

1

2λk

(λk

2eλkt − 1

eλk − 1

)2

〈α − ρ, ek〉2

�∞∑k=1

8λke2λk(t−1)〈α − ρ, ek〉2,

where we used that for λ � λ1 we have eλ � 2. We thus get

I1(π |ρ) � 2

δ20

∫ 1

0dt‖h(t)‖2

H−1� 8

δ20

∞∑k=1

〈α − ρ, ek〉2 = 8

δ20

‖α − ρ‖22

which concludes the proof of the lemma. ✷We can now prove the upper bound for the quasi potential and conclude the

proof of Theorem 2.1.


LEMMA 5.8. For each ρ ∈ M, we have V (ρ) � S(ρ).Proof. Fix 0 < ε < δ0/(32), ρ ∈ M and let ρ∗(t, u) be the solution of (5.11)

with initial condition ρ. By Lemma 5.6 there exists T1 = T1(ε) such that ‖ρ∗(t)−ρ‖∞ < ε for any t � T1. Let α := ρ∗(T1) and let π be the path which connects ρ

to α in the interval [0, 1] constructed in Lemma 5.7.Let T := T1 + 1 and π∗(t), t ∈ [0, T ] the path

π∗(t) ={π(t), for 0 � t � 1,ρ∗(T − t), for 1 � t � T .

(5.13)

By Remark 4.6, given ρ ∈ M as above, we can find a sequence {ρn, n � 1}withρn ∈ Mδn for some δn > 0 converging to ρ in M and such that S(ρn) converges toS(ρ). Let us denote by ρn,∗ the solution of (5.11) with initial condition ρn and set

πn,∗(t) ={π n,∗(t), for 0 � t � 1,ρn,∗(T − t), for 1 � t � T ,

(5.14)

where π n,∗(t) is the path joining ρ to αn := ρn,∗(T1) in the time interval [0, 1]constructed in Lemma 5.7. We claim that the path πn,∗ defined above converges inD([0, T ],M) to π∗, as defined in (5.13). Before proving this claim, we concludethe proof of the lemma.

By the lower semi continuity of the functional IT (·|ρ) on D([0, T ],M) we have

IT(π∗∣∣ρ)

� lim infn

IT(πn,∗∣∣ρ)

. (5.15)

On the other hand, by definition of the rate function and its invariance with respectto time shifts we get

IT(πn,∗∣∣ρ) = I1

(π n,∗∣∣ρ)+ IT1

(ρn,∗(T1 − ·)∣∣ρn,∗(T1)

). (5.16)

By Theorem 4.2, Fn := F(ρn) converges to F = F(ρ) in C1([−1, 1]) sothat .n(t), the solution of (2.9) with initial condition Fn, converges to .(t) inC2([−1, 1]) for any t > 0. Hence, by (2.13), ρn,∗(T1) converges to ρ∗(T1) inC([−1, 1]). Recalling that ‖ρ∗(T1) − ρ‖∞ < ε � δ0/(32), we can find N0 =N0(δ0) such that for any n � N0 we have ‖ρn,∗(T1)− ρ‖∞ < ε � δ0/(16). We canthus apply Lemma 5.7 and get, for n � N0

I1(π n,∗∣∣ρ)

� C‖ρn,∗(T1)− ρ‖22 (5.17)

for some constant C = C(δ0).By Lemma 5.5, ρn,∗(T1 − t), t ∈ [0, T1] is smooth and bounded away from 0

and 1, namely it belongs to DT1,δn for some δn > 0. We can thus apply Lemma 5.4and conclude, as ρn,∗(T1 − t) solves (5.9) with K = 0,

IT1

(ρn,∗(T1 − ·)∣∣ρn,∗(T1)

) = S(ρn)− S(ρn,∗(T1)) � S(ρn). (5.18)


From Equations (5.15)–(5.18), we now get

IT (π∗|ρ) � lim inf

n

[S(ρn)+ C‖ρn,∗(T1)− ρ‖2

2

]= S(ρ)+ C‖ρ∗(T1)− ρ‖2

2 � S(ρ)+ 2Cε2

and we are done by the arbitrariness of ε.We are left to prove that πn,∗ → π∗ in D([0, T ],M). We show that πn,∗ con-

verges to π∗ in C([0, T ];M). Pick ε1 ∈ (0, T1]; since ρn,∗(t) converges to ρ∗(t)in C([−1, 1]) uniformly for t ∈ [ε1, T1] we conclude easily that πn,∗ converges toπ∗ in C([1, T − ε1] × [−1, 1]). We recall that, by Lemma 4.3, ∇Fn(t) and ∇F(t)

are uniformly bounded. Moreover, πn,∗(T − t) and π∗(T − t), t ∈ [T − T1, T ] areweak solutions of (5.11); for each G ∈ C([−1, 1]) we thus get

limε1↓0

lim supn

supt∈[T−ε1,T ]

∣∣〈πn,∗(t),G〉 − 〈π∗(t),G〉∣∣ = 0.

This concludes the proof that ρn,∗ converges to ρ∗ in C([1, T ];M). Since ρ∗,n(T1)

converges to ρ∗,n(T1) in C2([−1, 1]) it is easy to show that π n,∗ converges to π∗ inC([0, 1] × [−1, 1]). Hence πn,∗ converges to π∗ in C([0, T ];M). ✷

Appendix: A Lower Bound on the Quasi-Potential (d �� 1)

In this appendix we prove a lower bound for the quasi potential in the d-dimensionalboundary driven simple exclusion process. For d = 1 this bound has been derivedfrom (2.11) in [8, 9].

Let � ⊂ Rd be a smooth bounded open set and define �N := Z

d∩N�. Let alsoγ (u) be a smooth function defined in a neighborhood of ∂�. The d-dimensionalboundary driven symmetric exclusion process is then the process on the state space�N := {0, 1}�N with generator

LNf (η) = N2

2

∑{x,y}⊂�N|x−y|=1

[f (σ x,yη)− f (η)

]++ N2

2

∑x∈�N ,y �∈�N|x−y|=1

(η(x) + [1 − η(x)]γ

(y

N

))[f (σ xη)− f (η)

],

where σ x,y and σ x have been defined in Section 2.The hydrodynamic equation is given by the heat equation in �, namely

∂tρ = 12(ρ, u ∈ �,

ρ(t, u) = α(u), u ∈ ∂�,

ρ(0, u) = ρ0(u),


where α(u) = γ (u)/[1 + γ (u)]. We shall denote by ρ = ρ(u), u ∈ � the uniquestationary solution of the hydrodynamic equation.

By the same arguments as the ones given in Section 3 it is possible to prove thedynamical large deviation principle for the empirical measure. The rate function isstill given by the variational formula (2.5), but we now have

JT,H,ρ(π) := ⟨π(T ),H(T )

⟩− 〈ρ,H(0)〉 −−

∫ T

0dt

⟨π(t), ∂tH(t)+ 1

2(H(t)⟩−

− 1

2

∫ T

0dt

⟨χ(π(t)), (∇H(t))2

⟩++ 1

2

∫ T

0dt

∫∂�

dσ (u)α(u)∂nH(t, u),

where ∂nH(t, u) is the normal derivative of H(t, u) (n being the outward normalto �) and σ (u) is the surface measure on ∂�.

Let us define the quasi potential V (ρ) as in (2.6) and set

S0(ρ) :=∫�

du

[ρ(u) log

ρ(u)

ρ(u)+ [1 − ρ(u)] log

1 − ρ(u)

1 − ρ(u)

].

THEOREM A.1. For each ρ ∈ M we have V (ρ) � S0(ρ).Proof. We shall prove that IT (π |ρ) � S0(ρ) for any π(·) such that π(0) = ρ

and π(T ) = ρ. Let us assume first that π ∈ C1,2([0, T ] ×�), π(t, u) = α(u) for(t, u) ∈ [0, T ] × ∂�, and π is bounded away from 0 and 1. Given such π we usethe variational characterization of IT and chose

H(t, u) = logπ(t, u)

1 − π(t, u)− log

ρ(u)

1− ρ(u).

Note that H(t, u) = 0 for (t, u) ∈ [0, T ] × ∂� since π and ρ satisfy the sameboundary condition. By dominated convergence and an explicit computation weget

S0(π(T ))− S0(π(0)) =∫ T

0dt

d

dtS0(π(t)) =

∫ T

0dt 〈∂tπ(t),H(t)〉.

Recalling that JT,H,ρ(π) has been defined above, a simple computation shows

JT,H,ρ(π)

= S0(π(T ))+ 1

2

∫ T

0dt 〈∇H(t),∇π(t)− π(t)[1− π(t)]∇H(t)〉

= S0(π(T ))+ 1

2

∫ T

0dt

⟨(∇uρ)

2

[ρ(1 − ρ)]2 (π(t)− ρ)2

⟩,


since the second term above is positive we conclude the proof of the lemma forsmooth paths. To get the general result it is enough to repeat the approximationused in Lemma 5.3. ✷

Acknowledgements

We are grateful to G. Dell’Antonio for a useful discussion on the variational prob-lem which led to Lemma 4.4. We also thank T. Bodineau and G. Giacomin forstimulating discussions.

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269

Algebras of Random Operators Associated toDelone Dynamical Systems �

DANIEL LENZ and PETER STOLLMANNFakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany.e-mail: {d.lenz,p.stollmann}@mathematik.tu-chemnitz.de

(Received: 29 November 2002)

Abstract. We carry out a careful study of operator algebras associated with Delone dynamicalsystems. A von Neumann algebra is defined using noncommutative integration theory. Features ofthese algebras and the operators they contain are discussed. We restrict our attention to a certainC∗-subalgebra to discuss a Shubin trace formula.

Mathematics Subject Classifications (2000): 46L60, 47B80, 82B44, 52C23.

Key words: operator algebras, groupoids, random operators, aperiodic tilings, quasicrystals.

Introduction

This paper is part of a study of Hamiltonians for aperiodic solids. Among them,special emphasis is laid on models for quasicrystals. To describe aperiodic order,we use Delone (Delaunay) sets. Here we construct and study certain operator alge-bras which can be naturally associated with Delone sets and reflect the aperiodicorder present in a Delone dynamical system. In particular, we use Connes non-commutative integration theory to build a von Neumann algebra. This is achievedin Section 2 after some preparatory definitions and results gathered in Section 1.Let us stress the following facts: it is not too hard to write down explicitly the vonNeumann algebra N (�, T ,µ) of observables, starting from a Delone dynamicalsystem (�, T ) with an invariant measure µ. As in the case of random operators,the observables are families of operators, indexed by a set � of Delone sets. Thisset represents a type of (aperiodic) order and the ergodic properties of (�, T ) canoften be expressed by combinatorial properties of its elements ω. The latter arethought of as realizations of the type of disorder described by (�, T ). The algebraN (�, T ,µ) incorporates this disorder and plays the role of a noncommutativespace underlying the algebra of observables. To see that this algebra is in fact a vonNeumann algebra is by no means clear. At that point the analysis of Connes [9]enters the picture.

� Research partly supported by the DFG in the priority program Quasicrystals.

270 DANIEL LENZ AND PETER STOLLMANN

In order to verify the necessary regularity properties we rely on work donein [29], where we studied topological properties of a groupoid that naturally comeswith (�, T ). Using this, we can construct a measurable (even topological)groupoid. Any invariant measure µ on the dynamical system gives rise to a transver-sal measure and the points of the Delone sets are used to define a random Hilbertspace H . This latter step specifically uses the fact that we are dealing with a dy-namical system consisting of point sets and leads to a noncommutative randomvariable that has no analogue in the general framework of dynamical systems. Weare then able to identify N (�, T ,µ) as End(H). While in our approach we usenoncommutative integration theory to verify that a certain algebra is a von Neu-mann algebra, we should also like to point out that at the same time we provideinteresting examples for the theory. Of course, tilings have been considered inthis connection quite from the start as seen on the cover of [10]. However, weemphasize the point of view of concrete operators and thus are led to a somewhatdifferent setup.

The study of traces on this algebra is started in Section 3. Traces are intimatelylinked to transversal functions on the groupoid. These can also be used to studycertain spectral properties of the operator families constituting the von Neumannalgebra. For instance, spectral properties are almost surely constant for the mem-bers of any such family. This type of results is typical for random operators. In fact,we regard the families studied here in this random context. An additional featurethat is met here is the dependence of the Hilbert space on the random parameterω ∈ �.

In Section 4 we introduce a C∗-algebra that had already been encountered in adifferent form in [6, 17]. Our presentation here is geared towards using the elementsof the C∗-algebra as tight binding Hamiltonians in a quantum mechanical descrip-tion of disordered solids (see [6] for related material as well). We relate certainspectral properties of the members of such operator families to ergodic features ofthe underlying dynamical system. Moreover, we show that the eigenvalue countingfunctions of these operators are convergent. The limit, known as the integrated den-sity of states, is an object of fundamental importance from the solid state physicspoint of view. Apart from proving its existence, we also relate it to the canonicaltrace on the von Neumann algebra N (�, T ,µ) in case that the Delone dynamicalsystem (�, T ) is uniquely ergodic. Results of this genre are known as Shubin’strace formula due to the celebrated results from [36].

We conclude this section with two further remarks.Firstly, let us mention that starting with the work of Kellendonk [17], C∗-

algebras associated to tilings have been subject to intense research within the frame-work of K-theory (see, e.g., [18, 19, 32]). This can be seen as part of a programoriginally initiated by Bellissard and his coworkers in the study of so-called gap-labelling for almost periodic operators [3–5]. While the C∗-algebras we encounterare essentially the same, our motivation, aims and results are quite different.

ALGEBRAS OF RANDOM OPERATORS 271

Secondly, let us remark that some of the results below have been announced in[28, 29]. A stronger ergodic theorem will be found in [30] and a spectral theoreticapplication is given in [20].

1. Delone Dynamical Systems and Coloured Delone Dynamical Systems

In this section we recall standard concepts from the theory of Delone sets and intro-duce a suitable topology on the closed sets in Euclidian space. A slight extensionconcerns the discussion of coloured (decorated) Delone sets.

A subset ω of Rd is called a Delone set if there exist 0 < r,R < ∞ such that

2r � ‖x − y‖ whenever x, y ∈ ω with x �= y, and BR(x) ∩ ω �= ∅ for all x ∈ Rd .

Here, the Euclidean norm on Rd is denoted by ‖ · ‖ and Bs(x) denotes the (closed)

ball in Rd around x with radius s. The set ω is then also called an (r, R)-set. We

will sometimes be interested in the restrictions of Delone sets to bounded sets. Inorder to treat these restrictions, we introduce the following definition.

DEFINITION 1.1. (a) A pair (,Q) consisting of a bounded subset Q of Rd and

⊂ Q finite is called a pattern. The set Q is called the support of the pattern.(b) A pattern (,Q) is called a ball pattern if Q = Bs(x) with x ∈ for

suitable x ∈ Rd and s ∈ (0,∞).

The pattern (1,Q1) is contained in the pattern (2,Q2)written as (1,Q1) ⊂(2,Q2) if Q1 ⊂ Q2 and 1 = Q1 ∩ 2. Diameter, volume, etc., of a pattern aredefined to be the diameter, volume, etc., of its support. For patterns X1 = (1,Q1)

and X2 = (2,Q2), we define �X1X2, the number of occurrences of X1 in X2, tobe the number of elements in {t ∈ R

d : 1 + t ⊂ 2,Q1 + t ⊂ Q2}.For further investigation we will have to identify patterns that are equal up to

translation. Thus, on the set of patterns we introduce an equivalence relation bysetting (1,Q1) ∼ (2,Q2) if and only if there exists a t ∈ R

d with 1 = 2 + t

and Q1 = Q2 + t . In this latter case we write (1,Q1) = (2,Q2) + t . Theclass of a pattern (,Q) is denoted by [(,Q)]. The notions of diameter, volume,occurrence, etc., can easily be carried over from patterns to pattern classes.

Every Delone set ω gives rise to a set of pattern classes, P (ω) viz P (ω) ={[Q ∧ ω] : Q ⊂ R

d bounded and measurable}, and to a set of ball pattern classesPB(ω)) = {[Bs(x) ∧ ω] : x ∈ ω, s > 0}. Here we set Q ∧ ω = (ω ∩Q,Q).

For s ∈ (0,∞), we denote by P sB(ω) the set of ball patterns with radius s; note

the relation with s-patches as considered in [21]. A Delone set is said to be of finitelocal complexity if for every radius s the set P s

B(ω) is finite. We refer the readerto [21] for a detailed discussion of Delone sets of finite type.

Let us now extend this framework a little, allowing for coloured Delone sets.The alphabet A is the set of possible colours or decorations. An A-coloured Deloneset is a subset ω ⊂ R

d × A such that the projection pr1(ω) ⊂ Rd onto the first

coordinate is a Delone set. The set of all A-coloured Delone sets is denoted by DA.


Of course, we speak of an (r, R)-set if pr1(ω) is an (r, R)-set. The notions ofpattern, diameter, volume of pattern, etc., easily extend to coloured Delone sets, e.g.

DEFINITION 1.2. A pair (,Q) consisting of a bounded subset Q of Rd and

⊂ Q×A finite is called an A-decorated pattern. The set Q is called the supportof the pattern.

A coloured Delone set ω is thus viewed as a Delone set pr1(ω) whose pointsx ∈ pr1(ω) are labelled by colours a ∈ A. Accordingly, the translate Ttω of acoloured Delone set ω ⊂ R

d × A is given by

Ttω = {(x + t, a) : (x, a) ∈ ω}.From [29] we infer the notion of the natural topology, defined on the set F (Rd) ofclosed subsets of R

d . Since in our subsequent study in [30] the alphabet is supposedto be a finite set, the following construction will provide a suitable topology forcoloured Delone sets. Define, for a ∈ A,

pa: DA → F (Rd), pa(ω) = {x ∈ Rd : (x, a) ∈ ω}.

The initial topolgy on DA with respect to the family (pa)a∈A is called the naturaltopology on the set of A-decorated Delone sets. It is obvious that metrizability andcompactness properties carry over from the natural topology without decorationsto the decorated case.

Finally, the notions of Delone dynamical system and Delone dynamical systemof finite local complexity carry over to the coloured case in the obvious manner.

DEFINITION 1.3. Let A be a finite set. (a) Let � be a set of Delone sets. Thepair (�, T ) is called a Delone dynamical system (DDS) if � is invariant under theshift T and closed in the natural topology.

(a′) Let � be a set of A-coloured Delone sets. The pair (�, T ) is called an A-coloured Delone dynamical system (A-DDS) if � is invariant under the shift T andclosed in the natural topology.

(b) A DDS (�, T ) is said to be of finite local complexity if⋃

ω∈� PsB(ω) is finite

for every s > 0.(b′) An A-DDS (�, T ) is said to be of finite local complexity if

⋃ω∈� P

sB(ω) is

finite for every s > 0.(c) Let 0 < r,R < ∞ be given. A DDS (�, T ) is said to be an (r, R)-system if

every ω ∈ � is an (r, R)-set.(c′) Let 0 < r,R < ∞ be given. An A-DDS (�, T ) is said to be an (r, R)-

system if every ω ∈ � is an (r, R)-set.(d) The set P (�) of pattern classes associated to a DDS � is defined by

P (�) = ⋃ω∈� P (ω).

In view of the compactness properties known for Delone sets, [29], we get that� is compact whenever (�, T ) is a DDS or an A-DDS.


2. Groupoids and Noncommutative Random Variables

In this section we use concepts from Connes noncommutative integration theory [9]to associate a natural von Neumann algebra with a given DDS (�, T ). To do so,we introduce

• a suitable groupoid G(�, T ),

• a transversal measure = µ for a given invariant measure µ on (�, T ),

• and a -random Hilbert space H = (Hω)ω∈�,

leading to the von Neumann algebra

N (�, T ,µ) := End(H)

of random operators, all in the terminology of [9]. Of course, all these objectswill now be properly defined and some crucial properties have to be checked. Partof the topological prerequisites have already been worked out in [29]. Note thatcomparing the latter with the present paper, we put more emphasis on the relationwith noncommutative integration theory.

The definition of the groupoid structure is straightforward see also [6], Sect. 2.5.A set G together with a partially defined associative multiplication ·: G2 ⊂ G ×G → G, and an inversion −1: G → G is called a groupoid if the followingholds:

• (g−1)−1 = g for all g ∈ G,

• If g1 · g2 and g2 · g3 exist, then g1 · g2 · g3 exists as well,

• g−1 · g exists always and g−1 · g · h = h, whenever g · h exists,

• h · h−1 exists always and g · h · h−1 = g, whenever g · h exists.

A groupoid is called a topological groupoid if it carries a topology makinginversion and multiplication continuous. Here, of course, G×G carries the producttopology and G2 ⊂ G × G is equipped with the induced topology.

A given groupoid G gives rise to some standard objects: The subset G0 ={g · g−1 | g ∈ G} is called the set of units. For g ∈ G, we define its range r(g) byr(g) = g · g−1 and its source by s(g) = g−1 · g. Moreover, we set Gω = r−1({ω})for any unit ω ∈ G0. One easily checks that g · h exists if and only if r(h) = s(g).

By a standard construction we can assign a groupoid G(�, T ) to a Delone dy-namical system. As a set G(�, T ) is just � × R

d . The multiplication is given by(ω, x)(ω − x, y) = (ω, x + y) and the inversion is given by (ω, x)−1 = (ω − x,

−x). The groupoid operations can be visualized by considering an element (ω, x)as an arrow ω − x

x−→ ω. Multiplication then corresponds to concatenation ofarrows; inversion corresponds to reversing arrows and the set of units G(�, T )0

can be identified with �.


Apparently this groupoid G(�, T ) is a topological groupoid when� is equippedwith the topology of the previous section and R

d carries the usual topology.The groupoid G(�, T ) acts naturally on a certain topological space X. This

space and the action of G on it are of crucial importance in the sequel. The spaceX is given by

X = {(ω, x) ∈ G : x ∈ ω} ⊂ G(�, T ).

In particular, it inherits a topology form G(�, T ). This X can be used to definea random variable or measurable functor in the sense of [9]. Following the latterreference, p. 50f, this means that we are given a functor F from G to the categoryof measurable spaces with the following properties:

• For every ω ∈ G0 we are given a measure space F(ω) = (Yω, βω).• For every g ∈ G we have an isomorphism F(g) of measure spaces, F(g): Ys(g)

→ Yr(g) such that F(g1g2) = F(g1)F (g2), whenever g1g2 is defined, i.e.,whenever s(g1) = r(g2).

• A measurable structure on the disjoint union

Y =⋃ω∈�

Yω

such that the projection π : Y → � is measurable as well as the naturalbijection of π−1(ω) to Yω.

• The mapping ω �→ βω is measurable.

We will use the notation F : G� Y to abbreviate the above.Let us now turn to the groupoid G(�, T ) and the bundle X defined above. Since

X is closed ([29], Prop. 2.1), it carries a reasonable Borel structure. The projectionπ : X → � is continuous, in particular measurable. Now, we can discuss the actionof G on X. Every g = (ω, x) gives rise to a map J (g): Xs(g) → Xr(g), J (g)(ω −x, p) = (ω, p + x). A simple calculation shows that J (g1g2) = J (g1)J (g2) andJ (g−1) = J (g)−1, whenever s(g1) = r(g2). Thus, X is an G-space in the senseof [27]. It can be used as the target space of a measurable functor F : G� X. Whatwe still need is a positive random variable in the sense of the following definition,taken from [29]. First some notation:

Given a locally compact space Z, we denote the set of continuous functions onZ with compact support by Cc(Z). The support of a function in Cc(Z) is denotedby supp(f ). The topology gives rise to the Borel-σ -algebra. The measurable non-negative functions with respect to this σ -algebra will be denoted by F +(Z). Themeasures on Z will be denoted by M(Z).

DEFINITION 2.1. Let (�, T ) be an (r, R)-system.(a) A choice of measures β: � → M(X) is called a positive random variable

with values in X if the map ω �→ βω(f ) is measurable for every f ∈ F +(X), βω


is supported on Xω, i.e., βω(X − Xω) = 0, ω ∈ �, and β satisfies the followinginvariance condition∫

Xs(g)

f (J (g)p) dβs(g)(p) =∫

Xr(g)

f (q) dβr(g)(q)

for all g ∈ G and f ∈ F +(Xr(g)).(b) A map � × Cc(X) → C is called a complex random variable if there exist

an n ∈ N, positive random variables βi , i = 1, . . . , n and λi ∈ C, i = 1, . . . , nwith βω(f ) = ∑n

i=1 βωi (f ).

We are now heading towards introducing and studying a special random vari-able. This variable is quite important as it gives rise to the *2-spaces on which theHamiltonians act. Later we will see that these Hamiltonians also induce randomvariables.

PROPOSITION 2.2. Let (�, T ) be an (r, R)-system. Then the map α: � →M(X), αω(f ) = ∑

p∈ω f (p) is a random variable with values in X. Thus thefunctor Fα given by Fα(ω) = (Xω, αω) and Fα(g) = J (g) is measurable.

Proof. See [29], Corollary 2.6. ✷Clearly, the condition that (�, T ) is an (r, R)-system is used to verify the mea-

surability conditions needed for a random variable. We should like to stress thefact that the above functor given by X and α• differs from the canonical choice,possible for any dynamical system. In the special case at hand this canonical choicereads as follows:

PROPOSITION 2.3. Let (�, T ) be a DDS. Then the map ν: � → M(G), νω(f )= ∫

Rd f (ω, t) dt is a transversal function, i.e., a random variable with values in G.

Actually, one should possibly define transversal functions before introducingrandom variables. Our choice to do otherwise is to underline the specific functorused in our discussion of Delone sets. As already mentioned above, the analogueof the transversal function ν from Proposition 2.3 can be defined for any dynamicalsystem. In fact this structure has been considered by Bellissard and coworkers in aC∗-context. The notion almost random operators has been coined for that; see [3]and the literature quoted there.

After having encountered functors from G to the category of measurable spacesunder the header random variable or measurable functor, we will now meet randomHilbert spaces. By that one designates, according to [9], a representation of G inthe category of Hilbert spaces, given by the following data:

• A measurable family H = (Hω)ω∈G0 of Hilbert spaces.• For every g ∈ G a unitary Ug: Hs(g) → Hr(g) such that

U(g1g2) = U(g1)U(g2)


whenever s(g1) = r(g2). Moreover, we assume that for every pair (ξ, η) ofmeasurable sections of H the function

G → C, g �→ (ξ |η)(g) := (ξr(g)|U(g)ηs(g))is measurable.

Given a measurable functor F : G � Y there is a natural representation L2 ◦ F ,where Hω = L2(Yω, βω) and U(g) is induced by the isomorphism F(g) of mea-sure spaces.

Let us assume that (�, T ) is an (r, R)-system. We are especially interested inthe representation of G(�, T ) on H = (*2(Xω, αω))ω∈� induced by the measur-able functor Fα: G(�, T ) � X defined above. The necessary measurable struc-ture is provided by [29], Proposition 2.8. It is the measurable structure generatedby Cc(X).

The last item we have to define is a transversal measure. We denote the setof nonnegative transversal functions on a groupoid G by E+(G) and consider theunimodular case (δ ≡ 1) only. Following [9], p. 41f, a transversal measure is alinear mapping : E+(G) → [0,∞] satisfying

• is normal, i.e., (sup νn) = sup(νn) for every increasing sequence (νn)in E+(G).

• is invariant, i.e., for every ν ∈ E+(G) and every kernel λ with λω(1) = 1we get (ν ∗ λ) = (ν).

Given a fixed transversal function ν on G and an invariant measure µ on G0 thereis a unique transversal measure = ν such that (ν ∗ λ) = µ(λ•(1)), see [9],Theorem 3, p. 43. In the next section we will discuss that in a little more detail inthe case of DDS groupoids.

We can now put these constructions together.

DEFINITION 2.4. Let (�, T ) be an (r, R)-system and let µ be an invariantmeasure on �. Denote by V1 the set of all f : X → C which are measurableand satisfy f (ω, ·) ∈ *2(Xω, αω) for every ω ∈ �.

A family (Aω)ω∈� of bounded operators Aω: *2(ω, αω) → *2(ω, αω) is calledmeasurable if ω �→ 〈f (ω), (Aωg)(ω)〉ω is measurable for all f, g ∈ V1. It is calledbounded if the norms of the Aω are uniformly bounded. It is called covariant if itsatisfies the covariance condition

Hω+t = UtHωU∗t , ω ∈ �, t ∈ R

d,

where Ut : *2(ω) → *2(ω+ t) is the unitary operator induced by translation. Now,we can define

N (�, T ,µ) := {A = (Aω)ω∈� | A covariant, measurable and bounded}/∼,where ∼ means that we identify families which agree µ almost everywhere.


As is clear from the definition, the elements of N (�, T ,µ) are classes of fami-lies of operators. However, we will not distinguish too pedantically between classesand their representatives in the sequel.

Remark 2.5. It is possible to define N (�, T ,µ) by requiring seemingly weakerconditions. Namely, one can consider families (Aω) that are essentially boundedand satisfy the covariance condition almost everywhere. However, by standard pro-cedures (see [9, 25]), it is possible to show that each of these families agrees almosteverywhere with a family satisfying the stronger conditions discussed above.

Obviously, N (�, T ,µ) depends on the measure class of µ only. Hence, foruniquely ergodic (�, T ), N (�, T ,µ) =: N (�, T ) gives a canonical algebra. Thiscase has been considered in [28, 29].

Apparently, N (�, T ,µ) is an involutive algebra under the obvious operations.Moreover, it can be related to the algebra End(H) defined in [9] as follows.

THEOREM 2.6. Let (�, T ) be an (r, R)-system and let µ be an invariant measureon �. Then N (�, T ,µ) is a weak-∗-algebra. More precisely,

N (�, T ,µ) = End(H),

where = ν and H = (*2(Xω, αω))ω∈� are defined as above.Proof. The asserted equation follows by plugging in the respective definitions.

The only thing that remains to be checked is that H is a square integrable represen-tation in the sense of [9], Definition, p. 80. In order to see this it suffices to showthat the functor Fα giving rise to H is proper. See [9], Proposition 12, p. 81.

This in turn follows by considering the transversal function ν defined in Propo-sition 2.3 above. In fact, any u ∈ Cc(R

d)+ gives rise to the function f ∈ F +(X)

by f (ω, p) := u(p). It follows that

(ν ∗ f )(ω, p) =∫

Rd

u(p + t) dt =∫

Rd

u(t) dt,

so that ν ∗ f ≡ 1 if the latter integral equals 1 as required by [9], Definition 3,p. 55. ✷

We can use the measurable structure to identify L2(X,m), where m =∫�αωµ(ω) with

∫ ⊕�*2(Xω, αω) dµ(ω). This gives the faithful representation

π : N (�, T ,µ) → B(L2(X,m)), π(A)f ((ω, x)) = (Aωfω)((ω, x))

and the following immediate consequence.

COROLLARY 2.7. π(N (�, T ,µ)) ⊂ B(L2(X,m)) is a von Neumann algebra.

Next we want to identify conditions under which π(N (�, T ,µ)) is a factor.Recall that a Delone set ω is said to be nonperiodic if ω+ t = ω implies that t = 0.


THEOREM 2.8. Let (�, T ) be an (r, R)-system and let µ be an ergodic invariantmeasure on �. If ω is nonperiodic for µ-a.e. ω ∈ � then N (�, T ,µ) is a factor.

Proof. We want to use [9], Corollaire 7, p. 90. In our case G = G(�, T ), G0 = �

and

Gωω = {(ω, t) : ω + t = ω}.

Obviously, the latter is trivial, i.e., equals {(ω, 0)} iff ω is nonperiodic. By our as-sumption this is valid µ-a.s. so that we can apply [9], Corollaire 7, p. 90. Thereforethe centre of N (�, T ,µ) consists of families

f = (f (ω)1Hω)ω∈�,

where f : � → C is bounded, measurable and invariant. Since µ is assumed to beergodic this implies that f (ω) is a.s. constant so that the centre of N (�, T ,µ) istrivial. ✷

Remark 2.9. Since µ is ergodic, the assumption of nonperiodicity in the theo-rem can be replaced by assuming that there is a set of positive measure consistingof nonperiodic ω.

Note that the latter result gives an extension of part of what has been announcedin [28], Theorem 2.1 and [29], Theorem 3.8. The remaining assertions of [29] willbe proved in the following section, again in greater generality.

3. Transversal Functions, Traces and Deterministic Spectral Properties

In the preceding section we have defined the von Neumann algebra N (�, T ,µ)

starting from an (r, R)-system (�, T ) and an invariant measure µ on (�, T ). Inthe present section we will study traces on this algebra. Interestingly, this ratherabstract and algebraic enterprise will lead to interesting spectral consequences. Wewill see that the operators involved share some fundamental properties with ‘usualrandom operators’.

Let us first draw the connection of our families to ‘usual random operators’, re-ferring to [7, 31, 39] for a systematic account. Generally speaking one is concernedwith families (Aω)ω∈� of operators indexed by some probability space and actingon *2(Zd) or L2(Rd) typically. The probability space � encodes some statisticalproperties, a certain kind of disorder that is inspired by physics in many situations.One can view the set � as the set of all possible realization of a fixed disorderedmodel and each single ω as a possible realization of the disorder described by �.Of course, the information is mostly encoded in a measure on � that describes theprobability with which a certain realization is picked.

We are faced with a similar situation, one difference being that in any familyA = (Aω)ω∈� ∈ N (�, T ,µ), the operators Aω act on the possibly different


spaces *2(ω). Apart from that we have the same ingredients as in the usual ran-dom business, where, of course, Delone dynamical systems still bear quite someorder. That is, we are in the realm of weakly disordered systems. For a first ideawhat this might have to do with aperiodically ordered solids, quasicrystals, as-sume that the points p ∈ ω are the atomic positions of a quasicrystal. In a tightbinding approach (see [6], Section 4 for why this is reasonable), the HamiltonianHω describing the respective solid would naturally be defined on *2(ω), its matrixelements Hω(p, q), p, q ∈ ω describing the diagonal and hopping terms for anelectron that undergoes the influence of the atomic constellation given by ω. Thedefinite choice of these matrix elements has to be done on physical grounds. Inthe following subsection we will propose a C∗-subalgebra that contains what weconsider the most reasonable candidates; see also [6, 17]. It is clear, however, thatN (�, T ,µ) is a reasonable framework, since translations should not matter. Put inother words, every reasonable Hamiltonian family (Hω)ω∈� should be covariant.

The remarkable property that follows from this ‘algebraic’ fact is that certainspectral properties of the Hω are deterministic, i.e., do not depend on the choice ofthe realization ω µ-a.s.

Let us next introduce the necessary algebraic concepts, taking a second lookat transversal functions and random variables with values in X. In fact, randomvariables can be integrated with respect to transversal measures by [9], i.e., for agiven nonnegative random variable β with values in X and a transversal measure, the expression

∫Fβ d is well defined. More precisely, the following holds:

LEMMA 3.1. Let (�, T ) be an (r, R)-system and µ be T -invariant.(a) Let β be a nonnegative random variable with values in X. Then∫

�βω(f (ω, ·)) dµ(ω) does not depend on f ∈ F +(X) provided f satisfies∫f ((ω + t, x + t) dt = 1 for every (ω, x) ∈ X and∫

�

βω(f (ω, ·)) dµ(ω) =∫Fβ d,

where Fβ : G � X is the measurable functor induced by Fβ(ω) = (Xω, βω) and = ν the transversal measure defined in the previous section.

(b) An analogous statement remains true for a complex random variable β =∑k λkβk, when we define∫

Fβ d =∑k

λk

∫Fβk d

and restrict to f ∈ F +(X) with suppf compact.Proof. Part (a) is a direct consequence of the definitions and results in [9]. Part

(b), then easily follows from (a) by linearity. ✷A special instance of the foregoing lemma is given in the following proposition.


PROPOSITION 3.2. Let (�, T ) be an (r, R)-system and let µ be T -invariant. Ifλ is a transversal function on G(�, T ) then

ϕ �→∫�

〈λω, ϕ〉 dµ(ω)

defines an invariant functional on Cc(Rd), i.e., a multiple of the Lebesgue measure.

In particular, if µ is an ergodic measure, then either λω(1) = 0 a.s. or λω(1) = ∞a.s.

Proof. Invariance of the functional follows by direct checking. By uniquenessof the Haar measure, this functional must then be a multiple of Lebesgue measure.If µ is ergodic, the map ω �→ λω(1) is almost surely constant (as it is obviouslyinvariant). This easily implies the last statement. ✷

Each random operator gives rise to a random variable as seen in the followingproposition whose simple proof we omit.

PROPOSITION 3.3. Let (�, T ) be an (r, R)-system and µ be T -invariant. Let(Aω) ∈ N (�, T ,µ) be given. Then the map βA: � → M(X), βωA(f ) = tr(AωMf )

is a complex random variable with values in X.

Now, choose a nonnegative measurable u on Rd with compact support and∫

Rd u(x) dx = 1. Combining the previous proposition with Lemma 3.1, f (ω, p) :=u(p), we infer that the map

τ : N (�, T ,µ) −→ C, τ (A) =∫�

tr(AωMu) dµ(ω)

does not depend on the choice of f viz u as long as the integral is one. Importantfeatures of τ are given in the following lemma.

LEMMA 3.4. Let (�, T ) be an (r, R)-system and µ be T -invariant. Then themap τ : N (�, T ,µ) → C is continuous, faithful, nonegative on N (�, T ,µ)+and satisfies τ(A) = τ(U ∗AU) for every unitary U ∈ N (�, T ,µ) and arbitraryA ∈ N (�, T ,µ), i.e., τ is a trace.

We include the elementary proof, stressing the fact that we needn’t rely on thenoncommutative framework; see also [27] for the respective statement in a differentsetting.

Proof. Choosing a continuous u with compact support we see that

|τ(A)− τ(B)| �∫

‖Aω − Bω‖ trMu dµ(ω) � ‖A− B‖C,

where C > 0 only depends on u and �. On the other hand, choosing u witharbitrary large support we easily infer that τ is faithful. It remains to show the laststatement.


According to [12], I.6.1, Cor. 1 it suffices to show τ(K∗K) = τ(KK∗) for everyK = (Kω)ω∈� ∈ N (�, T ,µ). We write kω(p, q) := (Kωδq |δp) for the associatedkernel and calculate

τ(K∗K) =∫�

tr(K∗ωKωMu) dµ(ω)

=∫�

tr(Mu

12K∗ωKωM

u12) dµ(ω)

=∫�

∑m∈ω

‖KωMu

12δm‖2µ(ω)

=∫�

∑l,m∈ω

|kω(l,m)|2u(m)∫

Rd

u(l − t) dt dµ(ω),

where we used that∫

Rd u(l − t) dt = 1 for all l ∈ ω. By covariance and Fubinistheorem we get

· · · =∫

Rd

∫�

∑l,m∈ω

|kω−t (l − t, m− t)|2u(m)u(l − t) dµ(ω) dt.

As µ is T -invariant, we can replace ω − t by ω and obtain

=∫

Rd

∫�

∑l,m∈ω+t

|kω(l − t, m− t)|2u(m)u(l − t) dt dµ(ω)

=∫�

∫Rd

∑l,m∈ω

|kω(l,m)|2u(m+ t)u(l) dt dµ(ω)

=∫�

tr(KωK∗ωMu) dµ(ω)

by reversing the first steps. ✷Having defined τ , we can now associate a canonial measure ρA to every self-

adjoint A ∈ N (�, T ,µ).

DEFINITION 3.5. For A ∈ N (�, T ,µ) self-adjoint, and B ⊂ R Borel measur-able, we set ρA(B) ≡ τ(χB(A)), where χB is the characteristic function of B.

For the next two results we refer to [27] where the context is somewhat different.

LEMMA 3.6. Let (�, T ) be an (r, R)-system and µ be T -invariant. Let A ∈N (�, T ,µ) self-adjoint be given. Then ρA is a spectral measure for A. In partic-ular, the support of ρA agrees with the spectrum @ of A and the equality ρA(F ) =τ(F (A)) holds for every bounded measurable F on R.

LEMMA 3.7. Let (�, T ) be an (r, R)-system and µ be T -invariant. Let µ be er-godic and A = (Aω) ∈ N (�, T ,µ) be self-adjoint. Then there exists @,@ac,@sc,

@pp,@ess ⊂ R and a subset � of � of full measure such that @ = σ (Aω) and


σ•(Aω) = @• for • = ac, sc, pp, ess and σdisc(Aω) = ∅ for every ω ∈ �. In thiscase, the spectrum of A is given by @.

We now head towards evaluating the trace τ .

DEFINITION 3.8. The number∫Fα d =: D�,µ is called the mean density of

� with respect to µ.

THEOREM 3.9. Let (�, T ) be an (r, R)-system and µ be ergodic. If ω is nonpe-riodic for µ-a.e. ω ∈ � then N (�, T ,µ) is a factor of type IID, where D = D�,µ,i.e., a finite factor of type II and the canonical trace τ satisfies τ(1) = D.

Proof. We already know that N (�, T ,µ) is a factor. Using Proposition 3.2and [9], Cor. 9, p. 51 we see that N (�, T ,µ) is not of type I. Since it admits afinite faithful trace, N (�, T ,µ) has to be a finite factor of type II.

Note that Lemma 3.1, the definition of τ and α give the asserted valuefor τ(1). ✷

Remark 3.10. It is a simple consequence of Proposition 4.6 below that

Dω = limR→∞

#(ω ∩ BR(0))

|BR(0)|exists and equals D�,µ for almost every ω ∈ �. Therefore, the preceding result isa more general version of the results announced as [28], Theorem 2.1 and [29],Theorem 3.8, respectively. Of course, existence of the limit is not new. It canalready be found, e.g., in [6].

4. The C∗-Algebra Associated to Finite Range Operators and the IntegratedDensity of States

In this section we study a C∗-subalgebra of N (�, T ,µ) that contains those oper-ators that might be used as Hamiltonians for quasicrystals. The approach is directand does not rely upon the framework introduced in the preceding sections.

We define

X ×� X := {(p, ω, q) ∈ Rd ×�× R

d : p, q ∈ ω},which is a closed subspace of R

d ×�× Rd for any DDS �.

DEFINITION 4.1. A kernel of finite range is a function k ∈ C(X ×� X) thatsatisfies the following properties:

(i) k is bounded.(ii) k has finite range, i.e., there exists Rk > 0 such that k(p, ω, q) = 0, whenever

|p − q| � Rk.


(iii) k is invariant, i.e.,

k(p + t, ω + t, q + t) = k(p, ω, q),

for (p, ω, q) ∈ X ×� X and t ∈ Rd .

The set of these kernels is denoted by Kfin(�, T ).

We record a few quite elementary observations. For any kernel k ∈ Kfin(�, T )

denote by πωk := Kω the operator Kω ∈ B(*2(ω)), induced by

(Kωδq |δp) := k(p, ω, q) for p, q ∈ ω.

Clearly, the family K := πk, K = (Kω)ω∈�, is bounded in the product (equippedwith the supremum norm) Cω∈�B(*2(ω)). Now, pointwise sum, the convolution(matrix) product

(a · b)(p, ω, q) :=∑x∈ω

a(p, ω, x)b(x, ω, q)

and the involution k∗(p, ω, q) := k(q, ω, p) make Kfin(�, T ) into a ∗-algebra.Then, the mapping π : Kfin(�, T ) → Cω∈�B(*2(ω)) is a faithful ∗-representation.We denote Afin(�, T ) := π(Kfin(�, T )) and call it the operators of finite range.The completion of Afin(�, T ) with respect to the norm ‖A‖ := supω∈� ‖Aω‖ isdenoted by A(�, T ). It is not hard to see that the mapping πω: Afin(�, T ) →B(*2(ω)), K �→ Kω is a representation that extends by continuity to a representa-tion of A(�, T ) that we denote by the same symbol.

PROPOSITION 4.2. Let A ∈ A(�, T ) be given. Then the following holds:

(a) πω+t (A) = Utπω(A)U∗t for arbitrary ω ∈ � and t ∈ R

d .(b) For F ∈ Cc(X), the map ω �→ 〈πω(A)Fω, Fω〉ω is continuous.

Proof. Both statements are immediate for A ∈ Afin(�, T ) and then can beextended to A(�, T ) by density and the definition of the norm. ✷

We get the following result that relates ergodicity properties of (�, T ), spectralproperties of the operator families from A(�, T ) and properties of the representa-tions πω.

THEOREM 4.3. The following conditions on a DDS (�, T ) are equivalent:

(i) (�, T ) is minimal.(ii) For any self-adjoint A ∈ A(�, T ) the spectrum σ (Aω) is independent of

ω ∈ �.(iii) πω is faithful for every ω ∈ �.


Proof. (i) ⇒ (ii) Choose φ ∈ C(R). We then get πω(φ(A)) = φ(πω(A)) sinceπω is a continuous algebra homomorphism. Set �0 = {ω ∈ � : πω(φ(A)) = 0}.By Proposition 4.2(a), �0 is invariant under translations. Moreover, by Proposi-tion 4.2(b) it is closed. Thus, �0 = ∅ or �0 = � by minimality. As φ is arbitrary,this gives the desired equality of spectra by spectral calculus.

(ii) ⇒ (iii) By (ii) we get that ‖πω(A)‖2 = ‖πω(A∗A)‖ does not depend onω ∈ �. Thus πω(A) = 0 for some A implies that πω(A) = 0 for all ω ∈ � whenceA = 0.

(iii) ⇒ (i) Assume that � is not minimal. Then we find ω0 and ω1 such thatω1 �∈ (ω0 + Rd).

Consequently, there is r > 0, p ∈ ω, δ > 0 such that

dH ((ω0 − p) ∩ Br(0), (ω1 − q) ∩ Br(0)) > 2δ

for all q ∈ ω1. Let ρ ∈ C(R) such that ρ(t) = 0 if t � 1/2 and ρ(0) = 1.Moreover, let ψ ∈ Cc(R

d) such that suppψ ⊂ Bδ(0) and φ ∈ Cc(Rd) and φ = 1

on B2r(0).Finally, let

a(x, ω, y) := ρ

(∥∥∥∥(∑p∈ω

Tpψ

)Txφ −

(∑q∈ω0

Tqψ

)Tyφ

∥∥∥∥∞+

+∥∥∥∥(∑p∈ω0

Tpψ

)Txφ −

(∑q∈ω

Tqψ

)Tyφ

∥∥∥∥∞

).

It is clear that a is a symmetric kernel of finite range and by construction the cor-responding operator family satisfies Aω1 = 0 but Aω0 �= 0, which implies (iii). ✷

Let us now comment on the relation between the algebra A(�, T ) definedabove and the C∗-algebra introduced in [6, 17] for a different purpose and in adifferent setting. Using the notation from [6] we let

Y = {ω ∈ � : 0 ∈ ω}and

GY = {(ω, t) ∈ Y × Rd : t ∈ ω} ⊂ X.

In [6] the authors introduce the algebra C∗(GY), the completion of Cc(GY) withrespect to the convolution

fg(ω, q) =∑t∈ω

f (ω, t)g(ω − t, q − t)

and the norm induced by the representations

Cω: Cc(GY) → B(*2(ω)),Cω(f )ξ(q) =∑t∈ω

f (ω − t, t − q)ξ(q), q ∈ ω.

The following result can be checked readily, using the definitions.


PROPOSITION 4.4. For a kernel k ∈ Kfin(�, T ) denote fk(ω, t) := k(0, ω, t).Then

J : Kfin(�, T ) → Cc(GY), k �→ fk

is a bijective algebra isomorphism and πω = Cω ◦ J for all ω. Consequently,A(�, T ) and C∗(GY) are isomorphic.

Note that the setting in [6] and here are somewhat different. In the tiling frame-work, the analogue of these algebras have been considered in [17].

We now come to relate the abstract trace τ defined in the last section with themean trace per unit volume. The latter object is quite often considered by physi-cists and bears the name integrated density of states. Its proper definition rests onergodicity. We start with the following preparatory result for which we need thenotion of a van Hove sequence of sets.

For s > 0 and Q ⊂ Rd , we denote by ∂sQ the set of points in R

d whose distanceto the boundary of Q is less than s. A sequence (Qn) of bounded subsets of R

d iscalled a van Hove sequence if |Qn|−1|∂sQn| → 0, n → 0 for every s > 0.

PROPOSITION 4.5. Assume that (�, T ) is a uniquely ergodic (r, R)-system withinvariant probability measure µ and A ∈ A(�, T ). Then, for any van Hovesequence (Qn) it follows that

limn∈N

1

|Qn| tr(Aω|Qn) = τ(A)

for every ω ∈ �.

Clearly, Aω|Q denotes the restriction of Aω to the subspace *2(ω∩Q) of *2(ω).Note that this subspace is finite-dimensional, whenever Q ⊂ R

d is bounded.We will use here the shorthand Aω(p, q) for the kernel associated with Aω.

Proof. Fix a nonnegative u ∈ Cc(Rd) with

∫Rd u(x) dx = 1 and support con-

tained in Br(0) and let f (ω, p) := u(p). Then

τ(A) =∫�

tr(AωMu) dµ(ω)

=∫�

(∑p∈ω

Aω(p, p)u(p)

)dµ(ω)

=∫�

F(ω) dµ(ω),

where

F(ω) :=∑p∈ω

Aω(p, p)u(p)


is continuous by virtue of [29], Proposition 2.5(a). Therefore, the ergodic theoremfor uniquely ergodic systems implies that for every ω ∈ �:

1

|Qn|∫Qn

F (ω + t) dt →∫�

F(ω) dµ(ω).

On the other hand,

1

|Qn|∫Qn

F (ω + t) dt = 1

|Qn|∫Qn

( ∑p∈ω+t

Aω+t (p, p)u(p))

dt

= 1

|Qn|∫Qn

(∑q∈ω

Aω(q, q)u(q + t)

)dt

︸︷︷︸In

by covariance of Aω. Since supp u ⊂ Br(0) and the integral over u equals 1, everyq ∈ ω such that q + Br(0) ⊂ Qn contributes Aω(q, q) · 1 in the sum under theintegral In. For those q ∈ ω such that q + Br(0) ∩ Qn = ∅, the correspondingsummand gives 0. Hence∣∣∣∣ 1

|Qn|( ∑

q∈ω∩Qn

Aω(q, q) − In

)∣∣∣∣ � 1

|Qn| · #{q ∈ ∂2rQn} · ‖Aω‖

� C · |∂2rQn||Qn| → 0

since (Qn) is a van Hove sequence. ✷A variant of this proposition also holds in the measurable situation.

PROPOSITION 4.6. Let µ be an ergodic measure on (�, T ). LetA∈ N (�, T ,µ)

and an increasing van Hove sequence (Qn) of compact sets in Rd with R

d = ⋃Qn,

0 ∈ Q1 and |Qn −Qn| � C|Qn| for some C > 0 and all n ∈ N be given. Then,

limn∈N

1

|Qn| tr(Aω|Qn) = τ(A)

for µ-almost every ω ∈ �.Proof. The proof follows along similar lines as the proof of the preceding propo-

sition after replacing the ergodic theorem for uniquely ergodic systems by theBirkhoff ergodic theorem. Note that for A ∈ N (�, T ,µ), the function F definedthere is bounded and measurable. ✷

In the proof we used ideas of Hof [14]. The following result finally establishesan identity that one might call an abstract Shubin’s trace formula. It says that theabstractly defined trace τ is determined by the integrated density of states. The lat-


ter is the limit of the following eigenvalue counting measures. Let, for self-adjointA ∈ A(�, T ) and Q ⊂ R

d :

〈ρ[Aω,Q], ϕ〉 := 1

|Q| tr(ϕ(Aω|Q)), ϕ ∈ C(R).

Its distribution function is denoted by n[Aω,Q], i.e., n[Aω,Q](E) gives the num-ber of eigenvalues below E per volume (counting multiplicities).

THEOREM 4.7. Let (�, T ) be a uniquely ergodic (r, R)-system and µ its er-godic probability measure. Then, for self-adjoint A ∈ A(�, T ) and any van Hovesequence (Qn),

〈ρ[Aω,Qn], ϕ〉 → τ(ϕ(A)) as n → ∞for every ϕ ∈ C(R) and every ω ∈ �. Consequently, the measures ρQn

ω convergeweakly to the measure ρA defined above by 〈ρA, ϕ〉 := τ(ϕ(A)), for every ω ∈ �.

Proof. Let ϕ ∈ C(R) and (Qn) be a van Hove sequence. From Proposition 4.5,applied to ϕ(A) = (ϕ(Aω))ω∈�, we already know that

limn∈N

1

|Qn| tr(ϕ(Aω)|Qn) = τ(ϕ(A))

for arbitrary ω ∈ �. Therefore, it remains to show that

limn∈N

1

|Qn|(

tr(ϕ(Aω)|Qn)− tr(ϕ(Aω|Qn

))) = 0. (∗)

This latter property is stable under uniform limits of functions ϕ, since bothϕ(Aω|Qn

) and ϕ(Aω)|Qnare operators of rank dominated by c · |Qn|.

It thus suffices to consider a polynomial ϕ.Now, for a fixed polynomial ϕ with degree N , there exists a constant C = C(ϕ)

such that

‖ϕ(A)− ϕ(B)‖ � C‖A− B‖(‖A‖ + ‖B‖)N

for any A,B on an arbitrary Hilbert space. In particular,

1

|Qn|∣∣ tr(ϕ(Aω)|Qn

)− tr(ϕ(Bω)|Qn)∣∣ � C‖Aω − Bω‖(‖Aω‖ + ‖Bω‖)N

and

1

|Qn|∣∣ tr(ϕ(Aω|Qn

))− tr(ϕ(Bω|Qn))

∣∣ � C‖Aω − Bω‖(‖Aω‖ + ‖Bω‖)N

for all Aω and Bω.Thus, it suffices to show (∗) for a polynomial ϕ and A ∈ Afin(�, T ), as this

algebra is dense in A(�, T ). Let such A and ϕ be given.


Let Ra the range of the kernel a ∈ C(X ×� X) corresponding to A. Since thekernel of Ak is the k-fold convolution product b := a · · · a one can easily verifythat the range of Ak is bounded by N · Ra . Thus, for all p, q ∈ ω ∩ Qn such thatthe distance of p, q to the complement of Qn is larger than N · Ra, the kernels ofAkω|Qn

and (A|Qn)k agree for k � N . We get:

((ϕ(Aω)|Qn)δq |δp) = b(p, ω, q) = (ϕ(Aω|Qn

)δq |δp).Since this is true outside {q ∈ ω ∩ Qn : dist(q,Qc

n) > N · Ra} ⊂ ∂N ·RaQn thematrix elements of (ϕ(Aω)|Qn

) and ϕ(Aω|Qn) differ at at most c · |∂N ·RaQn| sites,

so that

|tr(ϕ(Aω)|Qn)− tr(ϕ(Aω|Qn

))| � C · |∂N ·RaQn|.Since (Qn) is a van Hove sequence, this gives the desired convergence. ✷

The above statement has many precursors: [2–4, 31, 36] in the context of almostperiodic, random or almost random operators on *2(Zd) or L2(Rd). It generalizesresults by Kellendonk [17] on tilings associated with primitive substitutions. Itsproof relies on ideas from [2–4, 17] and [14]. Nevertheless, it is new in the presentcontext.

For completeness reasons, we also state the following result.

THEOREM 4.8. Let (�, T ) be an (r, R)-system with an ergodic probabiltiy mea-sure µ. Let A ∈ A(�, T ) be self-adjoint (Qn) be an increasing van Hove sequence(Qn) of compact sets in R

d with⋃Qn = R

d , 0 ∈ Q1 and |Qn −Qn| � C|Qn| forsome C > 0 and all n ∈ N. Then,

〈ρ[Aω,Qn], ϕ〉 → τ(ϕ(A)) as n → ∞for µ-almost every ω ∈ �. Consequently, the measures ρQn

ω converge weakly to themeasure ρA defined above by 〈ρA, ϕ〉 := τ(ϕ(A)), for µ-almost every ω ∈ �.

The Proof follows along similar lines as the proof of the previous theorem withtwo modifications: Instead of Proposition 4.5, we use Proposition 4.6; and insteadof dealing with arbitrary polynomials we choose a countable set of polynomialswhich is dense in Cc([−‖A‖ − 2, ‖A‖ + 2]).

The primary object from the physicists point of view is the finite volume limit:

N[A](E) := limn→∞ n[Aω,Qn](E)

known as the integrated density of states. It has a striking relevance as the numberof energy levels below E per unit volume, once its existence and independence ofω are settled.

The last two theorems provide the mathematically rigorous version. Namely,the distribution function NA(E) := ρA(−∞, E] of ρA is the right choice. It givesa limit of finite volume counting measures since

ρ[Aω,Qn] → ρA weakly as n → ∞.


Therefore, the desired independence of ω is also clear. Moreover, by standard argu-ments we get that the distribution functions of the finite volume counting functionsconverge to NA at points of continuity of the latter.

In [30] we present a much stronger result for uniquely ergodic minimal DDSthat extends results for one-dimensional models by the first named author, [26].Namely we prove that the distribution functions converge uniformly, uniform in ω.The above result can then be used to identify the limit as given by the tace τ . Letus stress the fact that unlike in usual random models, the function NA does exhibitdiscontinuities in general, as explained in [20].

Let us end by emphasizing that the assumptions we posed are met by all themodels that are usually considered in connection with quasicrystals. In particular,included are those Delone sets that are constructed by the cut-and-project methodas well as models that come from primitive substitution tilings.

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Moody (eds), Directions in Mathematical Quasicrystals, CRM Monogr. Ser., Amer. Math. Soc.,Providence, RI, 2000, pp. 143–159.

34. Senechal, M.: Quasicrystals and Geometry, Cambridge Univ. Press, Cambridge, 1995.35. Shechtman, D., Blech, I., Gratias, D. and Cahn, J. W.: Metallic phase with long-range

orientational order and no translation symmetry, Phys. Rev. Lett. 53 (1984), 1951–1953.36. Shubin, M.: The spectral theory and the index of elliptic operators with almost periodic

coefficients, Russian Math. Surveys 34 (1979).37. Solomyak, B.: Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997),

695–738.38. Solomyak, B.: Spectrum of a dynamical system arising from Delone sets, In: J. Patera

(ed.), Quasicrystals and Discrete Geometry, Fields Institute Monogr. 10, Amer. Math. Soc.,Providence, RI, 1998, pp. 265–275.

39. Stollmann, P.: Caught by Disorder: Bound States in Random Media, Progr. in Math. Phys. 20,Birkhäuser, Boston, 2001.


291

Macroscopic Dimension of 3-Manifolds

DMITRY V. BOLOTOVDepartment of Mathematics, Institute for Low Temperature Physics, Lenina Sqr. 47,Kharkov 61103, Ukraine. e-mail: [email protected]

(Received: 4 November 2002)

Abstract. In this paper we give the answer to the Gromov’s question about the macroscopic di-mension of universal coverings of closed manifolds in dimension 3. We prove that the macroscopicdimension of a universal covering of a closed Riemannian 3-manifold cannot be equal to 2.

Mathematics Subject Classifications (2000): 57M10, 55M10.

Key words: macroscopic dimension, closed 3-manifold.

1. Macroscopic Dimension of Manifolds

The concept of macroscopic dimension was introduced by M. Gromov. Despite theasymptotic dimension, which has been intensively studied in recent times [4], themacroscopic dimension of a Riemannian manifold is not greater than its coveringdimension. It is known that the macroscopic dimension of the universal coveringof a closed K(π, 1)-Riemannian manifold coincides with its covering dimension.So the well-known Gromov–Lawson conjecture reduces to the statement that themacroscopic dimension of the universal covering of a closed n-manifold with pos-itive scalar curvature is not greater than n− 1. Gromov’s conjecture studied belowsays that actually it is not greater than n − 2. It is true in dimension 3 as followsfrom [2] and the main theorem below.

DEFINITION 1.1 ([1]). A metric space V has the macroscopic dimension at mostk (dimmc V � k) if there is a k-dimensional polyhedron P and a proper uniformlyco-bounded map ϕ: V → P that is a proper map such that Diam(ϕ−1(p)) � ε forall p ∈ P and some possibly large ε < ∞. If k is minimal possible, we say thatdimmc V = k.

Remark 1.2. In the original definition by Gromov, the map ϕ doesn’t needto be proper. But we think this is exactly what was meant (see the proof of theProposition-Example on page 10 of [1]).

The next question was stated by M. Gromov in [1]:

CONJECTURE. If the universal isometric covering of a compact n-dimensionalRiemannian manifold M has dimmc M < n, then dimmc M < n− 1.

292 DMITRY V. BOLOTOV

Remark 1.3. The macroscopic dimension of M doesn’t depend on the Rie-mannian metric on M since any metrics on a compact manifold are equivalentin the following sense: if g1 and g2 are some metrics on M, then there exist pos-itive constants c and C such that cg1(X,X) � g2(X,X) � Cg1(X,X) for anyX ∈ TM, where TM is the tangent bundle ofM.

In this paper we give a positive answer to Gromov’s conjecture in the casen = 3.

MAIN THEOREM. The macroscopic dimension of the universal covering M of aclosed Riemannian 3-manifold M cannot be equal to 2.

2. Coarse Cohomology of a Metric Spaces

All facts in this section were taken from [3].

DEFINITION 2.1. LetM and N be metric spaces, and let f : M → N be a func-tion (not necessarily continuous). The function f is called uniformly bornologousif for every R > 0 there exists S > 0 such that d(f (x), f (x′)) < S for everyx, x′ ∈ M with d(x, x′) < R.

DEFINITION 2.2. The category UBB (uniformly bornologous Borel) is the cat-egory whose objects are proper metric spaces and whose morphisms are uniformlybornologous Borel maps which are proper in the sense that the inverse image of arelatively compact set is relatively compact.

DEFINITION 2.3. We will say that f, g: M → N is bornotopic, and write f ∼ gif there is a constant R > 0 such that for all x ∈ M,d(f (x), g(x)) < R. Further,if f : M → N is a morphism, and there exists a morphism h: N → M such thatf h ∼ 1N, hf ∼ 1M , we say that f is a bornotopy-equivalence, and that M and Nare bornotopy-equivalent.

DEFINITION 2.4. Let Pen(M,R) = {x ∈ N : d(x,M) � R}. We will say thatM is ω-dense in N if there is R > 0 such that N = Pen(M,R).

PROPOSITION 2.5. If M is ω-dense in N , then the inclusion map i: M → N isa bornotopy-equivalence.

Let M be a metric space. Then Mq+1 denotes the Cartesian product of M, and ⊂ Mq+1 denotes the multi-diagonal {(x, . . . , x) : x ∈ M}. The metric onMq+1

is defined by

d((x0, . . . , xq), (y0, . . . , yq)) = max{d(x0, y0), . . . , d(xq, yq)}.

MACROSCOPIC DIMENSION OF 3-MANIFOLDS 293

DEFINITION 2.6. Let M be a metric space. The coarse complex CX∗(M) isdefined as follows: CXq(M) is the space of locally bounded Borel functions φ:Mq+1 → R which satisfy the following support condition: for each R > 0, theset Supp(φ) ∩ Pen( ;R) is relatively compact inMq+1. The complex CX∗(M) isequipped with the usual coboundary map of Alexander–Spanier cohomology [6],that is

∂φ(x0, . . . , xq+1) =q+1∑i=0

(−1)iφ(x0, . . . , xi , . . . , xq+1).

The coarse cohomology HX∗(M) is the cohomology of this complex.

PROPOSITION 2.7. Bornotopic morphisms induce the same homomorphisms oncoarse cohomology. Bornotopy-equivalent spaces have isomorphic coarse coho-mology.

2.1. THE CHARACTER MAP

There is a natural map c from HX∗(M) to the ordinary cohomology of M withcompact supports, H ∗

c (M), where H ∗c (M) is Alexander–Spanier cohomology with

compact support of M. A q-cochain in this theory is just an equivalence class offunctions z: Mq+1 → R that are locally zero on the complement of a compact set,two such functions being considered equivalent if they agree on a neighborhood ofthe diagonal [6].

We define the map c: HXq(M) → Hqc (M) by sending a cocycle φ to its

truncation to any penumbra Pen( ;R) of the diagonal.When is c an isomorphism?

DEFINITION 2.8. A metric space M is uniformly contractible (UC-space) if forany R > 0 there is S > 0 such that any subset X of M of diameter less than Rcan be contracted (to a point) inside Pen(X;S). Equivalently, there is a functionS: [0,∞) → R such that every ball B(x, r) of radius r centered at x can becontracted to a point in the ball B(x, S(r)).

For example, the universal covering of a compact aspherical space is uniformlycontractible.

The following theorem holds:

THEOREM 2.9. IfM is a uniformly contractible space, then the character map cis an isomorphism.

2.2. FUNCTORIALITY

Coarse cohomology is contravariantly functorial on the category UBB. A mor-phism f : M → N induces a chain transformation f ∗: CX(N) → CX(M) by theusual formula


(f ∗φ)(x0, . . . , xq) = φ(f (x0), . . . , f (xq)).

It is easy to see that the character map c: HX∗(M)→ H ∗c (M) is functorial on the

subcategory UBC (uniformly bornologous continuous) of UBB. It is clear that thenext diagram is commutative:

HXi(M)

c

HXi(N)f ∗

c

H ic (M) H i

c (N)f ∗c

for each i.

3. Construction

It is known that an orientable closed 3-manifold can be decomposed as a connectsum [5]:

M = )1* . . . *)n*k(S2 × S1)*K1* . . . *Km, (1)

where )i is covered by a homotopic 3-sphere and each Ki is of K(π, 1).Using (1) decomposeM byMi \D3 and ‘handles’ which are homeomorphic to

S2 × I , whereMi are components of the decomposition (1).Equip M with a Riemannian metric such that the direct product metric is in-

duced on each handle.Attach the cylinderD3×I to each handle identifying ∂(D3)×I with the handle

by identity map. It will be called a full-handle. Extend the Riemannian metric fromS2 = ∂(D3) toD3 and define the direct product metric on each full-handle D3 × I .Denote the obtained polyhedron by L. Notice that L contains M and each Mi asclosed subsets. Using the Seifert–van Kampen theorem, it is not hard to show thatπ1(L) = π1(M).

Consider the isometric universal covering p: L → L. Clearly p−1(M) is theuniversal covering M of M and p−1(Mi) is a union of the universal coveringsMik (k ∈ N in the general case) ofMi .

Let Hl (l ∈ N in the general case) denote the full-handles in L, which are pre-images of the full-handles in L. Let sl denote the corresponding homeomorphismsl: D3 × I → Hl. We will denote Mpk, which are pre-images of Kp, by Kpk,and Mik will denote the pre-images of such Mi , whose universal coverings are notcontractible.

It is easy to see that each Mik has a macroscopic dimension not greater than1. Let ϕik: Mik → Pik be a map to the polyhedron of dimension not greater 1corresponding to the definition of macroscopic dimension. Let Mϕik be a cylinderof the map ϕik (i.e. the space obtained by attaching the cylinder Mik × I to Pik viax × 1 → ϕik(x) ∈ Pik).


Let us consider Mϕik in detail. In the case when Mik is homeomorphic to

S2 × S1 = S2 × R, we can take R as Pik and the projection to the second factor asϕik: S2 × R → R. One can see thatMϕik will be homeomorphic to D3 × R.

In the case when Mik is a homotopic sphere )ik, we can take a point ∗ as Pik .One can see thatMϕik will be homeomorphic to a cone C)ik .

Building up Mϕik over each Mik in L we obtain the polyhedron T . Using theSeifert–van Kampen theorem, it is not hard to show that π1(T ) = {1}.

Continue the isometric action of π1(M) on L to the action of π1(M) on T . Letg · Mik = Mil , where g ∈ π1(M). Then we define an action of g on Mϕik =Mik × I/(x × 1 ∼ ϕik(x)) as follows: g · (x × t) = g · x × t , if 0 � t < 1 andg · (ϕik(x)) = ϕil(g · x).

Extend the metric from L to T as follows: In the case whenMi is homeomorphicto S2 × S1 = ∂(D3 × S1), we can extend the metric induced on Mi from L to thecollar (by the direct product metric) and then to D3 × S1 (using, for example,a partition of unity). Lift the constructed metric from D3 × S1 to the universalcovering D3 × R = Mϕi1 and extend this metric to each Mϕik by the action ofπ1(M).

In the case whenMi is homeomorphic to )i , let us consider the induced metricon )i from L. Lift this metric to the universal covering )i1 and denote it by ds2

i ,then the metric dσ 2

i on C)i1 can be given as follows:

dσ 2i = dt2 + (1 − t)2 ds2

i , t ∈ I.Equip each C)ik with this metric by the action of π1(M). Let (T , ρ) be the con-structed metric space. Remark that the action of π1(M) on (T , ρ) is isometric withrespect to the constructed metric and the factor-space T /π1(M) is compact.

PROPOSITION 3.1. (T , ρ) is uniformly contractible polyhedron and M is ω-dense in (T , ρ).

Proof. ω-density of M in T follows from the construction of (T , ρ).Let 41: D3 × I × I → D3 × I be a homotopy which is immovable on the

second factor and radially contracts the ball D3 to a point, i.e.

41(r, s, t) = ((1 − t)r, s),where r is a position vector of a point in D3.

Let D31 be a ball concentrically containing D3. Define a homotopy 41:

D31 × I → D3

1, which coincides with 41 on D3 and immovable on ∂D31. If we

take D31 of radius 1 and D3 of radius r0, the homotopy will be the following:

41(r, t) =(1 − t)r, |r| � r0,(1 − t)r + r

( |r| − r01 − r0

)t, r0 < |r| � 1.

LetD31 × I1 be a height 1 cylinder such that D3

1 × 0 concentrically contains D3.


Define a homotopy 41: D31 × I1 × I → D3

1 × I1 as follows:

41(x, s, t) = 41(x, t (1 − s)),where x ∈ D3

1, s ∈ I1, t ∈ I .Construct a closed ε-neighborhood Ul of Hl in T . It is sufficient to construct a

neighborhood for sl(D3 × j), j ∈ {0, 1}. If sl(D3 × j) ⊂ Kpk for some Kpk andj ∈ {0, 1}, then we embed D3

1 in Kpk by sl: D31 → Kpk such that sl|D3 = sl|D3×j

and sl(D31) would be an ε-neighborhood of sl(D3 × j) in Kpk. If sl(D3 × j) ⊂ Mik

for some Mik and j ∈ {0, 1}, then we embed the height 1 cylinder D31 × I1 in

Mϕik by sl: D31 × I1 → Mϕik such that sl |D3 = sl|D3×j and sl(D3

1 × I1) would beε-neighborhood of sl(D3 × j) in Mik . We can choose ε small enough so that theε-neighborhoods of the full-handles do not intersect.

Define a homotopy F1: T × I → T by

F1(x, t) =

sl41(s

−1l (x), t), if x ∈ Hl,

sl41((sl)−1(x), t), if x ∈ Ul ∩ Kpk,

sl41(( sl )−1(x), t), if x ∈ Ul ∩Mϕik ,

x in other case.

Define f1 = F1(T , 1).Denote the intervals which are images of f1(Hl) by Il .Let 42:

⊔i,k Mϕik × I → T be a natural deformation retracting each Mϕik on

Pik. Decompose each interval Il into two equal intervals Il+∪Il−. Let Il∗∩Mϕik �= ∅(∗ ∈ {+,−}) for some l and {i, k}. Actually, Il∗ ∩Mϕik is a point which we denoteby pl∗. Continue the homotopy 42 on Il∗. Consider the subset

It × Iv = {(t, v) | t ∈ [0, 2], v ∈ [0, 1]}of the plane (t, v). Divide the interval It on three subintervals:

I1 = [0, 1], I2+ = [1, 112 ] and I2− = [11

2 , 2].Construct a continuous mapW : It × Iv → T as follows:

Wl∗(I2+, v) = Il∗,Wl∗(I2−, v) = Il∗, where {+, −} = {−,+},Wl∗(t, v) = 42(pl∗, 1 − t), if t ∈ [0, 1].

Let ψv carry out a homotopy of the interval It as it shown in Figure 1. One cansee that ψv is immovable on I2−. Then the required continuation of 42 on Il∗ is

42(Il∗, t) = Wl∗(ψt (I2+)).

Now we can continue 42 to a deformation F2: f1(T )× I → T as follows:

F2(x, t) ={42(x, t) if x ∈ {Il∗ ∪Mϕik |Il ∩Mϕik �= ∅ for some l, {i, k}}x in other case.


Figure 1.

Define f2(x) = F2(x, 1).Let ; be a factor space obtained from f2 ◦ f1(T ) by contracting each Kpk to a

point by itself. It is clear that f2 ◦ f1(T ) is homotopy equivalent to ;. ; is a treesince it is homotopy equivalent to T by construction and therefore π1(;) = {1}.Thus we conclude that T is contractible.

To prove that T is uniformly contractible, we recall that the factor spaceT /π1(M) is compact, and if B(x, r) is contractible inside B(x, S(r)) for somepoint x ∈ T , then B(z, r) is contractible inside B(z, S(r+D)) for any point z ∈ T ,where D = Diam(T /π1(M)). It finishes the proof of the proposition.

4. A Proof of the Main Theorem

If we suppose that the decomposition (1) does not contain any Ki , then f2 ◦ f1(T )

is a tree (see proof of the proposition above). Each of the maps f1|M and f2|f1(M)

are proper and uniformly co-bounded and so is f2 ◦ f1|M by construction. So inthis case the theorem is proved.

Let the decomposition (1) contain some Ki . Suppose that dimmc M = 2, thenthere is a uniformly co-bounded proper map ψ : M → P 2 to a polyhedron ofdimension 2.

LEMMA 4.1. There exists a map p: P 2 → T such that the composition of mapsp ◦ ψ is bornotopic to the inclusion i: M → T .

Sketch of proof. (1) Since ψ is a uniformly co-bounded proper map, we cantriangulate P 2 in small pieces so that the preimage of the star of each vertex wouldbe bounded by the same constant R.

(2) Firstly, define p on the 0-dimensional skeleton (P 2)(0) of P 2 as a sectionof ψ .

(3) Continue p consistently to all skeletons by appealing to the uniform con-tractibility of T and an obstruction theory such that p(σ i) is contained in the


Si-neighborhood of p((σ i)(0)) for each i-dimensional simplex σ i , where Si areconstants determined by the function S from the definition of UC property of T .

It is easy to see that ρ(x, p ◦ ψ(x)) � C for any x ∈ T , where C is a constant.

Remark 4.2. Since p ◦ ψ is proper, then p is also proper.

LEMMA 4.3. The inclusion i: M → T induces a nontrivial homomorphismin H 3

c .Proof. Consider a small enough ball D3 ⊂ Kik ⊂ M ⊂ T , which is open both

in M and in T . Such a ball exists by construction of T .The inclusion D3 ⊂ M induces an isomorphism in H 3

c since M is orientablemanifold, so the result follows from the next commutative diagram:

H 3c (D

3) H 3c (M)

H 3c (T )

idH 3c (T )

All homomorphisms are induced by inclusion. Remark that H ∗c is covariantly

functorial with respect to inclusions of open sets and one is controvariantly func-torial with respect to inclusions of closed sets.

To finish the proof it is enough to consider the next commutative diagram:

HX3(M)

c1

HX3(T )h∗

1 i∗1

c

HX3(M)

c1

H 3c (M) H 3

c (T )h∗

2 i∗2H 3c (M)

where i∗1 , i∗2 are induced by the inclusion i: M → T , the homomorphisms h∗

1, h∗2

are induced by h, which is the composition of proper maps p ◦ ψ , and c, c1 arecharacter maps.

Since T is uniformly contractible, M is ω-dense in T , and h is bornotopic to i,homomorphisms c, i∗1 , h

∗1 are isomorphisms. The map i∗2 is nontrivial by Lemma 4.3.

Therefore c1 is nontrivial.On the other hand, h∗

2 must be trivial, as it passes through a polyhedron ofdimension less than 3, therefore c1 must be trivial.

This contradiction finishes the proof of the main theorem.

References

1. Gromov, M.: Positive curvature, macroscopic dimension, spectral gaps and higher signatures,Preprint, 1996.


2. Gromov, M. and Lawson, H.: Spin and scalar curvature in the presence of a fundamental group,I, Ann. of Math. 111(2) (1980), 209–230.

3. Roe, J.: Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer.Math. Soc. 4(497) (1993).

4. Dranishnikov, A. N.: Asymptotic topology, Russian Math. Surveys 55(6) (2000), 71–116.5. Hempel, J.: 3-Manifolds, Ann. of Math. Stud. 86, Princeton Univ. Press, 1976.6. Spanier, E.: Algebraic Topology, McGraw-Hill, New York, 1996.


301

How to Find Separation Coordinates for theHamilton–Jacobi Equation: A Criterion ofSeparability for Natural Hamiltonian Systems

CLAES WAKSJÖ and STEFAN RAUCH-WOJCIECHOWSKIMatematiska institutionen, Linköpings universitet, SE-581 83 Linköping, Swedene-mail: {clwak, strau}@mai.liu.se

(Received: 28 June 2001; in final form: 4 November 2002)

Abstract. The method of separation of variables applied to the natural Hamilton–Jacobi equation12∑

(∂u/∂qi )2 + V (q) = E consists of finding new curvilinear coordinates xi (q) in which the

transformed equation admits a complete separated solution u(x) = ∑u(i)(xi ;α). For a potential

V (q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q)

exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmicsolution, which we present here. It is based on recursive use of the Bertrand–Darboux equations,which are linear second order partial differential equations with undetermined coefficients. The resultapplies to the Helmholtz (stationary Schrödinger) equation as well.

Mathematics Subject Classifications (2000): 70H20, 70G10, 35Q40.

Key words: separation of variables, Hamilton–Jacobi equation, integrability, Schrödinger equation.

1. Introduction

Natural Hamiltonians

H = T + V = 1

2

n∑i=1

p2i + V (q) (1.1)

are the sum of the kinetic energy function T = 12p

2 and a potential energy func-tion V . In (1.1) it is written in Cartesian coordinates q = (q1, . . . , qn) with canon-ical momenta coordinates p = (p1, . . . , pn). The corresponding Hamiltonian sys-tem

dqidt

= ∂H

∂pi

≡ pi,dpi

dt= −∂H

∂qi≡ −∂V

∂qi, i = 1, . . . , n (1.2)

is equivalent to the Newtonian system d2qi/dt2 = −∂V/∂qi, which describe themotion of a particle in R

n under the influence of a conservative force −∂V/∂q.The most effective way of solving these equations is by separation of vari-

ables in the Hamilton–Jacobi PDE. This amounts to finding curvilinear coordinates

302 CLAES WAKSJÖ AND STEFAN RAUCH-WOJCIECHOWSKI

x = (x1, . . . , xn) with canonical momenta coordinates y = (y1, . . . , yn) in whichthe transformed Hamiltonian

H(x, y) = 1

2

n∑i,j=1

gij (x) yi yj + V (q(x)) (1.3)

allows the equation H(x, ∂u/∂x) = E to be solved through an additive ansatzu(x;α) = ∑

u(i)(xi;α) depending nontrivially on a set of separation constantsα = (α1, . . . , αn). If separation can be accomplished, the original problem issolved since the function u(x;α) is a generating function for a canonical transfor-mation (x, y) → (ξ, η) to new canonical coordinates (ξ, η) in which the Hamil-tonian system (1.2) becomes trivial.

For a given potential V (q), the problem of finding separation coordinates ishighly nontrivial, and the first property to establish is the mere existence of suchcoordinates. The problem was originally formulated by Jacobi when he inventedelliptic coordinates and successfully applied them to solve several important me-chanical problems, such as the problem of geodesic motion on an ellipsoid, and theproblem of planar motion in a force field of two attracting centres.

Jacobi himself was rather sceptical about the possibility of finding a completesolution to this problem and in his work [21, pp. 198–199] he wrote:

Die Hauptschwierigkeit bei der Integration gegebener Differentialgleichung-en scheint in der Einführung der richtigen Variablen zu bestehen, zu derenAuffindung es keine allgemeine Regel giebt. Man muss daher das umgekehrteVerfahren einschlagen und nach erlangter Kenntniss einer merkwürdigen Sub-stitution die Probleme aufsuchen, bei welchen dieselbe mit Glück zu brauchenist.

The quotation is also referred to in the English translation of the book by Arnol’d[2, p. 266]: “The main difficulty in integrating a given differential equation lies inintroducing convenient variables, which there is no rule for finding. Therefore, wemust travel the reverse path and after finding some notable substitution, look forproblems to which it can be successfully applied.”

Jacobi’s remark had a profound influence on the further developments of sep-arability theory, which mainly focused on results in separation coordinates, suchas:

(a) Complete characterization of separable natural Hamiltonians (1.3) in terms ofseparation coordinates (Stäckel [30]; Levi-Civita [24]).

(b) Complete description and classification of all separable coordinate systems inEn and on S

n (Eisenhart [15]; Benenti [5]; Kalnins and Miller [23]).These results build the foundation of this work in which we present a complete,

effective and algorithmic solution to the Jacobi problem. More precisely, we shallformulate an algorithm that for a given potential determines if separation coor-dinates exist, and in that case, shows how to construct them. The algorithm can

A CRITERION OF SEPARABILITY FOR NATURAL HAMILTONIAN SYSTEMS 303

easily be programmed on a computer and thus turned into a practical tool. Thebasis of the algorithm is the characterization of each separable potential in terms of

Cartesian coordinates by a system of(n

2

)second-order linear PDEs with quadratic

coefficients.In two dimensions all separable potentials are fully characterized through the

following theorem, where we write ∂i = ∂/∂qi and ∂ij = ∂2/∂qi∂qj .

THEOREM 1.1 (Bertrand–Darboux). Let H = 12 (p

21 +p2

2)+V (q1, q2) be a naturalHamiltonian. The following are equivalent:

(a) There is an extra quadratic first integral

K = (− 12αq

22 − β2q2 + 1

2γ11)p2

1 + (− 12αq

21 − β1q1 + 1

2γ22)p2

2 ++ (αq1q2 + β1q2 + β2q1 + γ12)p1p2 + U(q1, q2) (1.4)

functionally independent of H .(b) The potential V satisfies the equation

(αq1q2 + β1q2 + β2q1 + γ12)(∂22V − ∂11V )++ (αq2

1 − αq22 + 2β1q1 − 2β2q2 + γ11 − γ22)∂12V +

+ 3(αq1 + β1)∂2V − 3(αq2 + β2)∂1V = 0 (1.5)

with nontrivial parameters: (α, β1, β2, γ11 − γ22, γ12) = (0, 0, 0, 0, 0).(c) The potential V is separable. A characteristic coordinate system for (1.5)

provides separation for V and can be taken as one of the following fourorthogonal coordinate systems: the elliptic, parabolic, polar or Cartesian.

Condition (b) can be interpreted as an effective criterion of separability. Byrequiring the Bertrand–Darboux (BD) Equation (1.5) to be satisfied identicallywith respect to q1 and q2, one gets a system of linear homogeneous equations forthe parameters α, β1, β2, γ11, γ12, γ22. For a generic potential this system is heavilyoverdetermined and has only the trivial solution, which means that V is nonsepa-rable. If a nontrivial solution exists, the BD equation can be reduced to canonicalform by transforming to characteristic coordinates, which appear to be separationcoordinates for the Hamilton–Jacobi equation related to the natural Hamiltonian(see Darboux [14], or Whittaker [33, §152] and Ankiewicz and Pask [1] for a fullproof of the BD theorem).

In order to present the main idea and indicate the nature of our solution we shallstart with a simple example due to Rauch-Wojciechowski [34]: examination of thegeneralized Hénon–Heiles [19] potential

V = 12 (q

21 + q2

2 ) + q21q2 + 2q3

2 . (1.6)

This potential satisfies the BD Equation (1.5) with (α, β1, β2, γ11 − γ22, γ12) =(0, 0, 2t, 3t, 0) for arbitrary t , which means that it is separable. In order to deter-


mine the separation coordinates, we insert these parameters into the original BDEquation (1.5) and divide through by t to find that (1.6) solves

2q1(∂22V − ∂11V ) + (−4q2 + 3)∂12V − 6∂1V = 0. (1.7)

This equation can be brought into canonical form by introducing the characteristic(translated parabolic) coordinates (x1, x2) defined by

q1 = √x1x2, q2 = 1

2(x1 − x2) + 34 . (1.8)

Equation (1.7) takes the form (x1 + x2)∂2V /∂x1∂x2 + ∂V /∂x1 + ∂V /∂x2 = 0,

which can be rewritten as ∂2/∂x1∂x2((x1 + x2)V ) = 0 to immediately find itsgeneral solution

V = f1(x1) + f2(x2)

x1 + x2. (1.9)

In parabolic coordinates, the Hamilton–Jacobi equation takes the form

2

x1 + x2

[x1

(∂u

∂x1

)2

+ x2

(∂u

∂x2

)2]+ V = E,

which after multiplication by x1 + x2 and insertion of the ansatz u = u(1)(x1) +u(2)(x2) separates to

2x1(u′(1))

2 + f1(x1) = α1x1 + α2, 2x2(u′(2))

2 + f2(x2) = α1x2 − α2,

where α1 = E and α2 are separation constants.Thus we have shown that (1.9) is separable in parabolic coordinates, and

since (1.6) is a special case of (1.9), we deduce that the generalized Hénon–Heilespotential indeed is separable in (x1, x2). [For (1.6), f1(x) = f (−x) and f2(x) =−f (x) with f (x) = 1

16(4x4 − 20x3 + 33x2 − 18x).]

In order to formulate an n-dimensional analogue of the two-dimensional crite-rion of separability given by (1.5), Marshall and Rauch-Wojciechowski [26] havederived generalized BD equations corresponding to elliptic, parabolic and coni-cal coordinates. These equations are given in a distinguished Euclidean referenceframe and are difficult to apply for testing separability of a given potential. More-over, they do not take into account the possibility of all degenerate separationcoordinates that appear in the classification by Kalnins and Miller [23]. Here wesolve all these problems connected with the usage of the BD equations and givea new unified generalization of (1.5). The new equations are shown to encompassall previously known equations, and allow also for determination of the unknownEuclidean reference frame. The analysis of the generalized BD equations constitutethe essential part of the algorithm presented here. The algorithm is simple, yetnontrivial, and splits into several subcases that correspond to the degenerationsspecified in the classification by Kalnins and Miller.


Closely connected with the (additive) separation of the Hamilton–Jacobi equa-tion, is the (multiplicative) separation of the Helmholtz equation∑ ∂2u

∂q2i

+ V u = Eu,

also known as the stationary Schrödinger equation. In the latter case one tries tofind coordinates such that the transformed equation

1√g

∑i,j

∂

∂xi

(√ggij ∂u

∂xj

)+ V u = Eu, g = det(gij ),

admits solutions of the form u(x) = ∏u(i)(xi). It is well known that in Euclidean

space, the orthogonal separable coordinate systems for the natural Hamiltoniancoincide with those for the Helmholtz equation; see Robertson [29] and Eisen-hart [15, §2]. This fact implies that a potential recognized as separable by ourcriterion also separates in the Helmholtz equation. If one also requires the sepa-rated solution of the Helmholtz equation to depend on 2n parameters (separationand integration constants) in a nontrivial way, the Hamilton–Jacobi and Helmholtzequations separate simultaneously, so that our criterion decides separability forthe Helmholtz equation too. A review of these separability issues can be found inBenenti et al. [11].

1.1. NOTATION AND CONVENTIONS

We denote by En the n-dimensional Euclidean space. An n-tuple of Cartesian co-

ordinates is written q and is regarded as an n× 1 matrix; Cartesian coordinates arecharacterized by the fact that the metric takes the form ds2 = dq2 = ∑

dq2i . The

unit sphere Sn−1 is the Riemannian subspace of E

n whose points have Cartesiancoordinates ω satisfying |ω| = √

ω2 = 1.A Euclidean transformation is the composition of a rotation and a translation.

It can be viewed as an affine mapping q �→ Aq + b, where the n × n matrix A

satisfies AtA = Id and detA = 1 and b is an n × 1 matrix.Generically, the metric takes the form ds2 = ∑

gij dxi dxj in curvilinear coor-dinates xi . The metric is diagonal precisely when the coordinates are orthogonal,and we write gii = H 2

i .The pair of Cartesian and momenta coordinates (q, p) are canonical coordinates

on the cotangent bundle T ∗En, and a change of coordinates q → x on E

n inducesa change of momenta coordinates p → y by requiring (x, y) to be canonical. Thusthe quadratic form 1

2

∑p2i transforms to 1

2

∑gij yiyj when the metric transforms

to ds2 = ∑gij dxi dxj .

Partial derivatives are always taken with respect to Cartesian coordinates un-less otherwise stated, and we write ∂i = ∂/∂qi . We define the radial and angularderivatives as

R =∑

qi∂i and Jij = qi∂j − qj ∂i; (1.10)


in polar coordinates they read R = r∂/∂r and Jij = ∂/∂ϕ where ϕ is an angularcoordinate in the {qi, qj }-plane. The operators R and Jij commute. We also definethe cyclic operator

Cλijk = λi∂iJjk + c.p. = λi∂iJjk + λj∂jJki + λk∂kJij , (1.11)

where λ = (λi) is an n-tuple of real parameters and “+ c.p.” means “plus all cyclicpermutations.”

We note some simple properties of the operators Jij and Cλijk . For all indices

i, j, k, %, we have the following identities:

Jij = −Jji , (1.12a)

qiJjk + qjJki + qkJij = 0, (1.12b)

Cijk = Cjki = Ckij , (1.12c)

Cijk = −Cikj , Cijk = −Ckji , Cijk = −Cjik, (1.12d)

qiCjk% − qjCk%i + qkC%ij − q%Cijk = 0, (1.12e)

where Cijk = Cλijk with fixed parameters λi . The antisymmetry relations (1.12a)

and (1.12d) show that both Jij and Cλijk vanish if two indices coincide. Further,

(1.12b) and (1.12e) show that these operators are linearly dependent over the fieldof rational functions depending on q.

1.2. SEPARATION OF VARIABLES IN THE NATURAL HAMILTON–JACOBI

EQUATION

We consider the Hamilton–Jacobi equation

1

2

n∑i,j=1

gij (x)∂u

∂xi

∂u

∂xj+ V (q(x)) = E (1.13)

of the natural Hamiltonian (1.3). A complete separated solution of (1.13) is a solu-tion u(x;α) = ∑

u(i)(xi;α) depending on n separation constants αj in such a waythat the matrix (∂2u/∂xi∂αj ) is nonsingular. The coordinate system {xi} is calledseparable if (1.13) admits a complete separated solution. In the same spirit, wecall a natural Hamiltonian or a potential separable if such a separable coordinatesystem exists.

The problem of finding a separable coordinate system for (1.3) naturally splitsinto two stages: that of finding all separable coordinate systems for the geodesicHamiltonian and that of selecting those compatible with the potential. This factwas observed already by Levi-Civita [24], who found a necessary and sufficientcondition for a coordinate system to be separable for a natural Hamiltonian (1.3).The condition is that the equations


1

2

∑i,j,k,%

yiyj yky%(

12g

rs∂rgij ∂sg

k% + ∂r∂sgij gkrg%s −

− ∂rgisgjr∂sg

k% − ∂sgirgjs∂rg

k%)+

+∑i,j

yiyj(

12g

rs∂rgij ∂sV + 1

2grs∂rV ∂sg

ij +

+ ∂r∂sV girgjs − ∂rgisgjr∂sV − ∂sg

irgjs∂rV)+

+ grs∂rV ∂sV = 0 (∂r = ∂/∂xr, ∂s = ∂/∂xs) (1.14)

are satisfied identically in yi for all r = s. By the geodesic Hamiltonian, we meanH = T with V = 0. From Equations (1.14) it is clear that a coordinate system isseparable for (1.3) if and only if it is separable for the geodesic Hamiltonian and,moreover, the expressions that are coefficients for the second and zeroth power ofyi in (1.14) vanish. The latter condition is a set of PDEs for V with coefficientsdepending on the metric. It can be thought of as a condition for the compability ofthe potential with the metric. The primary objects to study are thus the separablecoordinate systems for the geodesic Hamiltonian. In the next section we shall comeback to this, after which we are able to study separability of potentials.

An essential fact for our criterion of separability is the close relation betweenthe separability of a natural Hamiltonian and the existence of n quadratic firstintegrals for the Hamiltonian system. Eisenhart’s theorem [15] states that undercertain technical conditions, their existence are necessary and sufficient for orthog-onal separation. In the nonorthogonal case the situation becomes more delicatebecause of the existence of linear first integrals corresponding to cyclic coordi-nates, i.e., coordinates xα such that ∂H/∂xα = 0. Also in this more general casea theory has been developed in a series of papers by Benenti, Kalnins and Miller.A comprehensive reference is Benenti [9].

Instead of quadratic and linear first integrals, one can also use the geometricconcept of (symmetric) Killing tensors K = (Kij ) and Killing vectors L = (Li).They are related to each other by the fact that K and L are Killing if and only if12

∑Kijyiyj and

∑Liyi are first integrals for the geodesic Hamiltonian system.

2. Orthogonal Separation

We now turn to the classification of all orthogonal separable coordinate systems.

2.1. SEPARATION IS ORTHOGONAL

First of all, we notice that in any separable coordinate system it is necessary thatgrs(∂V /∂xr)(∂V /∂xs) = 0 for r = s, which is evident from the Levi-Civita


Equations (1.14). If the potential has no vanishing derivatives, it is clear that thecontravariant metric tensor (grs) has to be diagonal. Thus, for a potential thathas this property, the separable coordinates are orthogonal. If the potential is in-dependent of some variables in the separable coordinates, the analysis is muchmore intricate, but the conclusion is the same. We give an outline of the relevantideas.

From Benenti [9, Theorem 2] we have: A natural Hamiltonian H = T + V

is orthogonally separable if and only if there exists a Killing tensor K with point-wise simple real eigenvalues and orthogonally integrable eigenvectors, such thatd(K · dV ) = 0, where d denotes the exterior derivative.

Such a tensor K is called characteristic; its existence is equivalent to the exis-tence of a full set of n Killing tensors. The equation d(K ·dV ) = 0 is an integrabilitycondition for the existence of the “potential parts” that has to be added to thefunctions 1

2

∑Kijyiyj to generate first integrals in the nongeodesic case. Indeed,

if K is a Killing tensor and the differential form K · dV is exact, then there exists asolution U to the equation dU = K · dV such that 1

2

∑Kij yiyj +U is a quadratic

first integral associated with H = T + V .In the nonorthogonal case we have the following fact from Benenti [9, Theo-

rem 5]: If a natural Hamiltonian H = T + V is separable then there exists anr-dimensional Abelian algebra D of Killing vectors and a D-invariant Killingtensor K with n − r distinct real eigenvalues with orthogonally integrable eigen-vectors, such that DV = 0 and d(K · dV ) = 0.

Benenti shows, using the Levi-Civita equations, that it is possible to choosean equivalent separable coordinate system such that the metric tensor attains theblock-diagonal form(

gii 00 gαβ

),

where the block (gii) is diagonal. The nondiagonal block (gαβ) corresponds to ther cyclic coordinates xα . Two separable coordinate systems are equivalent if thesolutions to the Hamilton–Jacobi equation are the same apart from transformationsof the separation constants.

Further, it is known (see Benenti [6] and Kalnins and Miller [22, 23]) that inRiemannian spaces of constant curvature it is possible to diagonalize the block(gαβ), so that the metric is purely diagonal. This is accomplished by chosing anorthogonal basis {Xα} for D.

It follows that the tensor K∗ = K + ∑cα Xα ⊗ Xα , with appropriately cho-

sen real numbers cα, is a characteristic Killing tensor. Moreover, DV = 0 andd(K · dV ) = 0 implies d(K∗ · dV ) = 0, so that the natural Hamiltonian indeed isseparable in orthogonal coordinates.

In view of these results: If a natural Hamiltonian is separable in some co-ordinates, then it is possible to choose equivalent orthogonal separable coordi-nates.


2.2. STÄCKEL’S THEOREM

In case of orthogonal coordinates, the Levi-Civita equations (1.14) can be rewrittenusing logarithmic derivatives as

∂2 logH 2k

∂xi∂xj− ∂ logH 2

k

∂xi

∂ logH 2k

∂xj+

+ ∂ logH 2j

∂xi

∂ logH 2k

∂xj+ ∂ logH 2

i

∂xj

∂ logH 2k

∂xi= 0 (2.1)

together with

∂2V

∂xi∂xj+ ∂ logH 2

i

∂xj

∂V

∂xi+ ∂ logH 2

j

∂xi

∂V

∂xj= 0 (2.2)

(i = j ), for the coefficients for the fourth and second power of yi respectively.The characterization of orthogonal separable coordinate systems given through

the Levi-Civita Equations (2.1) and (2.2) can also be formulated in purely algebraicterms. This is the content of Stäckel’s theorem [30], which is crucial in the theoryof orthogonal separation.

The basic concept is a Stäckel matrix, which is a nonsingular matrix ϕ(x) =(ϕij (xi)) whose ith row depends on xi only. We will say that the metric ds2 =∑

H 2i dx2

i has Stäckel form in a coordinate system {xi} if there exists a Stäckelmatrix ϕ such that

∑H−2

i ϕij = δ1j ; i.e., (H−21 , . . . , H−2

n ) is the first row in ϕ−1.Further, a function V is called a Stäckel multiplier if there exist n functions fidepending on one variable only, such that V takes the form V (x) = ∑

H−2i fi(xi)

in a coordinate system {xi} that gives the metric Stäckel form.Stäckel’s theorem reads now: An orthogonal coordinate system {xi} is separable

for (1.3) if and only if

(a) the metric has Stäckel form in {xi}, and(b) the potential is a Stäckel multiplier.

Obviously, (a) corresponds to (2.1) and (b) corresponds to (2.2).

2.3. BASIC SEPARABLE COORDINATE SYSTEMS

All orthogonal separable coordinate systems can be viewed as an orthogonal sumof certain basic coordinate systems. We now introduce these, and discuss theirdegenerations.

DEFINITION 2.1. The elliptic coordinate system {xi} in En with parameters

λ1 < λ2 < · · · < λn is defined through the equation

1 +n∑

i=1

q2i

z − λi=

n∏j=1

(z − xj )

/ n∏k=1

(z − λk). (2.3)


The elliptic coordinate system was introduced by Jacobi in a note in Crelle’sJournal [20]. A thorough discussion of its general properties as well as of its usefor separation of variables in the Hamilton–Jacobi equation can be found in hislecture notes from Königsberg [21, Vorlesungen 26–29].

The defining Equation (2.3) should be interpreted as an identity with respect toz, and for each set of elliptic coordinates xi it is possible to solve (2.3) for q2

i bycalculating the residues at z = λi . The elliptic coordinates are uniquely definedonly in the open subsets E

1± × · · · × E1± of E

n, since all combinations of signs in(±q1, . . . ,±qn) define the same elliptic coordinates.

The elliptic coordinate system is orthogonal, and the coordinates take valuesonly in the intervals

x1 < λ1 < x2 < λ2 < · · · < xn < λn.

By a simultaneous rescaling of the coordinates and the parameters, xi �→ axi ,λi �→ aλi , it is always possible to take λ1 = 1.

The coordinate surfaces are geometrically an ellipsoid (x1 = const.), a one-sheeted hyperboloid (x2 = const.), a two-sheeted hyperboloid (x3 = const.), etc.,and the parameters λi determine the eccentricity of these surfaces.

It is possible to degenerate the elliptic coordinate system in a proper way byletting two or more of the parameters λi coincide. Then the ellipsoid will becomea spheroid, or even a sphere if all parameters coincide. Rotational symmetry ofdimension m is thus introduced if m + 1 parameters coincide. At the same timewill the intervening coordinates disappear, and need to be replaced by coordinateson an m-dimensional sphere.

EXAMPLE 2.2. Consider the elliptic coordinate system in E3 defined by (2.3).

When λ1 = λ2, we have

1 + r2

z − λ1+ q2

3

z − λ3= (z − x1)(z − x3)

(z − λ1)(z − λ3), r2 = q2

1 + q22 ,

with x1 < λ1 < x3 < λ3. This determines a mapping r = u1(x1, x3), q3 =u2(x1, x3) that defines elliptic coordinates in E

2 = {r, q3}. In order to get an or-thogonal coordinate system in E

3, it is possible to complement r with an angularcoordinate ϕ in the {q1, q2}-plane, for instance, through

q1 = u1(x1, x3) cos ϕ, q2 = u1(x1, x3) sin ϕ, q3 = u2(x1, x3).

These equations define the prolate spherical coordinate system {x1, x3, ϕ}.When λ2 = λ3, we get in a similar manner

q1 = u1(x1, x2), q2 = u2(x1, x2) cos ϕ, q3 = u2(x1, x2) sin ϕ,

which define the oblate spherical coordinate system {x1, x2, ϕ}.


When λ1 = λ2 = λ3, the only remaining coordinate is r = (q21 + q2

2 + q23 )

1/2,and two coordinates ϑ, ϕ have to be introduced on the unit sphere. We get

q1 = rω1(ϑ, ϕ), q2 = rω2(ϑ, ϕ), q3 = rω3(ϑ, ϕ),

where the mapping R2 � (ϑ, ϕ) �→ ω ∈ S

2 can be chosen in several different ways.For instance, spherical coordinates are defined by choosing

ω1 = cosϑ cos ϕ, ω2 = cosϑ sin ϕ, ω3 = sinϑ. (2.4)

DEFINITION 2.3. The parabolic coordinate system {xi} in En with parameters

λ1 < λ2 < · · · < λn−1 is defined through the equation

n−1∑i=1

q2i

z − λi+ 2qn − z = −

n∏j=1

(z − xj )

/ n−1∏k=1

(z − λk). (2.5)

Like the elliptic coordinates, the parabolic coordinates are also orthogonal andonly locally defined. They take values in the intervals

x1 < λ1 < x2 < λ2 < · · · < λn−1 < xn,

(where λ1 always can be taken as unity) and can be degenerated in the same wayas the elliptic coordinates.

The parabolic coordinate system can, in fact, be derived from the elliptic co-ordinate system by an improper degeneration. Indeed, introduce new Cartesiancoordinates q ′

i defined by

qi = q ′i/√λn, i = 1, . . . , n − 1, qn = (q ′

n − λn)/√λn,

in (2.3), let λn tend to infinity, and drop the primes to get (2.5).

EXAMPLE 2.4. There is only one possible degeneration of the parabolic coor-dinates in E

3, namely λ1 = λ2. It gives parabolic coordinates in the {r, q3}-plane,r = (q2

1 + q22 )

1/2, which can be complemented by an angular coordinate in the{q1, q2}-plane to give rotational parabolic coordinates.

DEFINITION 2.5. The elliptic coordinate system {xi} on Sn−1 with parameters

λ1 < λ2 < · · · < λn is defined through the equation

n∑i=1

ω2i

z − λi=

n−1∏j=1

(z − xj )

/ n∏k=1

(z − λk). (2.6)

Notice that (2.6) implies∑

ω2i = 1. Like the elliptic coordinates in E

n, theelliptic coordinates on S

n−1 are also orthogonal and only locally defined. Theytake values in the intervals

λ1 < x1 < λ2 < x2 < · · · < xn−1 < λn.


The coordinates and the parameters can be subjected to a simultaneous lineartransformation xi �→ axi + b, λi �→ aλi + b, so it is always possible to chooseλ1 = 0 and λ2 = 1. The coordinates can be degenerated by letting some, but notall, parameters λi coincide. No improper degenerations exist.

EXAMPLE 2.6. In the case of S2, the only possible degenerations are λ1 = λ2

and λ2 = λ3. They both correspond to the spherical coordinates (2.4) because ofthe arbitrariness in the choice of angles ϑ , ϕ.

DEFINITION 2.7. By letting all parameters coincide in the elliptic coordinates inEn we have polar coordinates (r, ω) ∈ E

1+ × Sn−1 defined by q = rω. The special

case, when elliptic coordinates are chosen on the sphere Sn−1, is called conical

coordinates.

2.4. ALL STÄCKEL FORMS AND CORRESPONDING COORDINATE SYSTEMS

All possible Stäckel forms for the metric in a Riemannian space of constant curva-ture were found by Eisenhart [15, §3]. He found that all solutions of the Levi-CivitaEquations (2.1) are of the form

H 2i = Xi(xi)

∏j =i

(σij (xi) + σji(xj )

), (2.7)

where Xi and σij are functions that depend on one variable only. He found furtherthat depending on whether the functions σij are constant, the indices i and j can bedivided into classes, which impose certain conditions on the σ ’s having i or j as anindex. The form (2.7) together with these conditions can then be used to eventuallyfind all separable coordinate systems. The most generic situation is when all σ ’sare nonconstant; it corresponds to elliptic or parabolic coordinates in E

n and toelliptic coordinates on S

n−1. Indeed, in these cases

H 2i = 1

Pλ(xi)

∏j =i

(xi − xj ). (2.8)

The denominator Pλ(x) is a polynomial with roots λi ; it is of degree n in the ellipticcases, and of degree n − 1 in the parabolic case, reflecting the absence of λn.

Eisenhart used these ideas to derive a complete list of all eleven orthogonalseparable coordinate system in E

3 [15, §4–7]; for a nice exposition see Morse andFeshbach [27, Chapter 5]. Later, Kalnins and Miller [22, 23] improved this result tofind a recursive method that generates all orthogonal separable coordinate systemsin E

n and on Sn−1 for arbitrary n. (The coordinate systems on the spheres are

essential for the construction of the coordinate systems in En due to the rotational

symmetries that arise when degenerating elliptic and parabolic coordinates.) Wewill use these results frequently, and give a succinct formulation in the followingtwo theorems. We call a metric having the Stäckel form simply a Stäckel form.


THEOREM 2.8. All Stäckel forms on Sn−1 are given recursively by

ds2 = dσ 2 +N∑I=1

r2I dω2

I , (2.9)

where

(a) N and n1, . . . , nN are some positive integers that satisfy 2 � N � n and∑NI=1 nI = n,

(b) (r1, . . . , rN) = r(x) is the mapping between Cartesian coordinates rI andelliptic coordinates xi on S

N−1 with metric dσ 2 = ∑NI=1 dr2

I ,(c) dω2

I is a Stäckel form on SnI−1 if nI � 2; otherwise dω2

I = 0.

Separable coordinates can be associated with (2.9) in the following way. Let4 = {4I ; I = 1, . . . , N} be a partition of {1, . . . , n} associated with the integersnI , i.e., a collection of sets satisfying⋃

4 = {1, . . . , n}, 4I ∩ 4J = ∅ if I = J, #4I = nI ,

where # denotes the cardinality. Define new coordinates recursively in terms of theCartesian coordinates ωi on S

n−1 by

ωi = rIωI,i (i ∈ 4I), I = 1, . . . , N, (2.10)

where rI are as in (b), and ωI,i (i ∈ 4I) are Cartesian coordinates on the spheresin (c), unless #4I = 1 in which case we set ωI,i = 1. Then define new coordinatesin terms of ωI,i in the same way to fulfill condition (c), i.e., so that

dω2I =

∑i∈4I

dω2I,i

are Stäckel forms too.This yields a coordinate system on S

n−1 that corresponds to (2.9), which iseasily seen by invoking the conditions

∑NI=1 r

2I = 1 and∑

i∈4I

ω2I,i = 1 and its differentiated version

∑i∈4I

ωI,i dωI,i = 0.

Indeed, the coordinates belong to the sphere since

n∑i=1

ω2i =

N∑I=1

r2I

∑i∈4I

ω2I,i = 1,

and the metric has the Stäckel form sincen∑

i=1

dω2i =

N∑I=1

(dr2

I

∑i∈4I

ω2I,i + 2rI drI

∑i∈4I

ωI,i dωI,i + r2I

∑i∈4I

dω2I,i

)

is of the form (2.9).


EXAMPLE 2.9. On S1, there is only the elliptic coordinate system. It can be

parametrized by an angle ϕ so that ω1 = cos ϕ, ω2 = sin ϕ, with ds2 = dϕ2.On S

2, there are two coordinate systems: the elliptic system with metric ds2 =dσ 2, and the system

ω1 = r1(x1) ω11(x2), ω2 = r1(x1) ω12(x2), ω3 = r2(x1), (2.11)

where r21 + r2

2 = 1 and ω211 + ω2

12 = 1. For instance, if r1(x1) = cos x1 andω11(x2) = cos x2, then we have the spherical coordinates (2.4) with ds2 = dr2

1 +dr2

2 + r21 (dω

211 + dω2

12) = dϑ2 + cos2 ϑ dϕ2, where x1 = ϑ and x2 = ϕ.

DEFINITION 2.10. A metric ds2 on En is in the basic elliptic form if

ds2 = dσ 2 +N∑I=1

r2I dω2

I , (2.12)

where


(b) (r1, . . . , rN) = r(x) is the mapping between Cartesian coordinates rI andelliptic coordinates xi in E

N with metric dσ 2 = ∑NI=1 dr2

I ,(c) dω2

I is a Stäckel form on SnI−1 if nI � 2; otherwise dω2

I = 0.

It is in the basic parabolic form if we instead of (a) and (b) have

(a′) 2 � N � n and∑N

I=1 nI = n and nN = 1,(b′) (r1, . . . , rN) = r(x) is the mapping between Cartesian coordinates rI and

parabolic coordinates xi in EN with metric dσ 2 = ∑N

I=1 dr2I .

THEOREM 2.11. All Stäckel forms on En are given by

ds2 =N∑I=1

dσ 2I , (2.13)

where


(b) dσ 2I are metrics on E

nI in the basic elliptic or the basic parabolic form.

It is almost as easy to associate coordinates with this metric as in the caseof the sphere. Note first that (2.13) reflects a decomposition of E

n into mutuallyorthogonal subspaces. On these subspaces we have metrics in the basic elliptic orparabolic forms for which (2.10) yields coordinates (with ωi replaced by qi). Theorthogonal sum of these basic coordinate systems then corresponds to (2.13).


EXAMPLE 2.12. In E1, there is only the elliptic coordinate system; it is equiva-

lent to the Cartesian.In E

2 we have first the totally decomposed Cartesian coordinate system (E2 =E

1 ⊕ E1); ds2 = dq2

1 + dq22 . Further, the elliptic coordinate system and its proper

degeneration, the polar coordinate system q1 = r cos ϕ, q2 = r sin ϕ, with metricds2 = dr2 + r2 dϕ2. Finally, there is the parabolic coordinate system.

In E3, there are four types of orthogonally decomposed systems (E3 = E

2 ⊕E1)

arising from the systems on E2 by adding an extra Cartesian coordinate. Then there

are the elliptic and the parabolic coordinate systems. Further, the prolate spheroidaland the rotational parabolic coordinate systems with ds2 = dσ 2 + r2

1 dϕ2 and theoblate spheroidal coordinate system with ds2 = dσ 2+r2

2 dϕ2. Finally, there are twotypes of polar coordinate systems (r, ω) ∈ E

1+ × S2; their metric is ds2 = dr2 +

r2 dω2. The difference is the choice of coordinates on S2; either elliptic coordinates,

which yields conical coordinates; or the coordinates (2.11), which yields sphericalcoordinates.

3. Equations Satisfied by Separable Potentials

From Theorems 2.8 and 2.11 it is evident that all orthogonal separable coordinatesystems are constructed of four fundamental coordinate systems:

(a) The multipolar coordinate system (to be defined below).(b) The elliptic coordinate system.(c) The parabolic coordinate system.(d) The elliptic coordinate system on the sphere.

Since (d) is the restriction of the conical coordinate system to the unit sphere, itcan be replaced by

(d′) The conical coordinate system.In order to construct a general system of equations that has the Stäckel mul-

tipliers as its solution, such equations for these four coordinate systems will bediscussed first. These equations are simply the Levi-Civita Equations (2.2) spe-cialized by inserting the relevant metric coefficients. The form of these equationsin Cartesian coordinates is also needed, and could in principle be obtained by achange of coordinates in the Levi-Civita equations. However, it turns out to bemore practical to derive these equations directly in Cartesian coordinates. Marshalland Rauch-Wojciechowski did this first, starting from the requirement of existenceof n quadratic first integrals with known kinetic part. The equations for V thenarise as an integrability condition for the existence of the corresponding potentialparts in the first integrals.

An alternative way of describing equations for separable potentials V has laterbeen obtained by Benenti through the equation d(K ·dV ) = 0. In this case, one hasto find the relevant characteristic tensor K expressed in Cartesian coordinates, and


simply write down the equations. Explicit expressions for K have been obtained incertain special cases by Benenti [7], who used inertia tensors to construct so-calledStäckel systems corresponding to K.

The aim of this section is to give an overview and a coherent presentation ofequations satisfied by separable potentials expressed both in Cartesian and separa-tion coordinates. They play a pivotal role in the criterion of separability that is theobjective of this work.

3.1. MULTIPOLAR COORDINATES

A multipolar coordinate system is simply a collection of polar coordinate systems.They have already been used in the construction of separable coordinate systems,see Equation (2.10). Here we give a formal definition.

DEFINITION 3.1. If 4 = {4I ; I = 1, . . . , N} is a partition of {1, . . . , n}, thenwe call the coordinate system {rI , ωI,i} defined by

qi = rIωI,i (i ∈ 4I), I = 1, . . . , N, (3.1)

the multipolar coordinate system associated with the partition 4 if (rI , ωI ) are po-lar coordinates in {qi; i ∈ 4I } for I = 1, . . . , N. If #4I = 1, so that{qi; i ∈ 4I } is one-dimensional, we set ωI,i = 1.

We allow the radial coordinates rI to take all real values. So if #4I = 1, thenqi = rI for i ∈ 4I .

Examples of multipolar coordinate systems in En are the Cartesian coordinate

system, which is associated with the total (N = n) partition 41 = {1}, . . . ,4n ={n}, and the polar coordinate system, which is associated with the trivial (N = 1)partition 41 = {1, . . . , n}.

Consider now a separable coordinate system as given by Theorem 2.11. It isconstructed by taking the orthogonal sum of a set of multipolar coordinate system,each of which is subjected to a transformation connecting its radii. Thus the coor-dinate system is also a multipolar coordinate system, but with the special featurethat some of the polar coordinate systems which form the components of it areconnected. If a potential is separable in this coordinate system, then we can writethe potential in the form of a sum corresponding to the orthogonal sum. This isalmost trivial and can be shown as follows.

PROPOSITION 3.2. Suppose that a potential V is separable in a coordinatesystem {xi}, which is the orthogonal sum of a set of coordinate systems, i.e.,

qi = fi(xj ; j ∈ 4I) (i ∈ 4I),

where {4I ; I = 1, . . . , N} is a partition of {1, . . . , n}. Then it holds that

∂ijV = 0, i ∈ 4I, j ∈ 4J , I = J. (3.2)


Proof. The metric is ds2 = ∑H 2

i dx2i = ∑N

I=1 dσ 2I , where dσ 2

I only depends onxi for i ∈ 4I . Hence ∂H 2

j /∂xi = 0 for i ∈ 4I and j ∈ 4J with I = J . The Levi-Civita Equation (2.2) for these indices is clearly equivalent to ∂2V /∂xi∂xj = 0,and (3.2) follows by the chain rule. ✷

3.2. ELLIPTIC COORDINATES

PROPOSITION 3.3. The potential V is separable in the elliptic coordinate system{xi} if and only if it satisfies

∂2

∂xi∂xj

((xi − xj )V

) = 0, i = j. (3.3)

Proof. Equations (3.3) are the Levi-Civita Equations (2.2) combinedwith (2.8). ✷THEOREM 3.4. Let H = 1

2p2 + V (q) be a natural Hamiltonian. The following

are equivalent.

(a) H admits n quadratic first integrals

Ki =∑j =i

(qipj − qjpi)2

λi − λj+ p2

i + Ui(q). (3.4)

(b) V satisfies the system of(n

2

)equations

(λi − λj )∂ijV − Jij (2 + R)V = 0, i = j. (3.5)

(c) V is separable in elliptic coordinates with parameters λi .

Note that∑

(Ki − Ui) = p2. Hence H = 12

∑Ki , so that the functions H, Ki

are dependent, but any set of n of them is functionally independent.The theorem is due to Marshall and Rauch-Wojciechowski [26, Section III];

cf. Appendix. See also Benenti [8, Section 5] (where the first integrals are ex-pressed through elementary symmetric polynomials).

3.3. PARABOLIC COORDINATES

PROPOSITION 3.5. The potential V is separable in the parabolic coordinatesystem {xi} if and only if it satisfies

∂2

∂xi∂xj

((xi − xj )V

) = 0, i = j. (3.6)


This is the same as in the elliptic case, Proposition 3.3. The reason is that themetric also in this case has the coefficients (2.8), the only difference being in thedenominator Pλ, which is irrelevant here.

THEOREM 3.6. Let H = 12p

2 + V (q) be a natural Hamiltonian. The followingare equivalent.

(a) H admits n − 1 quadratic first integrals

Ki =n−1∑j =i

(qipj − qjpi)2

λi − λj+ 2pi(qipn − qnpi) − λip

2i + Ui(q),

i = 1, . . . , n − 1. (3.7)


2

)equations

(λi − λj )∂ijV + Jij ∂nV + (δin∂j − δjn∂i)(2 + R)V = 0, i = j,(3.8)

where λn = 0.(c) V is separable in parabolic coordinates with parameters λ1, . . . , λn−1.

The theorem can be proved by degenerating the elliptic coordinates in The-orem 3.4 to parabolic coordinates as in Definition 2.3. As λn → ∞, one findsthat Kn → H and that K1, . . . , Kn−1 become as in (a). See Marshall and Rauch-Wojciechowski [26, Section IV], and Rauch-Wojciechowski [28, Section 3] for themore elegant formulation given here. See also Benenti [8, Section 8].

3.4. CONICAL COORDINATES

PROPOSITION 3.7. The potential V is separable in the conical coordinate system{r, x1, . . . , xn−1} if and only if it satisfies

∂2

∂r∂xi(r2V ) = 0,

∂2

∂xi∂xj

((xi − xj )V

) = 0, i = j, (3.9)

i, j = 1, . . . , n − 1.

The proposition follows as in the elliptic case, Proposition 3.3; here the metric isds2 = dr2+r2 dω2, where dω2 is a metric on the sphere S

n−1 with coefficients (2.8).The first n − 1 equations have the general solution V = f (r) + r−2g(x) with

arbitrary functions f and g. As a consequence, the radius r can be separated offfrom the potential. Further, V is a Stäckel multiplier for ds2 provided that g is a

Stäckel multiplier for dω2. This is guaranteed by the remaining(n−1

2

)equations,

which are satisfied if and only if g is separable in elliptic coordinates xi on thesphere S

n−1.


THEOREM 3.8. Let H = 12p

2 + V (q) be a natural Hamiltonian. The followingare equivalent.

(a) H admits n quadratic first integrals

Ki =∑j =i

(qipj − qjpi)2

λi − λj+ Ui(q). (3.10)


2

)equations

J1i (2 + R)V = 0, i = 2, . . . , n, (3.11a)

Cλ1ij V = 0, i = j, i, j = 2, . . . , 2n. (3.11b)

(c) V is separable in conical coordinates with parameters λi .

Note that∑

Ki = 0, so the first integrals Ki are dependent. Every set of n − 1of them is functionally independent, and they are also functionally independent ofthe Hamiltonian H .

The theorem can be proved in the same way as in the elliptic case; see Mar-shall and Rauch-Wojciechowski [26, Appendix B]; cf. Appendix. See also Benenti[8, Section 6].

There are two types of equations that characterize potentials separable in conicalcoordinates. Equations of the first type ensure that the form of the potential isf (r) + r−2g(ω) in polar coordinates, equations of the second type ensure that g isseparable in elliptic coordinates on the sphere. In Theorem 3.8, (3.11a) are of thefirst type and (3.11b) are of the second type. We show this by proving the following.

PROPOSITION 3.9. The general solution of (3.11a) is

V = f (r) + r−2g(ω), (3.12)

where f and g are arbitrary functions.Proof. Consider first the system

J1iW = 0, i = 2, . . . , n. (3.13)

In the coordinates

x1 = r2 = q21 + · · · + q2

n, xi = qi, i = 2, . . . , n,

the angular derivatives become J1i = q1 ∂/∂xi , which shows that W defined by(3.13) is independent of x2, . . . , xn.

Hence (3.11a) is equivalent to (2 + R)V = W(r2) for some function W . Inpolar coordinates this is a linear first order ODE since R = r ∂/∂r. The generalsolution is (3.12). ✷


4. Generalized Bertrand–Darboux Equations

All equations characterizing separability given in the previous section have beenformulated in a distinguished reference frame, the canonical Euclidean referenceframe. In order to be able to use these equations in the criterion of separability,we need to rewrite the equations in an arbitrary Euclidean reference frame since,for a given potential, it is a priori unknown. We shall do this now for all foursystems (3.2), (3.5), (3.8) and (3.11). It will turn out that the equations obtained forthe elliptic coordinate system encompass the equations obtained for the remainingthree coordinate systems.

4.1. TOWARDS A CRITERION OF SEPARABILITY

LEMMA 4.1. A necessary and sufficient condition for a potential V to be sep-arable in an elliptic coordinate system with respect to some Euclidean referenceframe, is that it satisfies∑

k

((αqiqk + βiqk + βkqi + γik)∂kjV −

− (αqjqk + βjqk + βkqj + γjk)∂kiV)+

+ 3((αqi + βi)∂jV − (αqj + βj)∂iV

) = 0, i = j, (4.1)

for some parameters α, βi, γij = γji , which fulfil the conditions

(a) α = 0, and(b) ββ t − αγ has simple eigenvalues,

where β = (βi) ∈ Rn and γ = (γij ) is a real symmetric n × n matrix.

Due to symmetries in the equations, there are(n

2

)linearly independent equa-

tions of type (4.1). This agrees with the number of Levi-Civita Equations (2.2).

Proof. Theorem 3.4 asserts that the potential V is separable in elliptic coordi-nates with parameters λi in the canonical Euclidean reference frame q ′ if and onlyif it satisfies

(λk − λ%)∂′k%V − (q ′

k∂′% − q ′

%∂′k)(

2 +∑

q ′i∂

′i

)V = 0, (4.2)

where ∂ ′i = ∂/∂q ′

i , etc. [Both (4.1) and (4.2) vanish if the free indices coincide,so we can consider all n2 equations.] Now introduce new Cartesian coordinatesdefined by the Euclidean transformation q �→ q ′ = At(q − b). By the chain rule

(λk − λ%)∑r,s

ArkAs%∂rsV −

−∑r,s

(Ar%Ask − ArkAs%)(qs − bs)∂r

(2 +

∑t

(qt − bt )∂t

)V = 0, (4.3)


which multiplied by AtkiA

t%j and summed over k and % takes the form∑

r

(Sri∂rj − Srj∂ri)V−

− ((qi − bi)∂j − (qj − bj )∂i)(

2 +∑t

(qt − bt)∂t

)V = 0, (4.4)

where S = (Sij ) = Adiag(λ1, . . . , λn)At. The last step combines the original

equations with free indices k, % in an invertible way to form a new system ofequations with free indices i, j . To get (4.1), multiply (4.4) by α, and set β = −αb

and γ = α(bbt − S).The system (4.2) is thus equivalent to (4.1) if and only if (a) holds. Moreover,

since S = α−2(ββ t −αγ ), it is clear that (b) is a necessary and sufficient conditionfor the parameters λi to be distinct. ✷

Note that the parameters α, βi, γij are free in (4.1), i.e., they do not have to obeyany constraints. This is important when we are going to use these equations in thecriterion of separability we will formulate later. The fact that the parameters arefree is rather surprising, because when we introduce the Euclidean transformationto get (4.3), a set of nonlinear algebraic constraints on the parameters Aij willarise since the rotation matrix A has to satisfy AtA = Id. The invertible linearcombination that carries (4.3) into (4.1) is therefore crucial for our purpose.

A direct consequence of the proof of Lemma 4.1 is the following corollary,which explains how to construct the separation coordinates for a potential thatfulfils the hypothesis of Lemma 4.1.

COROLLARY 4.2. Suppose that the potential V (q) satisfies the hypothesis ofLemma 4.1. Then it is separable in the elliptic coordinate system {xi} defined by 1+∑n

i=1 (q′i )

2/(z − λi) = ∏nj=1(z − xj )

/∏nk=1(z − λk). The Cartesian coordinates

q ′i are related to qi through q = Aq ′ + b where b = −α−1β. The parameters λk

are the eigenvalues of the symmetric matrix S = bbt − α−1γ sorted in increasingorder λ1 < λ2 < . . . < λn. The orthogonal matrix A is given by the relationS = A diag(λ1, . . . , λn)A

t.

The corollary shows how we can use the system (4.1) in a criterion of sepa-rability to test if potentials given in Cartesian coordinates are separable in ellipticcoordinates. We formulate a simple algorithm that shows how to proceed:

(1) Insert V (q) into (4.1), which has to be satisfied identically with respectto all qi . This gives a system of linear homogeneous algebraic equations for theparameters α, βi , γij , which should be solved. If α = 0, then V (q) is not separablein elliptic coordinates.

(2) If α = 0, put b = −α−1β and S = bbt − α−1γ , and diagonalize S, i.e.,find an orthogonal matrix A with detA = 1 such that S = A diag(λ1, . . . , λn)A

t.If some eigenvalues λi coincide, then V (q) is not separable in elliptic coordinates.


Otherwise V (q) is separable in elliptic coordinates with parameters λi . The changeof coordinates is given by Corollary 4.2.

Note that the algorithm above not only gives definite answers to the questionif a given potential is separable in elliptic coordinates; it also gives an explicitconstruction of these coordinates in the separable case.

4.2. BERTRAND–DARBOUX EQUATIONS

In the two-dimensional case, there is only one equation of type (4.1), namelyEquation (1.5) of the Bertrand–Darboux theorem. The following theorem showsthat (4.1) in fact serves as a generalization of (1.5), and we shall therefore refer toall these equations as the Bertrand–Darboux (BD) equations in the following.

THEOREM 4.3. The system of BD equations (4.1) can always be brought intoone of the following three canonical forms by a transformation to the canonicalEuclidean reference frame.

(a) If α = 0, the canonical form is the elliptic

(λi − λj )∂ijV − Jij (2 + R)V = 0. (4.5)

(b) If α = 0 and β = 0, it is the parabolic

(λi − λj )∂ijV + Jij ∂nV + (δin∂j − δjn∂i)(2 + R)V = 0. (4.6)

(c) If α = 0 and β = 0, it is the Cartesian

(λi − λj )∂ijV = 0. (4.7)

Note that the λ’s are not specified, and that they may coincide.The name ‘elliptic’ canonical form is motivated by the fact, that the general

solution of the system (4.5) is separable in elliptic coordinates if all λi are distinct.Likewise for the parabolic and the Cartesian forms. The Cartesian system (4.7) isin particular equivalent to (3.2) if λi = λj when i, j ∈ 4I and λi = λj wheni ∈ 4I and j ∈ �4I .

The content of this theorem is essentially the same as the idea of standard formsin the theory of (G, I ) cofactor pair systems; see Lundmark [25].

Proof. If α = 0, we get the equivalence of (4.1) and (4.5) from the proof ofLemma 4.1, so we only have to consider the last two cases.

If α = 0, we have∑k

((βiqk + βkqi + γik)∂kjV − (βjqk + βkqj + γjk)∂kiV

)++ 3(βi∂jV − βj∂iV ) = 0. (4.8)

Consider first the case β = 0. We can assume that |β| = 1, for otherwise it ispossible to multiply by 1/|β|. Introduce the Euclidean transformation q = Aq ′ +b


where the last column of A is β, multiply by AirAjs and sum over i, j . After havingdropped the primes, we have∑

t

(Ttr∂tsV − Tts∂trV ) + Jrs∂nV + (δrn∂s − δsn∂r)(2 + R)V = 0, (4.9)

where T = At(βbt + bβ t + γ )A = en(Atb)t + (Atb)et

n + AtγA and en = (δin)

denotes the nth standard basis vector. We now choose b and the n−1 first columnsof A in such a way that T becomes diagonal. Let R be any orthogonal matrixwith last column equal to β and detR = 1. Choose an orthogonal matrix P withdetP = 1 that diagonalizes the upper left (n − 1) × (n − 1) block Q in RtγR,

i.e., P tQP = diag(λ1, . . . , λn−1). The matrix R(P 00 1

)is orthogonal with unit

determinant and has β as its last column, so it is consistent to let A be this matrix,which gives AtγA = diag(λ1, . . . , λn−1, 0) + enc

t + cetn for some c ∈ R

n. Bychoosing b = −Ac, we get T = diag(λ1, . . . , λn−1, 0), and it follows that (4.9)takes the form (4.6).

In the case β = 0, (4.8) simplifies to∑

k(γik∂kjV −γjk∂kiV ) = 0. We introducethe Euclidean transformation q → Aq as above, multiply by AirAjs and sum overi, j to find

∑t (Ttr∂tsV − Tts∂trV ) = 0 where T = AtγA. Finally, we choose A

such that T = diag(λ1, . . . , λn) to get (4.7). ✷

4.3. CYCLIC BERTRAND–DARBOUX EQUATIONS

Suppose that a potential satisfies the BD equations with α = 0 so that they canbe transformed to the canonical elliptic form [case (a) of Theorem 4.3]. If all λicoincide, then we only have Jij (2+R)V = 0 in the canonical Euclidean referenceframe. We know that these equations say precisely that the radius can be separatedoff from the potential, and Proposition A.2 shows that there is a basis of n − 1 ofthem.

We need to be able to detect separability of the remaining ‘spherical part’ aswell. Therefore we rewrite also the second type of equations from Theorem 3.8 inan arbitrary Euclidean reference frame. This results in new equations that have a

basis of(n−1

2

)equations, which again gives a total of

(n

2

)equations when com-

bined with the previous n−1 equations. We shall refer to the new Equations (4.10)as the cyclic Bertrand–Darboux (CBD) equations.

Note that by applying the BD equations to a given potential, we already knowthe location of the origin, i.e., we know the vector b. Hence it is sufficient hereto introduce a linear orthogonal transformation q → Aq. Even though the newequations are dependent, it is convenient to use them all simultaneously in theproof below. In fact, it is possible to work with the whole set of n3 equations sinceCλijk = 0 if some indices coincide.


THEOREM 4.4. The system of CBD equations∑%

((γi%qj − γj%qi)∂k%V +

+ (γj%qk − γk%qj )∂i%V + (γk%qi − γi%qk)∂j%V) = 0, (4.10)

where γij = γji , can always be brought into the canonical cyclic form

CλijkV ≡ λi∂iJjkV + c.p. = 0 (4.11)

by a transformation to the canonical Euclidean reference frame.Proof. Set q = Aq ′ in (4.10), multiply by AirAjsAkt and sum over i, j , k.

After having dropped the primes, we have qr∑

m(Tms∂mtV − Tmt∂msV ) + c.p. =0, where T = AtγA. By choosing A such that T = diag(λ1, . . . , λn), we getqr(λs∂stV − λt∂tsV ) + c.p. = 0, which is (4.11). ✷

The proof shows that we can use the CBD equations to test if the ‘spherical part’of a potential is separable in elliptic coordinates on the sphere. The idea is the sameas for the BD equations: we find the matrix γ = (γij ) for which a given potentialsatisfies the CBD equations, and diagonalize it to get the rotation matrix A and theparameters λi .

5. General Solution of the Bertrand–Darboux Equations

We have derived and indicated the nature of the solution to the BD and CBDequations in generic situations, where we assumed distinct parameters λ1 < λ2 <

· · · < λn. We now show how to obtain information about the general solution in thedegenerate cases as well. It is impossible to give the solution explicitly in Cartesiancoordinates, but we are nevertheless able to give a description of it in terms of basicStäckel multipliers, which perfectly suits our needs.

We allow for the possibility of coinciding parameters λ1 � λ2 � · · · � λn. Inorder to be able to handle this situation, we need to define reduced distinct para-meters, which we will denote <I . This definition induces a partition of {1, . . . , n}as well, and so also an associated multipolar coordinate system.

More precisely, we define N to be the number of distinct λi , and let {4I ; I = 1,. . . , N} be a partition of {1, . . . , n} associated with the multiplicities of λi through

λi = λj ⇐⇒ i, j ∈ 4I,

λi < λj ⇐⇒ i ∈ 4I and j ∈ 4J where I < J.

Further, we let <1, . . . ,<N be defined by <I = λi for some i ∈ 4I . In theexceptional parabolic case, where λn = 0, we use n − 1 instead of n and N − 1instead of N ; then we let 4N = {n}. Thus <1 < <2 < · · · < <N , except for theparabolic case, where <1 < <2 < · · · < <N−1. Finally, we define a multipolarcoordinate system in E

n by qi = rI ωI,i (i ∈ 4I ).


We begin by giving ‘reduced’ BD systems, which together with some otherequations are equivalent to the original BD systems. The advantage of this newformulation is that the reduced systems are, in fact, lower-dimensional BD systemswith distinct parameters <I ; the solution of these is therefore known in view of ourprevious results.

LEMMA 5.1 (Reduced systems). The elliptic, parabolic and cyclic systems areequivalent to reduced systems according to the following, where the free indicesI, J,K range from 1 to N .

(a) The elliptic system

(<I − <J )∂ijV − Jij (2 + R)V = 0, i ∈ 4I, j ∈ 4J, (5.1)

is equivalent to

∂kJij V = 0, i, j ∈ 4I, k ∈ �4I, (5.2)

together with

Jij (2 + R)V = 0, i, j ∈ 4I, (5.3)

and the reduced elliptic system

(<I − <J )∂2V

∂rI ∂rJ−(rI

∂

∂rJ− rJ

∂

∂rI

)(2 +

N∑K=1

rK∂

∂rK

)V = 0. (5.4)

(b) If 4N contains only one index, i.e., if 4N = {n}, then the parabolic system

(<I − <J )∂ijV + Jij ∂nV + (δin∂j − δjn∂i)(2 + R)V = 0,

i ∈ 4I, j ∈ 4J, (5.5)

is equivalent to (5.2) together with (5.3) and the reduced parabolic system

(<I − <J )∂2V

∂rI ∂rJ+(rI

∂

∂rJ− rJ

∂

∂rI

)∂V

∂rN+

+(δIN

∂

∂rJ− δJN

∂

∂rI

)(2 +

N∑K=1

rK∂

∂rK

)V = 0. (5.6)

(c) The cyclic system

<I∂iJjkV + <J∂jJkiV + <K∂kJij V = 0,

i ∈ 4I, j ∈ 4J, k ∈ 4K, (5.7)

is equivalent to (5.2) together with the reduced cyclic system

<I

∂

∂rI

(rJ

∂

∂rK− rK

∂

∂rJ

)V + c.p. = 0. (5.8)


Proof. For the proof of these statements, we need to establish a few facts. Notefirst, that since R = r∂/∂r in the polar coordinates r, ωi , we have

R =N∑I=1

∑i∈4I

qi∂i =N∑I=1

rI∂

∂rI(5.9)

in the multipolar coordinates rI , ωI,i .Then we show that the system (5.2) implies

qiqj∂2V

∂rI ∂rJ= rI rJ ∂ijV , i ∈ 4I, j ∈ 4J, I = J, (5.10)

and

qiqj qk∂

∂rK

(rI

∂

∂rJ− rJ

∂

∂rI

)V = rI rJ rK∂kJij V ,

i ∈ 4I, j ∈ 4J, k ∈ 4K, I, J,K = . (5.11)

Equations (5.10) follow from

qiqj rI rJ∂2V

∂rI ∂rJ=

∑k∈4I

∑%∈4J

qiqj qkq%∂k%V

(†)=∑k∈4I

∑%∈4J

qkqjqkq% ∂i%V

(‡)=∑k∈4I

∑%∈4J

qkq%qkq% ∂ijV = r2I r

2J ∂ijV ,

where equality (†) is valid due to qi∂k∂%V = qk∂i∂%V which holds for i, k ∈ 4I

and % ∈ �4I , and similarly for equality (‡). In order to show (5.11), one has tostart with rI rJ rK times the left-hand side, and proceed as above.

In the same spirit, system (5.3) implies

qiqj

(rI

∂

∂rJ− rJ

∂

∂rI

)(2 + R)V = rI rJJij (2 + R)V ,

i ∈ 4I, j ∈ 4J, I = J, (5.12)

and, if 4N = {n},

qiqj

(δIN

∂

∂rJ− δJN

∂

∂rI

)(2 + R)V = rI rJ (δin∂j − δjn∂i)(2 + R)V ,

i ∈ 4I, j ∈ 4J, I = J. (5.13)

The proof of (5.12) is the same as that of (5.10) above; the idea is to use (5.3)written as qi∂j (2+R)V = qj∂i(2+R)V . For (5.13), we note that there is nothing


to prove unless I = N or J = N . Suppose that J = N . Then j = n follows dueto the requirement 4N = {n}, and we only have to show that

qiqj∂

∂rI(2 + R)V = rI rJ ∂i(2 + R)V , (5.14)

which is easily done in the same way as for (5.12) above. Likewise if I = N .We now proceed to the statements of the theorem.(a) Take distinct i, j ∈ 4I and k ∈ 4K where I = K. For this choice of indices,

there are three equations of type (5.1):

−Jij (2 + R)V = 0, (5.15a)

(<I − <K)∂jkV − Jjk(2 + R)V = 0, (5.15b)

(<K − <I)∂kiV − Jki(2 + R)V = 0, (5.15c)

and (5.15a) is (5.3).The linear combination qk(5.15a) + qi(5.15b) + qj (5.15c) reduces, because

of (1.12b), to (<I − <K)∂kJij V = 0, and we have (5.2).Using (5.9), (5.10) and (5.12), we find that (5.1), with I = J , is equivalent

to (5.4).The converse is immediate.(b) Take i, j, k as in (a), and write down all different forms of (5.5). Note that

I = N since we assume that #4I > 1. Hence i, j = n, and we get

Jij ∂nV = 0, (5.16a)

(<I − <K)∂jkV + Jjk∂nV − δkn∂j (2 + R)V = 0, (5.16b)

(<K − <I)∂kiV + Jki∂nV + δkn∂i(2 + R)V = 0. (5.16c)

The linear combination qk(5.16a) + qi(5.16b) + qj (5.16c) reduces in this case to

(<I − <K)∂kJij V − δknJij (2 + R)V = 0. (5.17)

When k = n, (5.17) yields ∂kJij V = 0, and together with (5.16a) we have(5.2). With k = n, (5.17) then implies (5.3).

Note that (5.11) with K = N reads

qiqj

(rI

∂

∂rJ− rJ

∂

∂rI

)∂V

∂rN= rI rJJij ∂nV (5.18)

after dividing by qn = rN . Using (5.9), (5.10), (5.18) and (5.13), we find that (5.5),with I = J , is equivalent to (5.6).

(c) Take i, j, k as in (a). Equation (5.7) implies

<I(∂iJjkV + ∂jJkiV ) + <K∂kJij V = 0, (5.19)

which is equivalent to (<I − <K)∂kJij V = 0, and (5.2) follows. Then, by using(5.11), we see that (5.7), with pairwise distinct I, J,K, is equivalent to (5.8). ✷


The system of Equations (5.2) and (5.3) is easily solved by introducing propercoordinates, as we show next.

LEMMA 5.2 (Radial solution). The general solution of the system

∂kJij V = 0, (5.20a)

Jij (2 + R)V = 0, (5.20b)

for i, j ∈ 4I and k ∈ �4I , where I = 1, . . . , N , is

V = f (r) +N∑I=1

r−2I gI (ωI ), (5.21)

where f and gI are arbitrary functions. If # 4I = 1, then gI = 0.

Notice the following two special cases. If N = n, then (5.20) is empty, and thesolution is V = f (r), where ri = qi . If N = 1, then (5.20a) vanishes, and thesolution is V = f (r) + r−2g(ω) where r, ωi are ordinary polar coordinates.

Systems (5.20a) and (5.20b) are linearly dependent since the operators Jij arelinearly dependent. As in the proof of Proposition A.2 it is always possible toconsider only the smaller systems obtained by fixing one index in the equations.For instance, we can let i be the minimal element of 4I and let j be any of theother elements in 4I .

Proof. We solve (5.20a) first. Because of linear dependence, it is sufficient toconsider

∂k(q�I�∂j − qj ∂�I�)V = 0, j ∈ 4I \ �I�, k ∈ 4K, I = K, (5.22)

where �I� denotes the minimal element of 4I . Let

xi ={r2I , if i = �I� for some I ,

qi, otherwise.

In these coordinates, (5.22) reads

2q�K�∂

∂x�K�

(− ∂

∂xj

)V = 0,(

2qk∂

∂x�K�+ ∂

∂xk

)(− ∂

∂xj

)V = 0, k = �K�,

which is equivalent to ∂2V /∂xj ∂xk = 0, where j ∈ 4I \ �I� and k ∈ 4K . Itfollows that the solution of (5.20a) is

V = f (x�1�, . . . , x�N�) +N∑I=1

gI (xi; i ∈ 4I) = f (r) +N∑I=1

gI (qi; i ∈ 4I),


where f and gI are arbitrary functions. If #4I = 1, then qi = rI , and we can setgI = 0.

Since Jij f = 0 and Jij gK = 0 for i, j ∈ 4I where I = K, (5.20b) is arestriction on gI only. Moreover, (5.20a) implies

RJij V =N∑

K=1

∑k∈4K

qk∂kJij V =∑k∈4I

qk∂kJij V = RIJij V ,

where RI is defined by the last equality. Hence (5.20b) is equivalent to

Jij (2 + RI )gI = 0, i, j ∈ 4I, I = 1, . . . , N.

The solution of these systems is given by Proposition 3.9 as gI = fI (rI )+r−2I gI (ωI ), which shows that (5.21) holds. ✷

We immediately have the following result, which is central for our theory.

THEOREM 5.3 (General solution of Bertrand–Darboux equations). The generalsolution of the BD Equations (4.1) with respect to the canonical Euclidean refer-ence frame is given by the following:

If α = 0, then

V = f (r) +N∑I=1

r−2I gI (ωI ), (5.23)

where f (r) is a Stäckel multiplier given by the elliptic coordinate system and thefunctions gI are arbitrary.

If α = 0 and β = 0, then (5.23) holds, but f (r) is now given by the paraboliccoordinate system.

If α = 0 and β = 0, then

V =N∑I=1

V(I)(qi; i ∈ 4I), (5.24)

where V(I) are arbitrary functions.Suppose that V = f (r)+ r−2g(ω) in some Euclidean reference frame, i.e., that

V satisfies the system

Jij (2 + R)V = 0. (5.25)

Then V solves the CBD Equations (4.10) if and only if

g = f (r) +N∑I=1

r−2I gI (ωI ), (5.26)


where f (r) is a Stäckel multiplier given by the elliptic coordinate system on the(N − 1)-dimensional sphere, gI are arbitrary functions and ωi = rI ωI,i (i ∈ 4I )define multipolar coordinates.

Proof. Theorem 4.3 shows that we can always assume that the BD equations aregiven in a canonical Euclidean reference frame. Thus we have to solve the canon-ical elliptic system (5.1), the canonical parabolic system (5.5) and the canonicalCartesian system

(<I − <J )∂ijV = 0, i ∈ 4I, j ∈ 4J, I, J = 1, . . . , N. (5.27)

Lemmas 5.1 and 5.2 show that the general solution of (5.1) must have the alge-braic form (5.23), and that V has to satisfy the reduced elliptic system (5.4). It iseasy to verify that (5.4) is a condition on f (r) only: note that

∂2

∂rI ∂rJ(r−2

K gK) = 0,

(2 +

N∑L=1

rL∂

∂rL

)(r−2

K gK) = 0 (5.28)

for all I, J,K with I = J . It then follows from Theorem 3.4 that f (r) is a Stäckelmultiplier given by the elliptic coordinate system with parameters <I .

By the same argument it follows that the general solution of the parabolic sys-tem (5.5) must have the algebraic form (5.23). To see that (5.6) is a condition onf (r) only, use (5.28) and

∂

∂rN(r−2

K gK) = 0, (5.29)

which trivially follows from the fact that gN = 0. Theorem 3.6 then shows thatf (r) is a Stäckel multiplier given by the parabolic coordinate system with parame-ters <I .

The function (5.24) is the general solution of the Cartesian system (5.27).Theorem 4.4 brings the CBD equations into the canonical form (5.7), which will

be solved together with (5.25). We first prove that (5.7) means that the algebraicform of g is given by (5.26). Since r∂i = ∂/∂ωi and qi∂i = ωi ∂/∂ωi, (5.25) and(5.7) can be formulated in terms of ωi instead of qi :(

ωi

∂

∂ωj

− ωj

∂

∂ωi

)(2 +

n∑k=1

ωk

∂

∂ωk

)V = 0, (5.30)

and by Lemma 5.1

∂

∂ωk

(ωi

∂

∂ωj

− ωj

∂

∂ωi

)V = 0, i, j ∈ 4I, k ∈ �4I, (5.31)

and

<I

∂

∂rI

(rJ

∂

∂rK− rK

∂

∂rJ

)V + c.p. = 0. (5.32)


Lemma 5.2 applied to (5.30) and (5.31) gives g as it appears in (5.26). Equa-tion (5.32) is a condition on f (r) only, which by Theorem 3.8 is a Stäckel mul-tiplier given by the elliptic coordinate system on S

N−1 with parameters <I . ✷It is an important fact that all separable potentials are generated by the three

forms above. Indeed, from the theorems by Stäckel (Section 2.2) and Kalnins andMiller (Theorems 2.8 and 2.11), we have: A potential is separable if and only ifit can be brought into the sum form (5.24) by a Euclidean transformation. In thissum, all functions V(I) should be of the radial form (5.23), where the functions f areStäckel multipliers given by the elliptic or the parabolic coordinate systems. The‘spherical parts’ gI should be of the radial form (5.26), where the functions f areStäckel multipliers given by the elliptic coordinate system on the sphere. Finally,the finer ‘spherical parts’ gI should also be of the radial form (5.26), where thenew lower-dimensional functions f and gI are Stäckel multipliers and of radialform respectively, as above. This description continues recursively by specifyingnew finer ‘spherical parts’ of radial form.

We use this fact to prove the following corollary, which in particular shows thatthe BD equations are a natural starting point for an investigation of the separabilityproperties of a given potential; a separable potential has to satisfy the BD equations.

COROLLARY 5.4 (Necessary condition for separability). Let V be a potentialin E

n.

(a) If V is separable, then V satisfies the BD equations with nontrivial parame-ters, i.e., parameters other than

α = 0, β = 0, γ = t (δij ), t ∈ R.

(b) If V is separable and satisfies the system (5.25) in some Euclidean referenceframe, then V satisfies the CBD equations with nontrivial parameters, i.e.,parameters not satisfying γ = t (δij ), t ∈ R.

Trivial parameters represent the parameter kernel of the equations in the sensethat the equations will cancel for trivial parameters. By linearity it is thus alwayspossible to add a multiple of the identity to any given parameters γ without chang-ing the equations. A consequence of this fact is that a potential will not determinethe parameters uniquely; our parameters α, β, γ will always depend on at least oneparameter (of the t-kind).

Proof. (a) If V is separable, then V is a Stäckel multiplier and the metric ds2

has the Stäckel form. Theorem 2.11 gives the form of ds2: it is either in the basic(elliptic or parabolic) form, or a sum of metrics in the basic forms.

If ds2 is in the basic form, then ds2 = ∑NI=1 dr2

I + ∑NI=1 r

2I dω2

I . The corre-sponding Stäckel multiplier is (5.23), where f (r) is a Stäckel multiplier given by


the elliptic or the parabolic coordinate system, and gI (ωI ) are Stäckel multiplierson the spheres of dimension #4I − 1 (or identically zero if #4I = 1).

Hence if V is of the elliptic or parabolic type (5.23), it follows from Theorem 5.3that it satisfies the BD equations with α = 0 or β = 0 respectively.

If ds2 = ∑NI=1 dσ 2

I , then the Stäckel multiplier is (5.24), where V(I) are Stäckelmultipliers for the metrics dσ 2

I on the Euclidean spaces {qi; i ∈ 4I }. Accordingto Theorem 5.3, any such V satisfies the BD equations with α = 0, β = 0 butγ = t (δij ), since <I are the eigenvalues of γ (see the proof of Theorem 4.3) andthere are N distinct <I .

(b) Lemma 5.2 shows that V = f (r) + r−2 g(ω), where r, ωi are polar co-ordinates in some suitable Euclidean reference frame. From Theorems 2.11 and2.8 we infer that V is a Stäckel multiplier if g(ω) is a Stäckel multiplier fords2 = ∑N

I=1 dr2I +∑N

I=1 r2I dω2

I , where ωi = rI ωI,i (i ∈ 4I) defines a multipolarcoordinate system. Hence if V is separable, then g(ω) is of the form (5.26), wheref (r) is a Stäckel multiplier given by the elliptic coordinate system on S

N−1, andgI (ωI ) are Stäckel multipliers on the spheres of dimension #4I − 1 (or identicallyzero if #4I = 1).

So if V is separable, then by Theorem 5.3 it satisfies the CBD equations withnontrivial parameters, since <I are the eigenvalues of γ (see the proof of Theo-rem 4.4) and there are N distinct <I . ✷

6. A Criterion of Separability

Having knowledge of the general solution of the Bertrand–Darboux equations, weare now able to explain the algorithm that constitutes the criterion of separability.

Given a potential in Cartesian coordinates, we begin by determining the pos-sible choices of the parameters α, β, γ that allow the potential to satisfy the BDequations: insertion of the potential into the BD equations yields a system of linearhomogeneous equations for the parameters by the requirement that the BD equa-tions should be satisfied identically with respect to qi . This system will alwayshave infinitely many solutions depending on one or more parameters that arisesfrom homogeneity (these parameters are of the t-kind in Corollary 5.4 and shouldnot be confused with α, β, γ ). If the only solution for the parameters α, β, γ isthe trivial (one-parametric), the potential is not separable and the study is finished.Otherwise, if a nontrivial (many-parametric) solution exists, the potential might beseparable, and there are three different cases to consider depending on α and β:

(a) The elliptic case if α = 0.(b) The parabolic case if α = 0 and β = 0.(c) The Cartesian case if α = 0 and β = 0.

The next step is to calculate the transformation to the canonical Euclidean refer-ence frame as well as to find the parameters λi that essentially are the eigenvalues ofγ . Note that if some λ’s coincide, then there still remains the freedom of choosing a


Euclidean reference frame within the subspaces corresponding to multiple λ’s. Byrecursive use of the BD and CBD equations, canonical frames will be determinedin these subspaces too. This procedure will eventually yield a complete canonicalEuclidean reference frame in all of E

n.If all λ’s are distinct, then the potential is separable in the elliptic, parabolic

or Cartesian coordinate systems respectively – all with respect to the canonicalEuclidean reference frame. The study is thus complete. Otherwise we have to dealwith degenerate cases according to the following (with respect to a Euclidean ref-erence frame that is only partially determined; the choice of Euclidean referenceframe in the subspaces corresponding to multiple λ’s is insignificant).

Cases (a) and (b): The potential has the correct main structure for separation indegenerate elliptic or degenerate parabolic coordinates respectively. Moreover, mcoinciding λ’s indicate the presence of an (m−1)-dimensional spherical symmetry,which means that the potential has a part living on a sphere S

m−1. We have toinvestigate if all such ‘spherical parts’ with m � 3 are separable; this is a necessaryand sufficient condition for the full separability of the potential. This investigationis accomplished by applying the CBD equations to the restriction of the potentialto the (smallest) Euclidean subspace in which a sphere is embedded. Recall thatthe CBD equations test a radial potential for separability in conical coordinates, orequivalently, they test if its ‘spherical part’ is separable in elliptic coordinates onthe sphere. Since the restricted potential has this radial structure, it is possible touse these equations.

Case (c): The potential has the correct main structure for separation in the or-thogonal sum of basic (elliptic or parabolic) coordinate systems. We can imaginethis as that E

n decomposes into mutually orthogonal Euclidean subspaces, and thatthe potential decomposes into a sum of lower-dimensional subpotentials living inthese. The potential is separable if and only if all such subpotentials are separable.To test if this is the case, we apply the appropriate lower-dimensional BD equationsto all restrictions of the potential to such subspaces.

The CBD equations are applied in a similar manner as the BD equations. Weinsert the appropriate restriction of the potential, and calculate the parameters γ . Anecessarily trivial (one-parametric) solution for γ means nonseparability in whichcase the study is finished, while the possibility of a nontrivial (many-parametric)solution for γ means that the restricted potential might be separable.

We next calculate the transformation to the canonical Euclidean reference frame(in the subspace) and the parameters λi that are eigenvalues of γ . If all λ’s aredistinct, then the present ‘spherical part’ is separable in elliptic coordinates onthe sphere, and the study is finished. If some λ’s coincide again, it means thatthe ‘spherical part’ has the correct main structure for separability in degeneratedelliptic coordinates, and that there are finer spherical symmetries present with cor-responding finer ‘spherical subparts’. As above, the restricted potential is separableif and only if all of these are separable. We apply the CBD equations again (in arecursive manner) to the appropriate restrictions of the restricted potential to test


these new ‘spherical subparts’ for separability. This is possible since this new finerrestriction also has the correct radial structure.

6.1. A FORMAL DESCRIPTION

The preceding considerations provide all ingredients for the formal formulation ofthe algorithm given below. It is split in two parts, Algorithms 6.1 and 6.2, whichexplains how to apply the BD and CBD equations respectively.

First, a remark on diagonalization: Let M be a symmetric matrix. In the follow-ing algorithms, we shall write ‘diagonalize M = A diag(λ1, . . . , λn)A

t’ wheneverwe want to find an orthogonal matrix A with detA = 1 such that the equality holds(appealing to the spectral theorem). The eigenvalues λi are supposed to be sortedin nondecreasing order λ1 � λ2 � · · · � λn.

ALGORITHM 6.1 (BD). Let V (q) be a potential in En with Cartesian coordi-

nates qi . The potential V is separable if and only if the following algorithm ter-minates without encountering ‘V is not separable.’ The separable coordinates areconstructed during execution of the algorithm.

Begin.

Step 1. If n = 1, then q1 is arbitrary – stop.

Step 2. Insert V into the BD equations. This gives a system of linear homogeneousalgebraic equations for α, β and γ as coefficients at linearly independent functions.Solve the system. If

α = 0, β = 0, γ = t (δij )

for some t ∈ R, then V is not separable – stop.

(a) If α = 0, we have the elliptic case denoted (a) below.(b) If α = 0 and β = 0, we have the parabolic case denoted (b) below.(c) If α = 0 and β = 0, we have the Cartesian case denoted (c) below.

Step 3. Calculate A, b and λi:

(a) Let b = −α−1β. Diagonalize bbt − α−1γ = A diag(λ1, . . . , λn)At.

(b) Normalize β and γ so that |β| = 1. Choose an orthogonal matrix R withdetR = 1 and last column β. Let Q be the upper left (n − 1) × (n − 1) blockin RtγR. Diagonalize Q = P diag(λ1, . . . , λn−1)P

t. Let

A = R

(P 00 1

).


Calculate c = (ci) ∈ Rn from

0 . . . 0 c1

. . . . . . . . . . . . . . . . . . .0 . . . 0 cn−1

c1 . . . cn−1 2cn

= AtγA − diag(λ1, . . . , λn−1, 0).

Let b = −Ac.(c) Diagonalize γ = A diag(λ1, . . . , λn)A

t. Let b = 0.Set q = Aq ′ + b.

Step 4. Calculate N,4I and reduced parameters <I :

(a) Let N be the number of distinct λi . Define a partition {4I ; I = 1, . . . , N} of{1, 2, . . . , n} associated with the multiplicities of λi through

λi = λj ⇐⇒ i, j ∈ 4I,

λi < λj ⇐⇒ i ∈ 4I and j ∈ 4J where I < J.

Let <1, . . . ,<N be defined by <I = λi for some i ∈ 4I .(b) As in (a), but use n − 1 instead of n and N − 1 instead of N . Let 4N = {n}.(c) As in (a).

Step 5.

(a) Set

q ′i = rI (x)ωI,i (i ∈ 4I),

where xi are elliptic coordinates with parameters <I in EN with Cartesian

coordinates rI , and ωI,i are Cartesian coordinates on the unit sphere in E#4I =

{q ′i; i ∈ 4I }. If #4I � 2, then execute Algorithm 6.2 (CBD) on the restriction

of V to the spaces E#4I to determine ωI ; otherwise take ωI,i = 1 and stop.

(b) As in (a), except that xi are parabolic coordinates with parameters<1, . . . ,<N−1.

(c) Execute Algorithm 6.1 (BD) on the restriction of V to the spaces E#4I =

{q ′i; i ∈ 4I } to determine q ′

i .

End.

Notice that the recursive use of this algorithm and the next always terminates,since the restricted potentials are of strictly lower dimension.

If the algorithm passes Step 2 but later arrives at ‘V is not separable’ at somepoint, then V is partially separable, i.e., it is possible to separate off some coor-dinates but not all. If we arrive at the Cartesian case in Step 5, then it is alwayspossible to separate off some Euclidean subspace, and if we arrive at the ellipticor the parabolic case, then it is always possible to separate off some spheres. Thesame remark applies to Algorithm 6.2, but in that case it is only possible to separateoff spheres.


ALGORITHM 6.2 (CBD). Let V (q) be a potential in En, n � 2, with polar

coordinates defined by q = rω. Suppose that V satisfies Jij (2 + R)V = 0for i, j = 1, . . . , n. The ‘spherical part’ of V is separable if and only if thefollowing algorithm terminates without encountering ‘V is not separable.’ Theseparable coordinates xi(ω) on the sphere are constructed during the executionof the algorithm.

Begin.

Step 1. If n = 2, then set ω = ω(x), where x is the elliptic coordinate on S1 and

stop.

Step 2. Calculate γ using the CBD equations in a manner similar to Step 2 inAlgorithm 6.1 (BD). If γ = t (δij ) for some t ∈ R, then V is not separable – stop.

Step 3. Diagonalize γ = A diag(λ1, . . . , λn)At. Set ω = Aω′.

Step 4. Calculate N,4I and <I as in Step 4(a) in Algorithm 6.1 (BD).

Step 5. Set

ω′i = rI (x)ωI,i (i ∈ 4I),

where xi are elliptic coordinates with parameters <I on SN−1 embedded in E

N

with Cartesian coordinates rI , and ωI,i are Cartesian coordinates on the unit spherein E

#4I = {q ′i; i ∈ 4I }. If #4I � 2, then execute Algorithm 6.2 (CBD) on the

restriction of V to the spaces E#4I to determine ωI ; otherwise take ωI,i = 1 and

stop.

End.

If we pass Step 2, then γ has at least one pair of distinct eigenvalues, and wehave N � 2 in Step 4. This is important in Step 5.

6.2. COMPLETENESS OF THE ALGORITHM

Let V be a potential in En. Step 1. If n = 1, it is clear that V is separable in

any choice of coordinates. Step 2. Now n � 2, and we get α, β and γ from theBD equations. If the only possible choice of these parameters is the trivial, thenCorollary 5.4 shows that V is nonseparable.

Otherwise we have a nontrivial choice of the parameters, and Theorem 4.3shows that Step 3 gives a Euclidean transformation that brings the BD equationsand V into canonical form. As a by-product we get a set of parameters λi , whichwe reduce to pairwise distinct parameters <I in Step 4.

Step 5. Theorem 5.3 now shows that V is either of the form (5.23) with a suitableStäckel multiplier f (r), or of the form (5.24). In the former case we test if allgI (ωI ) are Stäckel multipliers, in the latter case if all V(I) are Stäckel multipliers –this is a necessary and sufficient condition for the separability of V .


Suppose now that that V satisfies the system

Jij (2 + R)V = 0, (6.1)

and we apply Algorithm 6.2 to test if the ‘spherical part’ g(ω) of V = f (r) +r−2g(ω) is separable. Step 1. If n = 2, it is clear that V is separable in the polarcoordinates defined by q = rω(x) because the radius r can be separated off, andthe choice of the coordinate x on S

1 is irrelevant. Step 2. Now n � 3, and weapply the CBD equations to get γ . If there is only the trivial choice of γ , thenCorollary 5.4 shows that V is nonseparable.

Otherwise there is a nontrivial choice of γ , and Theorem 4.4 shows that Step 3gives a Euclidean transformation that brings the CBD equations and V into canon-ical form. We also get all λi which are reduced to <I in Step 4.

Step 5. By Theorem 5.3, V is given by (5.26) where f (r) is a Stäckel multiplier.The potential V is separable if and only if all gI are Stäckel multipliers, hence weuse the present algorithm again on the restriction of V to the subspaces {qi; i ∈ 4I }in order to test if gI are Stäckel multipliers. This is possible since (6.1) holds alsoon each of these subspaces.

6.3. REMARKS

Since the Levi-Civita condition for separable V consists of(n

2

)PDEs, there should

be the same number of PDEs in Cartesian coordinates. It can be shown that the al-

gorithms in fact check that V satisfies(n

2

)linearly independent PDEs; see Waksjö

[32, Lemma 4.6].In some exceptional cases it might happen that a potential is separable in more

than one coordinate system. For instance, the potential V (q) = q−21 + q−2

2 + q−23

is separable both in the spherical and the Cartesian coordinate systems, as wellas in many others; see Evans [17]. This potential satisfies the BD equations witharbitrary α and γ = diag(λ1, λ2, λ3) but β = 0. If we choose α = 0 and λ1 =λ2 = λ3, we see that the radius can be separated off, and the CBD equations willeventually determine the spherical coordinate system. If we choose α = 0 andλ1 < λ2 < λ3 instead, then we immediately get the Cartesian coordinate system.We infer that if there are more than one separable coordinate system, then we canfind all of them by choosing different types of specific values for the parameterswhenever it is possible.

7. Applications of the Criterion of Separability

We shall apply the criterion of separability to some three-dimensional potentialsto illustrate various aspects of our algorithm. Higher-dimensional potentials do notpose any essential challenges, but the amount of calculations grows rapidly and


they are best performed by a computer program. We leave a study of such examplesto subsequent publications.

7.1. AN ELLIPTIC EGG

The rational potential

V = κ

/[1 −

(q2

1

η1+ q2

2

η2+ q2

3

η3

)], (7.1)

where 0 < η1 < η2 < η3, has the ellipsoid q21/η1+q2

2/η2+q23/η3 = 1 as a singular

surface. For κ > 0, this singular surface is repelling inwards, so that all trajectoriesstarting inside the ellipsoid remain there forever.

This potential belongs to the Jacobi family of elliptic separable potentials de-fined recursively by Rauch-Wojciechowski [35] as V (m) = 1

2

∑ni=1 U

(m)i , where

U(1)i = q2

i and U(m+1)i = 2q2

i V(m) − ηi U

(m)i for m = 1, 2, . . . . This yields the

infinite sequence of symmetric polynomial potentials

V (1) = 1

2

∑q2i , V (2) = 1

2

(∑q2i

)2 − 1

2

∑ηiq

2i ,

etc. The recursion can be inverted: U(0)i = 1 and

U(m−1)i = −(q2

i /ηi)(∑

U(m)j /ηj

)/(1 −

∑q2j /ηj

)− U

(m)i /ηi

for m = 0,−1, . . . defines a backwards infinite sequence of rational potentials.The potential (7.1) is the first nontrivial of these: V = V (−1) with κ = − 1

2

∑η−1i .

The particular value of κ is however of no significance since any potential can bemultiplied by a nonzero constant without affecting its property of being separableor not.

Let us apply the criterion of separability to V . After inserting V into the BDequations, we find the solution for the parameters to be

α = s, β = 0, γ = t (δij ) + s diag(η3 − η1, η3 − η2, 0), t, s ∈ R.

The solution is two-parametric (it depends on t and s), implying that V mightbe separable. It is no loss of generality to set t = 0 here since γ = t (δij ) iscontained in the parameter kernel (see the remark following Corollary 5.4), butit proves practical to keep t as a free parameter in the subsequent. Depending onwhether we take s to be nonzero or zero, we have the elliptic or the Cartesian casesrespectively. However, in the latter case we have trivial parameters, proving that Vis not separable in Cartesian coordinates. Therefore, we now suppose that s = 0.

Since β = 0, also b = 0. In the next step we observe that −α−1γ = diag(η1 −η3, η2 − η3, 0) + t ′(δij ), where t ′ = −ts−1, already is diagonal. This shows thatwe already are in the canonical Euclidean reference frame, and that we can take A


to be the identity. Now, N = 3 (the partition is total) and the reduced parameterscoincide with the eigenvalues:

<1 = η1 − η3 + t ′, <2 = η2 − η3 + t ′, <3 = t ′.

It follows that the potential is separable in the elliptic coordinate system withparameters as above. The parameter t ′ can be chosen arbitrarily. E.g., if t ′ = 1 +η3−η1, we have normalized parameters <1 = 1, <2 = 1+η2−η1, <3 = 1+η3−η1,while if t ′ = η3, we have <1 = η1, <2 = η2, <3 = η3, which explains the meaningof the constants occurring in the potential; they are simply the parameters of theelliptic coordinate system.

7.2. THE CALOGERO INVERSE-SQUARE SYSTEM

The potential

V = (q1 − q2)−2 + (q2 − q3)

−2 + (q3 − q1)−2 (7.2)

describes the motion of three identical particles along a line. They interact witheach other due to forces depending on their relative distances. This potential wasseparated by Calogero [13] in cylindrical coordinates, and recently Benenti et al.[10] showed that it is actually separable in four other types of coordinate systems.We shall now give an efficient reconstruction of their results using our criterion ofseparability.

The BD equations give the four-parametric solution

α = v, β = w

( 111

), γ = t

( 1 0 00 1 00 0 1

)+ s

( 0 1 11 0 11 1 0

),

v,w, t, s ∈ R.

Here we can choose v and w independently in order to get all of the three cases (a),(b) and (c) of the algorithm. Before we examine these cases closer, we define twoorthogonal matrices A∗ and A∗∗ related by a permutation matrix:

A∗ = 1√6

( 1 −√3

√2

1√

3√

2−2 0

√2

), A∗∗ = A∗

( 0 1 00 0 11 0 0

). (7.3)

(a) The elliptic case. Suppose that v = 0. We find b = −α−1β =v−1w(1, 1, 1)t, and that the matrix bbt − α−1γ has the simple eigenvaluev−1(s − t) + 3v−1(v−1w2 − s) and the double eigenvalue v−1(s − t). Supposenow that v and w are fixed. Depending on the sign of v and whether s is assumedto be less than, greater than or equal to v−1w2, we have three subcases: in the firsttwo cases we have separability in spheroidal coordinates, while the third case isonly an indication of a radial component in the potential thus calling for use of theCBD equation.


(a′) The oblate spheroidal subcase v−1(v−1w2 − s) < 0. The two smaller eigen-values coincide, and it proves possible to diagonalize bbt − α−1γ using A = A∗.Furthermore, N = 2, 41 = {1, 2}, 42 = {3}, and

<1 = v−1(s − t) + 3v−1(v−1w2 − s), <2 = v−1(s − t).

The transformation to the separation coordinates can thus be written(q1

q2

q3

)= A∗

(r1(x1, x2)ω1,1(x3)

r1(x1, x2)ω1,2(x3)

r2(x1, x2)

)+ v−1w

( 111

),

where x1, x2 are elliptic coordinates with parameters <1,<2 defined in terms ofthe Cartesian coordinates r1, r2, and x3 is a coordinate on S

1 defined in terms of theCartesian coordinates ω1,1, ω1,2.

The value of v−1w determines the location of the origin of the separation co-ordinates; the origin should be placed somewhere on the line % going through(0, 0, 0)t in the direction (1, 1, 1)t . Once this value is fixed, it is possible to assignthe difference <2 −<1 any positive number by choosing s appropriately. Then, wecan still adjust t in order to give <1 any value. We conclude that neither the ellipticparameters nor the placement of the origin are of any significance, implying thatthe potential is separable in infinitely many different oblate spheroidal coordinatesystems.

(a′′) The prolate spheroidal subcase v−1(v−1w2 − s) < 0. This case is close tothe previous case (a′). Here, the two larger eigenvalues coincide, and we diagonal-ize bbt − α−1γ by using A = A∗∗. Furthermore, N = 2, 41 = {1}, 42 = {2, 3},and the parameters <1 and <2 have exchanged roles. The transformation to theseparation coordinates can be written(

q1

q2

q3

)= A∗∗

(r1(x1, x2)

r2(x1, x2)ω2,2(x3)

r2(x1, x2) ω2,3(x3)

)+ v−1w

( 111

),

where x1, x2 are elliptic coordinates with arbitrary parameters defined in terms ofthe Cartesian coordinates r1, r2, and x3 is a coordinate on S

1 defined in terms ofthe Cartesian coordinates ω2,2, ω2,3. The remark above applies in this case as well:the potential is separable in infinitely many different prolate spheroidal coordinatesystems too.

(a′′′) The radial subcase v−1(v−1w2 − s) = 0. Since all eigenvalues coincide inthis case, we can let A be the identity and set q = q ′ + b, where q ′ = rω and ω

is to be determined. For this purpose, we use the CBD equation applied to V (q ′),which yields the two-parametric solution

γ = t

( 1 0 00 1 00 0 1

)+ s

( 0 1 11 0 11 1 0

), t, s ∈ R.


The matrix γ has two coinciding eigenvalues; t + 2s is simple and t − s is double.This proves immediately that the potential is separable in spherical coordinates.It does not matter here which eigenvalues are smaller or larger, so let us supposethat the simple eigenvalue is the larger. Using A = A∗ it is possible to diagonalizeγ . We also have N = 2, 41 = {1, 2}, 42 = {3}, but the particular values of theeigenvalues are of no significance since the rotational symmetry corresponding to41 is one-dimensional. The transformation to the separation coordinates can bewritten(

q1

q2

q3

)= rA∗

(r1(x1)ω1,1(x2)

r1(x1)ω1,2(x2)

r2(x1)

)+ v−1w

( 111

),

where r is the radial coordinate, x1, x2 are coordinates on two different copies ofS

1 defined in terms of the Cartesian coordinates r1, r2 and ω1,1, ω1,2 respectively;e.g., it is possible to take

r1(x1) = cos x1, r2(x1) = sin x1,

ω1,1(x2) = cos x2, ω1,2(x2) = sin x2.

The location of the origin along the line % of the oblate spheroidal subcase (a′) isinsignificant, giving infinitely many different spherical coordinate systems.

(b) The parabolic case. Suppose that v = 0 and w = 0. We have to takew = ±1/

√3 in order to normalize β. Next, we see that R = A∗ will do, giving

RtγR = diag(t − s, t − s, t + 2s). Since this matrix is diagonal, it is possibleto take P to be the identity; thus A = A∗. We also have λ1 = λ2 = t − s. Theequality AtγA− diag(t − s, t − s, 0) = diag(0, 0, t + 2s) gives c = (0, 0, 1

2 t + s)t.Hence b = −Ac = −( 1

2 t + s)/√

3 (1, 1, 1)t . Furthermore, N = 2, 41 = {1, 2},42 = {3} and <1 = t − s. The transformation to the separation coordinates cannow be written(

q1

q2

q3

)= A∗

(r1(x1, x2)ω1,1(x3)

r1(x1, x2)ω1,2(x3)

r2(x1, x2)

)−

12 t + s√

3

( 111

),

where x1, x2 are parabolic coordinates with parameter <1 defined in terms of theCartesian coordinates r1, r2, and x3 is a coordinate on S

1 defined in terms of theCartesian coordinates ω1,1, ω1,2.

As in the oblate spheroidal subcase (a′), it is possible to choose any location ofthe origin along the line % by specifying the value of 1

2 t + s. It is also possible tochoose the value of <1 = t − s freely, since these two different choices depend ontwo independent parameters. Thus we have separability in infinitely many differentrotational parabolic coordinates systems.

(c) The Cartesian case. Suppose that v = 0 and w = 0. The matrix γ has twocoinciding eigenvalues: t + 2s is simple and t − s is of multiplicity two, and wesuppose the former eigenvalue is the larger. We can diagonalize γ using A = A∗.


We have N = 2, 41 = {1, 2}, 42 = {3}, which means that the three-dimensionalspace E

3 decomposes into the orthogonal sum E2 ⊕ E

1. The transformation q =Aq ′ yields the coordinate systems {q ′

1, q′2} and {q ′

3} respectively on these subspaces.The potential takes the corresponding form V = V(1) + V(2), where

V(1) = 9((q ′1)

2 + (q ′2)

2)2

2(q ′2)

2(3(q ′1)

2 − (q ′2)

2)2(7.4)

and V(2) = 0. We have, in fact, recovered the centre-of-mass and relative coordi-nates known from dynamics and used for the separation of V by Calogero: that Vis independent of q ′

3 means that the mass centre 13(q1 + q2 + q3) could be separated

off.We now have to apply the algorithm to V restricted to E

2 = {q ′1, q

′2} and E

1 ={q ′

3}, but the latter is trivial. From the single BD equation applied to V(1), we have

α = v, β = 0, γ = t (δij ), t, v ∈ R,

which shows that V(1) is separable in polar coordinates. The complete transforma-tion to the separation coordinates can thus be written(

q1

q2

q3

)= A∗

(rω1(x)

rω2(x)

q ′3

),

where r is the radial coordinate in the {q ′1, q

′2}-plane E

2, and x is a correspondingcoordinate on S

1 defined in terms of the Cartesian coordinates ω1, ω2. The coordi-nate q ′

3 in E1 is arbitrary, and in particular we can let q ′

3 = q ′′3 +s for some constant

s in order to get(q1

q2

q3

)= A∗

(rω1(x)

rω2(x)

q ′′3 + s

)= A∗

(rω1(x)

rω2(x)

q ′′3

)+ s√

3

( 111

).

This shows that it is also in this case possible to translate the origin along the line% of the oblate spheroidal subcase (a′).

The assumption initially made on the relative sizes of the eigenvalues is norestriction, since exchanging their roles amounts to a renumbering of the coordi-nates (we exclude here the trivial case corresponding to nonseparability that occursif all eigenvalues are assumed to coincide). Indeed, if we assume that the simpleeigenvalue is the smaller, we would have to use A = A∗∗ instead of A∗ in order todiagonalize γ . Accordingly, we would get E

1 = {q ′1} and E

2 = {q ′2, q

′3} since the

permutation matrix occurring in the definition of A∗∗ [Equation (7.3)] multipliedby (q ′

1, q′2, q

′3)

t yields (q ′2, q

′3, q

′1)

t.


7.3. THE TODA LATTICE

The three-particle periodic Toda [31] lattice potential is

V = exp(q1 − q2) + exp(q2 − q3) + exp(q3 − q1). (7.5)

It is well known that the Hamiltonian system governed by this potential is inte-grable in the Liouville sense; the potential is, however, not separable.

To prove this, we observe that if V is inserted into the BD equations, then weget

α = 0, β = 0, γ = t

( 1 0 00 1 00 0 1

)+ s

( 0 1 11 0 11 1 0

), t, s ∈ R,

which is the situation of case (c) of the Calogero system example (Section 7.2).Referring to the analysis carried out there, we conclude that it is possible to assignsuitable numerical values to t and s without loss of generality (we are only inter-ested in which eigenvalues that coincide, not their particular values, and since s

simply is a scale factor, it is possible to assign any nonzero value to it). E.g., t = 0and s = 1 leads to γ = A∗diag(−1,−1, 2)At∗, where A∗ is defined by (7.3).

Upon setting q = A∗q ′, the potential takes the form V = V(1) + V(2), where

V(1) = exp(−√

2 q ′2

)+ exp 12

(√6 q ′

1 + √2 q ′

2

)++ exp 1

2

(−√6 q ′

1 + √2 q ′

2

), (7.6)

and V(2) = 0. The BD equation applied to V(1) yields the trivial solution

α = 0, β = 0, γ = t (δij ), t ∈ R;hence the potential is not separable.

Note, however, that the potential is partially separable since it is possible toseparate off the coordinate q ′

3.

7.4. THE CAHO AND PAUL TRAP SYSTEMS

The integrability and separability properties of the CAHO (Coulomb anisotropicharmonic oscillator) system [18] and the Paul (ion) trap system [12, 3, 4] havebeen studied in these references. Both systems come from a natural Hamiltonian,and their potentials are special cases of

V = r−1 + η1q21 + η2q

22 + η3q

23 + η4q3, r = (q2

1 + q22 + q2

3 )1/2, (7.7)

corresponding to particular values of the constants η1, η2, η3, η4. Here we shall usethe criterion of separability to determine those values of η1, η2, η3, η4 for which V

is separable.


When we insert V into the BD equations, we get, as usual, a system of linearhomogeneous equations by requiring the equations to be satisfied identically. Thedifference is that we here consider this to be a system determining admissiblevalues of the η’s parametrized by α, β, γ .

All solutions that correspond to separability are

(a) η1 = η2 = η3 and η4 = 0,(b) η1 = η2 = η3 and η4 arbitrary,(c) η1 = η2 = 1

4η3 and η4 arbitrary;

we investigate them closer below.(a) The radial case. We have V = r−1 + η1r

2, which clearly is radial andseparable in any polar coordinate system, among them both the spherical and theconical coordinate system.

(b) The elliptic case. We have V = r−1 + η1r2 + η4q3, which requires

α = 2η1η−14 v, β = (0, 0, v)t, γ = t (δij ), v, t ∈ R.

We find b = (0, 0,−η′)t and bbt − α−1γ = diag[−η′v−1t,−η′v−1t,−η′v−1t +(η′)2], where η′ = 1

2η−11 η4, which proves separability in the prolate spheroidal

coordinate system defined by(q1

q2

q3

)=(r1(x1, x2)ω1,1(x3)

r1(x1, x2)ω1,2(x3)

r2(x1, x2)

)− η′

( 001

)

with parameters <1 = 1 and <2 = 1 + (η′)2 [using t = −(η′)−1v].(c) The parabolic case. We have V = r−1 + η1(q

21 + q2

2 + 4q23 ) + η4q3; note

that this case also encompasses the separable potentials r−1 + η1(4q21 + q2

2 + q23 )

and r−1 + η1(q21 + 4q2

2 + q23 ) by a renumbering of the coordinates. The potential V

requires

α = 0, β = (0, 0, w)t, γ = t (δij ), w, t ∈ R,

which gives λ1 = λ2 = t, A = (δij ) and b = (0, 0,− 12 t)

t, proving separability inrotational parabolic coordinates defined by(

q1

q2

q3

)=(r1(x1, x2)ω1,1(x3)

r1(x1, x2)ω1,2(x3)

r2(x1, x2)

)− 1

2t

( 001

)

with parameter <1 = t . Observe the special feature here that the location of theorigin depends on the parabolic parameter <1 (or, vice versa).

7.5. THREE-DIMENSIONAL EUCLIDEAN SPACE

We classify here all possible results of an application of the criterion of separabilityto an arbitrary potential in three-dimensional Euclidean space E

3.


Table I. Classification of the results of the criterion of separability in three dimensions.

α β γ E3 =

= 0 λ1 < λ2 < λ3 Elliptic

λ1 = λ2 < λ3 Prolate

spheroidal

λ1 < λ2 = λ3 Oblate

spheroidal

λ1 = λ2 = λ3 E1+ × S

2 γ ◦λ1 < λ2 < λ3 Conical

λ1 < λ2 = λ3

λ1 = λ2 < λ3 Spherical

λ1 = λ2 = λ3 Partially

separable

= 0 = 0 λ1 < λ2, 0 Parabolic

λ1 = λ2, 0 Rotational

parabolic

= 0 λ1 < λ2 < λ3 E1 ⊕ E

1 ⊕ E1 Cartesian

λ1 = λ2 < λ3

λ1 < λ2 = λ3 E1 ⊕ E

2 α β γ

= 0 λ1 < λ2 Elliptic

cylindrical

λ1 = λ2 Cylindrical

= 0 = 0 Parabolic

cylindrical

= 0 λ1 = λ2 Partially

separable

λ1 = λ2 = λ3 Not

separable

The classification is illustrated in Table I, and reconstructs the list of all three-dimensional separable potentials obtained by Eisenhart [16]. The columns α, β andγ indicate the outcome of the first application of the BD equations. Under γ , wesee the parameters λ1, λ2, λ3 that are determined in Step 3 in the algorithm.

In the elliptic case (α = 0) we have first the generic case with distinct parame-ters, corresponding to the elliptic coordinate system. If two parameters coincide,we have the prolate and the oblate spheroidal coordinate systems respectively. Fi-nally, if all parameters coincide, then we have polar coordinates, and it is necessaryto use the CBD equation to test separability of the ‘spherical part’ of the potential.This results in a new set of parameters shown under γ ◦. The generic situation


corresponds to the conical coordinates, while if two parameters coincide, we havethe spherical coordinate system. If all parameters coincide, then it is only possibleto separate off the radial coordinate.

In the parabolic case (α = 0 and β = 0) there are only two possibilities.The first is the generic case of parabolic coordinates, the second, with coincidingparameters, is the rotational parabolic coordinates.

In the Cartesian case (α = 0 and β = 0), the generic case is that of Cartesiancoordinates. If two parameters coincide, one Cartesian coordinate can be separatedoff, and another application of a single BD equation to the ‘remaining’ part ofthe potential is necessary. This gives new parameters α, β and γ , and an analysisof these demonstrates the recursive structure of the criterion of separability. As aresult we obtain three different cylindrical coordinate systems and the case whenthe remaining part is not separable. Finally, if all parameters coincide, then thepotential is not separable at all.

Appendix: On Bertrand–Darboux Equations

In the paper by Marshall and Rauch-Wojciechowski [26] an excessive, linearlydependent set of 1

2n(n − 1)2 Bertrand–Darboux equations were derived from therequirement that a natural Hamiltonian should admit n quadratic first integrals.Here we show that these equations are equivalent to the equations in Theorems 3.4and 3.8.

In the case of elliptic separable potentials, the requirement for the existence ofthe functions Ui(q) in (3.4) leads to

CλijkV = 0, i, j, k =, (A.1a)

∂ijV −∑k =i

qkBik∂jV − qiBij ∂iV − 3Bij V = 0, i = j, (A.1b)

where Bij = (λi − λj )−1Jij .

PROPOSITION A.1. The systems (A.1) and (3.5) are equivalent.

Proof. By using (A.1a) written in the form Bij ∂kV = Bik∂jV it is easy to showthat (A.1b) is equivalent to (3.5). Thus (A.1) implies (3.5).

For the converse, multiply (3.5) by qk and sum over all cyclic permutations ofthe indices i, j, k. The result is (A.1a) since qkJij (2+R)+c.p. = 0, which followsfrom (1.12b). Hence (3.5) implies (A.1a), and (A.1b) follows from the above. ✷

In the case of conical separable potentials, the requirement for the existence ofthe functions Ui(q) in (3.10) leads to

Jij (2 + R)V = 0, i = j, (A.2a)

CλijkV = 0, i, j, k = . (A.2b)


PROPOSITION A.2. The systems (A.2a) and (3.11a) are equivalent, and so arethe systems (A.2b) and (3.11b).

Proof. This follows from the identities (1.12b) and (1.12e), which imply thateach operator Jij can be expressed in terms of J1i through q1Jij = qiJ1j − qjJ1i ,and that each operator Cijk = Cλ

ijk can be expressed in terms of C1ij throughq1Cijk = qiC1jk + qjC1ki + qkC1ij . ✷Acknowledgements

We would like to thank our colleagues H. Lundmark and K. Marciniak for activeparticipation in our seminars, which considerably improved our understanding ofthe separability theory.

We appreciate valuable discussions with S. Benenti as well as his critical read-ing of an early version of this work. His substantial contributions to the subjecthave been of much use, and are also highly appreciated.

S. R.-W. would also like to thank F. Calogero, F. Magri, I. Marshall and P. Win-ternitz with whom he over several years had opportunities to discuss differentpieces of this work. His research has been supported by NFR grant M5105–20005093/2000, which is hereby gratefully acknowledged.

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30. Stäckel, P.: Ueber die Integration der Hamilton–Jacobi’schen Differentialgleichung mittelstSeparation der Variabeln, Habilitationsschrift, Halle, 1891.

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Linköping university, 2000.33. Whittaker, E. T.: Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge

University Press, 4th edn, 1988.34. Wojciechowski, S.: Separability of an integrable case of the Hénon–Heiles system, Phys. Lett. A

100 (1984), 277–278.35. Wojciechowski, S.: Review of the recent results on integrability of natural Hamiltonian systems,

In: Systèmes Dynamiques Non Linéaires: Intégrabilité et Comportement Qualitatif, PressesUniv. Montréal, Montreal, PQ, 1986, pp. 294–327.


349

Singular Perturbations of Self-Adjoint Operators

VLADIMIR DERKACH1, SEPPO HASSI2 and HENK DE SNOO3

1Department of Mathematics, Donetsk National University, Universitetskaya str. 24,83055 Donetsk, Ukraine. e-mail: [email protected] of Mathematics and Statistics, University of Vaasa, PO Box 700, 65101 Vaasa,Finland. e-mail: [email protected] of Mathematics, University of Groningen, Postbus 800, 9700 AV Groningen,The Netherlands. e-mail: [email protected]

(Received: 2 July 2002)

Abstract. Singular finite rank perturbations of an unbounded self-adjoint operator A0 in a Hilbertspace H0 are defined formally as A(α) = A0 +GαG∗, where G is an injective linear mapping fromH = Cd to the scale space H−k(A0), k ∈ N, of generalized elements associated with the self-adjoint operator A0, and where α is a self-adjoint operator in H . The cases k = 1 and k = 2 havebeen studied extensively in the literature with applications to problems involving point interactionsor zero range potentials. The scalar case with k = 2n > 1 has been considered recently by variousauthors from a mathematical point of view. In this paper, singular finite rank perturbations A(α) in thegeneral setting ranG ⊂ H−k(A0), k ∈ N, are studied by means of a recent operator model inducedby a class of matrix polynomials. As an application, singular perturbations of the Dirac operator areconsidered.

Mathematics Subject Classifications (2000): Primary: 47A55, 47B25, 47B50; secondary: 34L40,81Q10, 81Q15.

Key words: singular finite rank perturbations, extension theory, Kreın’s formula, boundary triplet,Weyl function, generalized Nevanlinna function, operator model.

1. Introduction

Let A0 be an unbounded self-adjoint operator in a Hilbert space H0 and let H+2(A0)

⊂ H0 ⊂ H−2(A0) be the triplet of Hilbert spaces, where H+2(A0) is domA0

equipped with the graph inner product and where H−2(A0) is the correspondingdual space of generalized elements, cf. [6]. Let G be an injective linear mappingfrom H = C

d to H−2(A0). For each self-adjoint operator α in H there is a(singular) finite rank perturbation A(α) of A0 formally given by

A(α) = A0 +GαG∗. (1.1)

Such perturbations can be found in many areas, especially in the theory of pointinteractions or zero range potentials, see [1, 3, 40]. In order to give a meaning toA(α) in (1.1) introduce a restriction S0 of A0 via

dom S0 = domA0 ∩ kerG∗. (1.2)

350 VLADIMIR DERKACH ET AL.

Then S0 is a closed symmetric operator with defect numbers (d, d). A natural in-terpretation for the perturbation A(α) in (1.1) is now as the self-adjoint extension ofS0 corresponding to the self-adjoint operator α in H via Kreın’s formula [28, 42].If the operator A0 is semibounded and ranG ⊂ H−1(A0), the (singular) pertur-bation (1.1) is said to be form-bounded and the operator A(α) can be constructeddirectly via the first representation theorem [32, 36, 42]. For an extension of thisapproach to the case of a nonsemibounded operator A0, cf. [2, 20, 24, 26]. Moregeneral singular finite rank perturbations of A0, where ranG belongs to the scalespace H−k(A0), k > 2, of generalized elements have received a lot of attention re-cently. For an extensive list of references, see [3]. Here H−k(A0), k ∈ N, is the dualspace corresponding to the space H+k(A0) = dom |A0|k/2 equipped with the graphnorm. Singular perturbations with k > 2 cannot be treated in terms of the extensiontheory of the operator S0 in the original space H0, since now the restriction of A0

to domA0 ∩ kerG∗ is in general essentially self-adjoint. However, there exists aninterpretation for the singular perturbations A(α) in (1.1), in the general settingwhere k > 2 and d � 1, as exit space extensions of an appropriate restrictionof A0. These extensions act in a space which is a finite-dimensional extension ofH0. They are nonself-adjoint with respect to the underlying Hilbert space innerproduct, but become self-adjoint when a suitable Pontryagin space scalar productis introduced.

Singular rank one perturbations (d = 1) in the case k = 2n + 2, n � 1, havebeen recently studied in [17, 18, 41]. The approach in these papers is based ona construction involving the Hilbert space H0, a sequence of vectors in the scalespaces H−2k(A0), k = 0, 1, . . . , n + 1, and some auxiliary set of parameters in C.After certain restrictions on these parameters, a Pontryagin space �n is constructedand the operator A0 is lifted (in the notation of the present paper) to a self-adjointrelation H0 in �n. Then a one-dimensional restriction S of H0 in �n is introduced.These constructions are related to the model for generalized Nevanlinna func-tions in [31]. The Q-function M of the pair (S,H0) is a generalized Nevanlinnafunction, cf. [35], which characterizes this pair up to unitary equivalence; it has arepresentation of the form

M = r + q�M0q, (1.3)

cf. [18, Proposition 3.1]. Here q(λ) = (λ − i)n, q�(λ) = q(λ)∗, r is a polynomialwith real coefficients of degree at most 2n−1, and M0 is the Q-function of A0 anda one-dimensional (densely defined) restriction S0 of A0, so that M0 is an ordinaryNevanlinna function.

In the present paper, singular finite rank perturbations of the form (1.1) are con-sidered with G an injective linear mapping from H = C

d to the space H−2n−2(A0)

or H−2n−1(A0) with n � 1. These perturbations are interpreted by means of ageneral operator model which was given for a class of matrix polynomials in [13],see also [8]. The construction is as follows. Select an nth order monic d×d matrixpolynomial q, and define G0 = q(A0)

−1G. Then G0 maps H = Cd into H−2(A0)

SINGULAR PERTURBATIONS OF SELF-ADJOINT OPERATORS 351

or H−1(A0), respectively. Introduce the restriction S0 of A0 to domA0 ∩ kerG∗0,

so that S0 is a closed symmetric operator in H0 with defect numbers (d, d). Thepolynomial q, together with a self-adjoint d × d matrix polynomial r of degree atmost 2n − 1, determine a matrix polynomial Q of the form

Q =(

0 q

q� r

). (1.4)

The function Q gives rise to a model involving a reproducing kernel Pontryaginspace HQ and a corresponding multiplication operator SQ in it, cf. [13]. Via G0

the polynomial q determines the operator S0 in H0 and the coefficients of thepolynomials q and r serve as parameters for the model space HQ. The orthogonalcoupling of the symmetric operator S0 in the Hilbert space H0 and the symmetricoperator SQ in the Pontryagin space HQ leads to a symmetric extension S of S0⊕SQand its self-adjoint extension H0 in the Pontryagin space H0 ⊕ HQ, such that thecorresponding Weyl function M is given by (1.3), cf. [13]. The symmetric operatorS associated to M is maximally nondensely defined in the sense that the dimensionof the multivalued part of S∗ is maximal, and the extension H0 is the generalizedFriedrichs extension of S in the sense of [10]. The self-adjoint parameters τ in Hgenerate self-adjoint extensions Hτ of S in H0⊕HQ via Kreın’s formula relative toS and H0. The pair (S,H0) in H0⊕HQ is the lifting of (S0, A0) in H0. The singularperturbations A(α) of A0 in (1.1) are now ‘identified’ with those extensions Hτ ofS for which the parameter τ is a self-adjoint operator in H . The motivation forthis identification is obtained from a perturbation result for the extended resolventacting in the rigging of H0 generated by A0 (see Theorem 4.8). Now the singularperturbations A(α) can be seen as exit space extensions of S0, whose compressedresolvents are characterized by the exit space version of Kreın’s formula. This givesthe connection between the singular finite rank perturbations A(α) and the self-adjoint extensions Hτ as perturbations in H0 ⊕ HQ that were studied in [10–12].Since the extensions Hτ are described by means of abstract boundary conditionsfor the adjoint S∗ in H0 ⊕ HQ as well as via interface conditions for the adjoint S∗0in H0, the results in this paper are directly applicable for studying singular finiterank perturbations of differential operators.

In the case of rank one perturbations (d = 1) with q(λ) = (λ− i)n, the modelin this paper with k = 2n + 2 is unitarily equivalent to the model in [18] sincethe Weyl function M coincides with the Q-function in [18]. For n = 1 a similardescription for the model operator S, based on abstract boundary conditions, wasgiven in [12, Theorem 3.1]. Therefore the results in [12] can be used to analysesingular rank one perturbations with ranG ⊂ H−3 or ranG ⊂ H−4; see also [38]for a different approach.

The paper is organized as follows. Some preliminary results are given in Sec-tion 2. They include necessary facts concerning boundary triplets, Weyl functions,and generalized resolvents of symmetric operators. In addition, the model con-cerning a class of matrix polynomials from [13] is briefly recalled. In Section 3


the factorization model from [13] is presented and the self-adjoint extensions Hτ

of the model operator S are defined via abstract boundary conditions. The com-pressed resolvents PH0(Hτ − λ)−1�H0, and the corresponding Štraus extensionsin H0 are described in terms of ‘interface conditions’ which in general are λ-depending. Singular finite rank perturbations (1.1) of a self-adjoint operator A0 areconsidered in Section 4. In the case where ranG ⊂ H−1(A0) or ranG ⊂ H−2(A0)

the boundary triplets for S∗0 are expressed in terms of G and A0. The general caseranG ⊂ H−2n−j (A0), j = 1, 2, is reduced to the previous two by replacing G byG0 = q(A0)

−1G. In Section 5 certain two-dimensional perturbations A0 +GαG∗of the Dirac operator A0 = D with

D = −ic d

dx⊗ σ1 + (c2/2)⊗ σ3, σ1 =

(0 11 0

), σ3 =

(1 00 −1

),

are considered. In the case where Gh = δ ⊗ h, h ∈ C2, an application of Theo-

rem 3.1 leads to a description of the perturbations A(α) in H−2(A0),

y ∈ domA(α) ⇔ y(0+) = !y(0−),where ! is a linear-fractional transformation of α given by

! = (2icσ1 − α)−1(2icσ1 + α).

This coincides with the descriptions of A(α) in [1–3, 5, 21]. The case Gh = −icδ′⊗σ1h+ (c2/2)δ⊗σ3h, h ∈ C

2, leads to perturbations in H−4(A0). Then the functiony = PH0(Hτ − λ)−1z is shown to be a solution of a boundary-value problem withthe λ-depending interface conditions of the form

y(0+) = !(λ)y(0−), !(λ) = (2icσ1 − λ2τ)−1(2icσ1 + λ2τ).

Some further applications of the model for singular perturbations will be studiedelsewhere.

2. Preliminaries

The necessary ingredients for the present paper are briefly reviewed in this sec-tion. They involve the extension theory of symmetric linear relations in Pontryaginspaces, and the construction of operator models for a class of polynomials.

2.1. BOUNDARY TRIPLETS AND ABSTRACT WEYL FUNCTIONS

Let H be a Pontryagin space with negative index κ , cf. [4]. Let S be a not neces-sarily densely defined closed symmetric relation in H with equal defect numbersd+(S) = d−(S) <∞ and let S∗ be the adjoint linear relation of S. The symmetry ofS can be expressed by S ⊂ S∗. Here and later operators will be identified with theirgraphs. In the rest of this paper [H] stands for the set of all bounded everywhere


defined linear operators in H. If T is a closed linear relation in H, i.e. T ∈ C(H),then dom T , ker T , ran T , and mul T indicate the domain, kernel, range, and multi-valued part of T , respectively. Moreover, ρ(T ) denotes the set of regular points ofthe linear relation T . Recall (see [7, 23]) that a triplet � = {H , '0, '1} of a Hilbertspace H with dim H = n±(S) and two linear mappings 'j , j = 0, 1, from S∗ toH is called a boundary triplet for S∗, if ' = ('0, '1)

�: f → ( '0f , '1f )� is asurjective linear mapping from S∗ onto H ⊕H and the abstract Green’s identity

(f ′, g)− (f, g′) = ('1f , '0g)H − ('0f , '1g)H

= i('g)∗J ('f ), J =(

0 −iIH

iIH 0

), (2.1)

holds for all f = {f, f ′}, g = {g, g′} ∈ S∗. The adjoint S∗ of any closed symmet-ric relation S with equal defect numbers has a boundary triplet � = {H , '0, '1}.Every other boundary triplet � = {H , '0, '1} is related to � via a J -unitarytransformation W : ' = W'. In particular, the transposed boundary triplet �� ={H , '�0 , '

�1 }, is defined by '� = iJ'. When S is densely defined, S∗ can be iden-

tified with its domain dom S∗, in which case the boundary mappings are interpretedas mappings from dom S∗ to H .

Let � = {H , '0, '1} be a boundary triplet for S∗. The mapping '�: f →{'1f ,−'0f } from S∗ onto H ⊕H establishes a one-to-one correspondence be-tween the set of all self-adjoint extensions of S and the set of all self-adjoint linearrelations τ in H via

Aτ := ker('0 + τ'1)

= {f ∈ S∗ : {'1f ,−'0f } ∈ τ } = {f ∈ S∗ : '�f ∈ τ }. (2.2)

When the parameter τ is an operator in H , equation (2.2) takes the form

'0f + τ'1f = 0. (2.3)

For τ = ∞, meaning that τ−1 = 0 or τ = {0, IH }, the equation in (2.2) reads as'1f = 0. More generally, there is a similar interpretation, when τ is decomposedorthogonally in terms of an operator part and a multivalued part. To each boundarytriplet � one may naturally associate two self-adjoint extensions of S by A0 =ker'0, A1 (= A∞) = ker'1, corresponding to the linear relations τ = 0 andτ = ∞ via (2.2).

Let Nλ(S∗) = ker(S∗ − λ), λ ∈ ρ(S), be the defect subspace of S and let

Nλ(S∗) := {{fλ, λfλ} : fλ ∈ Nλ(S

∗)}; here the notations Nλ and Nλ are usedwhen the context is clear. Associated with the boundary triplet � are two operatorfunctions

γ (λ) = p1('0� Nλ)−1 (∈ [H ,Nλ]), M(λ) = '1('0� Nλ)

−1 (∈ [H ]),λ ∈ ρ(A0) ( �= ∅), (2.4)


which are holomorphic on ρ(A0). Here p1 denotes the orthogonal projection ontothe first component of H ⊕H . The functions γ and M are called the γ -field andthe Weyl function of S corresponding to the boundary triplet �, cf. [7, 15, 16, 39](or the Q-function corresponding to the pair (S,A0), cf. [35]). The γ -field γ � andthe abstract Weyl function M� corresponding to the transposed boundary triplet�� are related to γ and M via

γ �(λ) = γ (λ)M(λ)−1, M(λ)� = −M(λ)−1, λ ∈ ρ(A1) ( �= ∅).If H is a Hilbert space, a Weyl function M of S is a so-called Nevanlinna func-

tion, that is, M is holomorphic in the upper halfplane C+, ImM(λ) � 0 for allλ ∈ C+, and M satisfies the symmetry condition M(λ)∗ = M(λ) for λ ∈ C+∪C−.In the case where H is a Pontryagin space of negative index κ , the Weyl functionM of S belongs to the class Nk, k � κ , of generalized Nevanlinna functions whichare meromorphic on C+ ∪ C−, satisfy M(λ)∗ = M(λ), and for which the kernel

NM(λ,µ) = M(λ)−M(µ)

λ− µ, NM(λ, λ) = d

dλM(λ), λ, µ ∈ C+, (2.5)

has k negative squares [35]. When S is simple, that is,

H = span {Nλ(S∗) : λ ∈ ρ(A0) ( �= ∅)}, (2.6)

then S is an operator without eigenvalues. Moreover, in this case the Weyl functionM belongs to the class Nκ , so that k = κ , and the domain of holomorphy ρ(M) ofM coincides with the resolvent set ρ(A0).

The resolvent of the extension Aτ and its spectrum σ (Aτ ) can be expressed interms of τ and the Weyl function M via Kreın’s formula. In the terminology ofboundary triplets the result can be formulated as follows, see [7, 15, 16].

PROPOSITION 2.1. Let S be a closed symmetric relation in the Pontryagin spaceH with equal defect numbers (d, d), d < ∞, let � = {H , '0, '1} be a boundarytriplet for S∗ with the Weyl function M, let τ be a linear relation in H connectedwith Aτ via (2.2). Then the resolvent of Aτ is given by

(Aτ − λ)−1 = (A0 − λ)−1 − γ (λ)(τ−1 +M(λ))−1γ (λ)∗,λ ∈ ρ(Aτ ) ∩ ρ(A0). (2.7)

Moreover, for every λ ∈ ρ(A0) the following equivalences hold:

(i) λ ∈ ρ(Aτ ) if and only if τ−1 +M(λ) is invertible;(ii) λ ∈ σp(Aτ ) if and only if ker(τ−1 +M(λ)) is nontrivial.

In a similar way, for a (generalized) Nevanlinna family τ (λ) the function

(A0 − λ)−1 − γ (λ)(τ(λ)+M(λ))−1γ (λ)∗,


is the compressed resolvent of an exit space extension of S in a Hilbert (or aPontryagin) space, cf. [7, 9, 15, 35, 39, 43].

2.2. A MODEL FOR A CLASS OF MATRIX POLYNOMIALS

The construction of a model for a class of matrix polynomials as given in [13] isnow briefly reviewed. Let q be a monic d × d matrix polynomial of the form

q(λ) = IHλn + qn−1λn−1 + · · · + q1λ+ q0, (2.8)

and let r be a self-adjoint d × d matrix polynomial of the form

r(λ) = r2n−1λ2n−1 + r2n−2λ

2n−2 + · · · + r1λ+ r0,

rj = r∗j , j = 0, . . . , 2n − 1. (2.9)

Observe, that the function Q in

Q(λ) =(

0 q(λ)

q�(λ) r(λ)

), (2.10)

is a 2d × 2d matrix polynomial whose leading coefficient is, in general, noninvert-ible. Let the n× n block matrices Bq and Cq be defined by

Bq =

q1 q2 . . . qn−1 IH

q2 . . . qn−1 IH 0... . .

.

. .. 0 0

qn−1 IH . ..

. .. ...

IH 0 0 . . . 0

,

Cq =

0 IH 0 . . . 0

0 0 IH. . .

......

.... . .

. . . 00 0 . . . 0 IH

−q0 −q1 . . . −qn−2 −qn−1

. (2.11)

Define the operators

B =(

0 Bq

Bq� Br

), C =

(Cq� C12

0 Cq

),

Br = (rj+k+1)n−1j,k=0, C12 = B−1

q�D, (2.12)

where

D =

rnrn+1...

r2n−1

(q0, q1, . . . , qn−1)−

IH

0...

0

(r0, r1, . . . , rn−1). (2.13)


Denote ! = (IH , λIH , . . . , λn−1IH ), and define

!1 = λn!B(r)B−1q , B(r) =

rn+1 . . . r2n−1 0... . .

. 0 0

r2n−1 . ..

. .. ...

0 0 . . . 0

. (2.14)

In terms of these notions the kernel NQ(0, λ) has the following factorization

NQ(0, λ) =(

L 0L1 L

)B

(! 0!1 !

)∗, (2.15)

where L and L1 are defined similar to ! and !1. Hence, Q is a strict generalizedmatrix Nevanlinna function with dn negative (and dn positive) squares. The rep-resentation (2.15) leads to an explicit form for the reproducing kernel Pontryaginspace H(Q) associated with Q in (2.10) and the corresponding operator S(Q) ofmultiplication by the independent variable in H(Q), cf. [13].

THEOREM 2.2. Let the matrix polynomial Q be given by (2.10) with q and r asin (2.8), (2.9). Let B and C be given by (2.12). Then:

(i) The reproducing kernel Pontryagin space H(Q) is isometrically isomorphicto the space HQ = Hn ⊕ Hn (= C

2dn) equipped with the inner product〈·, ·〉HQ

= (B ·, ·).(ii) The operator C is self-adjoint in HQ. Its restriction SQ to the subspace

dom SQ ={F =

(f

f

)∈ HQ : f1 = f1 = 0

}is a closed simple symmetric operator in HQ with defect numbers (2d, 2d),which is unitarily equivalent to S(Q).

(iii) The adjoint linear relation SQ∗ of SQ takes the form

SQ∗ =

{F =

{F,CF +B−1

(ϕ ⊗ e1

ϕ ⊗ e1

)}: F ∈ HQ, ϕ, ϕ ∈ H

}.

(iv) A boundary triplet �Q = {H ⊕H , 'Q

0 , 'Q

1 } for SQ∗ can be defined by

'Q

0 F =(f1

f1

), '

Q

1 F =(ϕ

ϕ

), F ∈ SQ

∗.

(v) The Weyl function of SQ associated with �Q coincides with Q and the corre-sponding γ -field is given by

γQ(λ)h =(!� !�10 !�

)(h1

h2

), h1, h2 ∈ H .


3. Construction of the Factorization Model

3.1. THE MODEL FOR SYMMETRIC OPERATOR

Let S0 be a closed symmetric operator in a Hilbert space H0 with defect numbers(d, d) whose Weyl function is M0. Let SQ be a symmetric operator in a Pontryaginspace HQ with the Weyl function (2.10) where q and r are d × d matrix polyno-mials, q is monic and r is self-adjoint. In [13] a Pontryagin space symmetric linearrelation S was constructed as a coupling of the operators S0 and SQ, such that thefollowing function is a Weyl function for S:

M(λ) = r(λ)+ q�(λ)M0(λ)q(λ). (3.1)

THEOREM 3.1 ([13, Theorem 4.2]). Let S0 be a closed symmetric operator in theHilbert space H0 and let �0 = {H , '0

0, '01} be a boundary triplet for S∗0 with the

Weyl function M0 and the γ -field γ0. Let SQ be the symmetric operator in HQ asdefined in Theorem 2.2 with the boundary triplet �Q = {H ⊕ H , '

Q

0 , 'Q

1 } andwith q, r, and Q as in (2.8), (2.9), and (2.10), respectively. Then

(i) The linear relation

S ={{

f0 ⊕(f

f

), f ′0 ⊕

(C

(f

f

)+B−1

('0

0 f0 ⊗ e1

0

))}∈ S∗0 ⊕ S∗Q : f1 = '0

1 f0

f1 = 0

}is closed and symmetric in H0 ⊕ HQ and has defect numbers (d, d).

(ii) The adjoint S∗ is given by

S∗ ={{

f0 ⊕(f

f

), f ′0 ⊕

(C

(f

f

)+B−1

('0

0 f0 ⊗ e1

ϕ ⊗ e1

))}∈ S∗0 ⊕ S∗Q:

f1 = '01 f0

ϕ ∈ H

}.

(iii) A boundary triplet � = {H , '0, '1} for S∗ is determined by

'0(f0 ⊕ F ) = f1, '1(f0 ⊕ F ) = ϕ, f0 ⊕ F ∈ S∗.

(iv) The corresponding Weyl function M is of the form (3.1) and the γ -field γ isgiven by

γ (λ)h = γ0(λ)q(λ)h⊕ ((!�M0(λ)q(λ)+!�1 )h � !�h),h ∈ H . (3.2)

If the operator S0 is densely defined in H0, then S is an operator. When r = 0the formulas for S and S∗ in Theorem 3.1 can be simplified and the Weyl functionis factorized as

M(λ) = q�(λ)M0(λ)q(λ). (3.3)


Theorem 3.1 was obtained earlier in [12, Section 3] in the special case that d =n = 1 and q(λ) = λ− α, α ∈ C. The problem of simplicity of the model operatorS was investigated in [12, 13].

3.2. SELF-ADJOINT EXTENSIONS OF THE MODEL OPERATOR

The model in Theorem 3.1 leads to an explicit form for the extension Hτ =ker('0 + τ'1).

PROPOSITION 3.2. Let the assumptions be as in Theorem 3.1, and let γ and M

be given by (3.2) and (3.1), respectively. Then:

(i) The self-adjoint extensions Hτ of S in H = H0 ⊕ HQ are in a one-to-onecorrespondence with the self-adjoint relations τ in H via

Hτ ={{

f0 ⊕(f

f

), f ′0 ⊕

(C

(f

f

)+B−1

('0

0 f0 ⊗ e1

ϕ ⊗ e1

))}

∈ S∗0 ⊕ S∗Q : f1 = '01 f0

f1 + τ ϕ = 0

}.

(ii) The resolvent (Hτ − λ)−1 is given by

(Hτ − λ)−1 = (H0 − λ)−1 − γ (λ)(τ−1 +M(λ))−1γ (λ)∗,λ ∈ ρ(Hτ) ∩ ρ(H0). (3.4)

(iii) For every λ ∈ ρ(H0) the following equivalences hold:

λ ∈ σp(Hτ ) ⇔ 0 ∈ σp(τ−1 +M(λ)),

λ ∈ ρ(Hτ) ⇔ 0 ∈ ρ(τ−1 +M(λ)).

Proof. By part (iii) of Theorem 3.1 the condition f0 ⊕ F ∈ ker('0 + τ'1) isequivalent to {ϕ, f1} ∈ −τ , or to f1 + τ ϕ = 0, when correctly interpreted if τ ismultivalued. The representation of Hτ now follows from (ii) of Theorem 3.1. Thisproves (i). The form of the resolvent of Hτ in (ii) is obtained from Proposition 2.1and Theorem 3.1. The statement (iii) is immediate from Proposition 2.1. ✷

Define the block matrices

Xn =

0 0 . . . 0

I 0. . .

......

. . .. . . 0

λn−2 · · · I 0

, Xn−1 = I . . . 0

.... . .

...

λn−2 · · · I

. (3.5)

The following properties of the companion matrix Cq are useful and easily checked,e.g., the last one is a simple corollary of the Frobenius formula.


LEMMA 3.3. Let Cq be the companion matrix corresponding to the polynomial qof the form (2.8). Then:

(i) (Cq − λ)!�(n)h = (0, . . . , 0,−q(λ)h)� for all λ ∈ C, h ∈ H ;(ii) σ (Cq) = σ (q) and ker(Cq − λ) = {!�(n)h : h ∈ ker q(λ)};

(iii) (Cq − λ)Xn = IHn −(

0 0qλXn−1 I

), where qλ = (q1, . . . , qn−2, qn−1 + λ);

(iv) For every λ ∈ C \ σ (q), g ∈ Hn,

(Cq − λ)−1g = Xng − 1

q(λ)!�(n)(gn + qλXn−1(g1, . . . , gn−1)

�). (3.6)

PROPOSITION 3.4. Let the assumptions be as in Theorem 3.1 and let H0 =ker'0 be as in Proposition 3.2 (with τ = 0). Then:

(i) ρ(H0) = ρ(A0);(ii) the compression of the resolvent of H0 to the subspace H0 is given by

PH0(H0 − λ)−1�H0 = (A0 − λ)−1, λ ∈ ρ(H0); (3.7)

(iii) the subspace L = {0} ⊕Hn ⊕ {0} of H = H0 ⊕ HQ is maximal neutral andinvariant under the resolvent (H0 − λ)−1. It satisfies (H0 − λ)−nL = {0},λ ∈ ρ(H0).

Proof. (i) Let G = (g0, g, g)� ∈ H and let f0 = {f0, f

′0} ∈ S∗0 . By Proposi-

tion 3.2 the relation G ∈ ran(H0 − λ) can be rewritten as a system of equalities

f ′0 − λf0 = g0,

(Cq� − λ)f + C12f + ϕ ⊗ en −B−1q�

Br('00 f0 ⊗ en) = g, (3.8)

(Cq − λ)f + '00 f0 ⊗ en = g, f1 = '0

1 f0, f1 = 0.

Since f1 = 0 the third identity in (3.8) and Lemma 3.3(iii) yield

(f2, . . . , fn)� = Xn−1(g1, . . . , gn−1)

�, (3.9)

'00 f0 = gn +

n−1∑j=1

qj fj+1 + λfn. (3.10)

Clearly, h0 = f0 − γ0(λ)'00 f0 ∈ A0. The first equality in (3.8) implies h′0 − λh0 =

f ′0 − λf0 = g0. This means that {h0, g0} ∈ A0 − λ, or equivalently, that {g0, h0} ∈(A0 − λ)−1. Now assume that λ ∈ ρ(A0). Then h0 = (A0 − λ)−1g0 and

f0 = (A0 − λ)−1g0 + γ0(λ)'00 f0, f ′0 = λf0 + g0. (3.11)

The second equality in (3.8) can be rewritten as

(Cq� − λ)f + ϕ ⊗ en = k, (3.12)


where k = g − C12f + B−1q�

Br ('00 f0 ⊗ en). Using f1 = '0

1 f0 and applyingLemma 3.3(i), (iii) to (3.12) one obtains

(f2, . . . , fn)� = Xn−1(k1, . . . , kn−1)

� + λ!�(n−1)'01 f0, (3.13)

ϕ = kn +n−1∑j=0

q∗j fj+1 + λfn. (3.14)

This shows that λ ∈ ρ(H0), and thus ρ(A0) ⊂ ρ(H0).Conversely, assume that λ ∈ ρ(H0). Then with G = (g0, 0, 0)� one obtains

from the third identity in (3.8) and Lemma 3.3(iii) that f = 0 and '00 f0 = 0. Now

the first identity in (3.8) gives ran (A0 − λ) = H0 and, therefore, λ ∈ ρ(A0). Infact, Lemma 3.3(i) yields

(H0 − λ)−1(g0, 0, 0)� = ((A0 − λ)−1g0,!�(n)'

01 f0, 0)�. (3.15)

(ii) The equality (3.7) follows immediately from (3.15).(iii) Clearly, L is a neutral subspace of H0 ⊕HQ and has dimension dn, so that

it is maximal neutral, cf. [4]. Moreover, again using Lemma 3.3(iii) one obtainsfrom (3.8) that for G = (0, g, 0)� ∈ L,

(H0 − λ)−1(0, g, 0)� = (0, Xng, 0)�,(H0 − λ)−n(0, g, 0)� = (0, Xn

ng, 0)� = 0. ✷A more complete description of the structure of root subspaces in the scalar

case can be found in [12]. The self-adjoint extensions Hτ = ker('0 + τ'1) of Sdescribed in Proposition 3.2 can be interpreted as standard range perturbations ofthe self-adjoint extension H∞ = ker'1 in the Pontryagin space H = H0⊕HQ, see[14]; cf. also [27, 29] for the Hilbert space case. These perturbations can be seen asliftings of the singular perturbations A(α) of A0 from H0 to the extended space H,cf. Corollary 3.6. Various properties of range perturbations in a Pontryagin spacesetting were considered in [10–12]. A more detailed study of this connection leadsto intermediate symmetric extensions of S and their generalized Friedrichs exten-sions which can be described by means of so-called extremal boundary conditions,cf. [14].

3.3. ŠTRAUS EXTENSIONS

Let S0 be a closed symmetric operator in H0 and let H be a self-adjoint extensionof S0 in an exit space H (⊃ H0). A family {T (λ) : λ ∈ C} of extensions of S0 in theoriginal space H0 defined by

T (λ) = {{PH0f, PH0f′} : {f, f ′} ∈ H, f ′ − λf ∈ H0

}(3.16)


is called the family of Štraus extensions of S0 corresponding to the self-adjointextension H , cf. [9, 19, 43]. Recall that S0 ⊂ T (λ) ⊂ S∗0 for all λ ∈ C. It followsfrom (3.16) that the compressed resolvent of H can be expressed by means of thefamily T (λ) as follows:

PH0(H − λ)−1�H0 = (T (λ)− λ)−1, λ ∈ ρ(H). (3.17)

In fact, the family of Štraus extensions can be characterized in terms of boundaryoperators. Let {H , '0

0, '01} be a boundary triplet for S∗0 and let the extension H of

S0 be related to a generalized Nevanlinna family τ via Kreın’s formula

PH0(H − λ)−1�H0 = (A0 − λ)−1 − γ0(λ)(τ (λ)+M0(λ))−1γ0(λ)

∗,λ ∈ ρ(H) ∩ ρ(A0). (3.18)

Then the family T (λ) of Štraus extensions is given by the equality

'0T (λ) = {{'00 f0, '

01 f0} : f0 ∈ T (λ)

} = −τ (λ), (3.19)

see [7, 16].

THEOREM 3.5. Let the assumptions be as in Theorem 3.1. Then the compressedresolvent and the Štraus family Tτ (λ) of the extension Hτ in Proposition 3.2 aregiven by

PH0(Hτ − λ)−1�H0

= (A0 − λ)−1 − γ0(λ)(τ(λ)−1 +M0(λ)

)−1γ0(λ)

∗, (3.20)

and

Tτ (λ) ={f0 = {f0, f

′0} ∈ S∗0 : ('0

0 + τ (λ)'01)f0 = 0

},

λ ∈ ρ(Hτ ) ∩ ρ(A0), (3.21)

where τ (λ) = q(λ)(τ−1 + r(λ))−1q�(λ).

Proof. The resolvent of Hτ is given by (3.4) in Proposition 3.2. In view of theidentity (3.7) and the form of the γ -field in (3.2) the compression of this formulato H0 gives

PH0(Hτ − λ)−1�H0

= PH0(H0 − λ)−1�H0 − PH0γ (λ)(τ−1 +M(λ)

)−1γ (λ)∗�H0

= (A0 − λ)−1 − γ0(λ)q(λ)(τ−1 +M(λ)

)−1q�(λ)γ0(λ)

∗.

Taking into account (3.1) this leads to (3.20) with τ = q(τ−1+ r)−1q�. The secondstatement follows now from (3.19), since

Tτ (λ) ={f0 ∈ S∗0 : {'0

0 f0, '01 f0} ∈ −τ (λ)−1 },

and this coincides with (3.21). ✷


The next result gives a connection between the self-adjoint extensions of S inH0 ⊕HQ and the self-adjoint extensions of S0 in H0. A similar result was obtainedin [29, Theorem 3.2] in a simpler situation.

COROLLARY 3.6. The self-adjoint extensions Hτ of S in H0 ⊕ HQ and the self-adjoint extensions Aτ of S0 in H0 are connected by

Aτ = ker('00 + τ'0

1) = {{PH0F,G} : {F,G} ∈ Hτ,G ∈ H0},where τ = q0(τ

−1+r0)−1q∗0 and this product is understood in the sense of relations.

Proof. When 0 ∈ ρ(Hτ)∩ ρ(A0) this result follows directly from Theorem 3.5.To prove it in the general case one can proceed as in the proof of Proposition 3.4.Consider the first three equalities in (3.8) with λ = 0 and g = g = 0. Then itfollows from the third equality in (3.8) that '0

0 f0 = q0f1 and f2 = · · · = fn = 0.Next a simple calculation using (2.12), (2.13) shows that

k = −C12f +B−1q�

Br('00 f0 ⊗ en) = r0f1 ⊗ en.

Now the second equality in (3.8), or equivalently (3.12), implies that Cq�f =(r0f1 − ϕ) ⊗ en. This gives q∗0f1 = ϕ − r0f1 and f2 = · · · = fn = 0. Hence,together with the description of Hτ in Proposition 3.2, one arrives at the followingconditions for f0:

'00 f0 = q0f1, q∗0'

01 f0 = ϕ − r0f1, {ϕ, f1} ∈ −τ.

It can be checked that these three conditions are equivalent to

{'01 f0, '

00 f0} ∈ −q0(τ

−1 + r0)−1q∗0 .

The linear relation τ = q0(τ−1+r0)

−1q∗0 (where the products and inverses are to beunderstood in the sense of relations) is self-adjoint. Therefore, Aτ is a self-adjointextension of S0 and the claim follows. ✷

Of course, when r0 = 0 and q0 = I the ‘inverse compression’ of Hτ in Corol-lary 3.6 gives the extension Aτ with precisely the same parameter τ = τ . In thissense the self-adjoint extensions Hτ of S can be seen as liftings of the self-adjointextensions of S0.

According to (3.20) the exit space for S0 is determined by the d × d matrixfunction τ−1 = q−�(τ−1 + r)q−1. This observation yields another construction ofthe model space associated with M. Namely, one may use the coupling methods aspresented in [9, 25] of the model spaces corresponding to the sum of two Nevan-linna functions M0 and τ−1. Here the degree of the rational matrix function τ−1

is equal to 2n and therefore the corresponding exit space will have the dimension2nd. However, it is not clear if the exit spaces Hτ−1 can be taken to be equal fordifferent values of τ ∈ C(H). In the present approach the situation is different. Tosee this observe that τ−1 = q−�(τ−1 + r)q−1 is obtained from Q given in (2.10)


by using a Schur complement and a transposed boundary triplet via the followingsteps,

Q→(

0 q

q� r + τ−1

)→ −q(r + τ−1)−1q� → τ−1 = q−�(r + τ−1)q−1.

This shows that for each τ the exit space determined by τ−1 can be taken to be HQ.Moreover, the model operator Sτ−1 for τ−1 is a closed symmetric extension of themodel operator SQ in Theorem 2.2 with smaller defect numbers (nd, nd) in HQ.

4. Singular Finite Rank Perturbations

Let A0 be a self-adjoint operator in the Hilbert space H0 and let G be a linearinjective mapping from H = C

d into H0. For a d × d matrix α = α∗ define theoperator A(α) by (1.1) so that A(α) is a finite rank perturbation of A0 (cf. [32]). LetS0 be the restriction of A0 defined by (1.2). Then S0 is a closed, symmetric, andnondensely defined operator with defect numbers (d, d). Its adjoint S∗0 is a closedlinear relation, given by

S∗0 = {f = {f,A0f −Gh} : f ∈ domA0, h ∈ H}. (4.1)

A boundary triplet for S∗0 can be defined by

H = Cd, '0

0 f = h, '01 f = G∗f, f ∈ S∗0 , (4.2)

where '00 is well defined, since kerG = {0}. The corresponding γ -field and the

Weyl function are given by

γ0(λ) = (A0 − λ)−1G, M0(λ) = G∗(A0 − λ)−1G, λ ∈ ρ(A0). (4.3)

The perturbations A(α) in (1.1) are now self-adjoint operator extensions of S0. TheWeyl function characterizes A0 and S0, up to unitary equivalence, cf. [35]. It alsocan be used to describe the spectrum of each perturbation A(α), cf. Proposition 2.1.Such and more general perturbations have been considered in several recent papers(see [3, 22, 27, 33, 34, 42]).

The perturbations A(α) in (1.1) with ranG ⊂ H0 are ordinary (range) per-turbations of the self-adjoint operator A0. To introduce perturbations of A0 of amore general type consider a rigging of the Hilbert space H0, generated by theoperator |A0|:

H+k ⊂ · · · ⊂ H+2 ⊂ H+1 ⊂ H0 ⊂ H−1 ⊂ H−2 ⊂ · · · ⊂ H−k, (4.4)

where H+k = dom |A0|k/2, k ∈ N, equipped with the graph inner product andH−k is the corresponding dual space, cf. [6]. Here the notation H±k for H±k(|A0|)is used for simplicity. If ranG ⊂ H−k\H0, the perturbing term GαG∗ becomesunbounded in H0, and the expression in (1.1) needs an interpretation. In the sequel,


interpretations for such (singular) perturbations will be presented for each of thefollowing cases, respectively:

ranG ⊂ H−1, ranG ⊂ H−2, ranG ⊂ H−k, k > 2.

4.1. PERTURBATIONS IN H−1

Let G be an injective linear mapping from H = Cd into H−1 and denote by G∗ its

adjoint operator from H+1 into H . The identity (1.2) gives again rise to a symmetricoperator S0 in H0. Let A0 be the [H+1,H−1]-continuation of A0 to all of H+1. Thenthe expressions for the γ -field γ0 and the Weyl function M0 in (4.3) are still welldefined, after A0 is replaced by A0. The connection of the finite rank perturbationsA(α) to the extension theory in this case can be given in terms of boundary tripletsas follows, cf. [10, Theorem 6.2] for the scalar case.

THEOREM 4.1. Let A0 be a self-adjoint operator in the Hilbert space H0 andlet A0 be its [H+1,H−1]-continuation. Let G be an injective linear mapping fromH = C

d into H−1 and define the restriction S0 of A0 by (1.2). Then:

(i) The operator S0 is closed and symmetric in H0 and has defect numbers (d, d).(ii) The adjoint linear relation S∗0 of S0 is given by

S∗0 = {f = {f, A0f −Gh} : f ∈ H+1, A0f −Gh ∈ H0, h ∈ H}. (4.5)

(iii) A boundary triplet for S∗0 can be defined by (4.2).(iv) The corresponding γ -field and Weyl function are given by

γ0(λ) = (A0 − λ)−1G, M0(λ) = G∗(A0 − λ)−1G. (4.6)

(v) The perturbation

A(α) = {{f, (A0 +GαG∗)f } : f ∈ H+1, (A0 +GαG∗)f ∈ H0} (4.7)

coincides with the self-adjoint extension Aτ = ker('00 + τ'0

1) of S0 withα = τ = τ ∗ ∈ [H ] and the resolvent of A(α) is given by (2.7).

Proof. As a restriction of A0, S0 is symmetric and its closedness follows fromthe closedness of kerG∗ in H+1 (⊃ H+2). The defect numbers are equal and theycannot be greater than (d, d), since kerG∗ has co-dimension d in H+1. The con-tinuation A0 is a self-adjoint operator from H+1 into H−1 and, in particular, in thesense of the duality between these spaces, the equality (A0f, g) = (f, A0g) holdsfor all f, g ∈ H+1, cf. [24]. The resolvent Rλ = (A0 − λ)−1 of A0 is a [H−1,H+1]-continuous operator for λ ∈ ρ(A0). Therefore, it follows from the definition (1.2)that for all f ∈ dom S0 and all λ ∈ ρ(A0):

((A0 − λ)−1Gh, (S0 − λ)f )H0 = (Gh, f )H0 = (h,G∗f )H = 0. (4.8)


Hence, Rλ(ranG) ⊂ Nλ(S∗0 ) and a dimension argument shows that

Rλ(ranG) = Nλ(S∗0 ), λ ∈ ρ(A0). (4.9)

In particular, the defect numbers of S0 are (d, d), and hence (i) has been proved.To see (ii), recall the decomposition

S∗0 = A0 + Nλ(S∗0 ), λ ∈ ρ(A0). (4.10)

It follows from (4.9) and (4.10) that every {f, f ′} ∈ S∗0 admits the representation

{f, f ′} = {f0 + RiGh,A0f0 + iRiGh} = {f, A0f −Gh},where f0 ∈ domA0, h ∈ H and, hence,

f = f0 + RiGh ∈ H+1, A0f −Gh ∈ H0.

This gives (4.5).As to (iii), it is clear from (4.5) that the mapping '0: S∗0 → H ⊕H determined

by (4.2) is surjective. With the vectors

{f, f ′} = {f, A0f −Gh} ∈ S∗0

and

{g, g′} = {g, A0g −Gk} ∈ S∗0

one obtains

(f ′, g)− (f, g′) = (A0f −Gh, g)− (f, A0g −Gk)

= (G∗f, k)H − (h,G∗g)H ,

so that the abstract Green’s identity holds.Each vector fλ ∈ Nλ(S

∗0 ) admits the representation

{fλ, λfλ} = {RλGh, λRλGh} = {RλGh, A0RλGh −Gh}.This implies

'00 fλ = h, '0

1 fλ = G∗(A0 − λ)−1G,

which gives (iv) in view of (2.4).Finally to prove (v), observe that with f ∈ S∗0 ,

'00 f + τ'0

1 f = h+ τG∗f.

Thus, f ∈ ker('00 + τ'0

1) precisely when h = −τG∗f . Substituting this into (4.5)gives the representation (4.7) for the extension ker('0

0 + τ'01) with α = τ . ✷


If ranG ⊂ H0, then the statements in Theorem 4.1 clearly reduce to the factspresented in the introduction of the present section. When ranG ⊂ H−1, theoperator in the right-hand side of (4.7) will be written shortly as

A(α) = A0 +GαG∗, α ∈ [H ].Observe, that if ranG ⊂ H−1\H0, then the operator S0 in Theorem 4.1 is denselydefined and its adjoint S∗0 in (4.5) is an operator. In the case where A0 � 0, theoperator A(α) is a form-bounded perturbation of A0 in the sense of [2]. When theoperator A0 is not semibounded, but ranG ⊂ H−1\H0, the lifting of the extensionsAτ = ker('0

0 + τ'01) to the space triplet H+1 ⊂ H0 ⊂ H−1 gives rise to a situation

where the lifted extensions Aτ behave like usual finite rank perturbations of A0

in H0 and they give rise to a generalized Friedrichs extension of S0 in the originalspace H0. Such results, involving so-called Kac subclasses of Nevanlinna functions,have been obtained in [24, 26], and then extended in [10] to Pontryagin spaces.The next results shows that perturbations in H−1 as described in Theorem 4.1 areadditive with respect to the parameter α ∈ [H ].PROPOSITION 4.2. For each self-adjoint τ ∈ [H ] the space triplets H+1(Aτ ) ⊂H0 ⊂ H−1(Aτ ) are (topologically) independent of τ and Aτ is a representation ofthe additive group [H ], i.e.

Aτ1+τ2 = (Aτ1)τ2, τj = τ ∗j ∈ [H ], j = 1, 2. (4.11)

Proof. The equality of the domains dom |Aτ |1/2 for the extensions Aτ

= ker('00 + τ'0

1) in (4.7) corresponding to the self-adjoint (operator) parametersτ ∈ [H ] can be proved along the lines of [24, 29]. It follows then from theclosed graph theorem that the norms on the spaces H±1(Aτ ) are equivalent, andtherefore the space triplets H+1(Aτ ) ⊂ H0 ⊂ H−1(Aτ ) for τ ∈ [H ] coincide, up toequivalent inner products.

In view of Theorem 4.1 the extension Aτ1 ⊂ S∗0 , τ1 ∈ [H ], is given by

Aτ1f = A0f +Gτ1G∗f, f ∈ H+1, (4.12)

with f ∈ domAτ1 if and only if A0f +Gτ1G∗f ∈ H0. Now, applying (4.12) again

with Aτ1 and τ2 ∈ [H ] yields

(Aτ1)τ2 = Aτ1 +Gτ2G∗ = A0 +G(τ1 + τ2)G

∗ = Aτ1+τ2 ,

and clearly f ∈ dom(Aτ1)τ2 if and only if f ∈ domAτ1+τ2 . This proves (4.11). ✷


Let G be an injective linear mapping from H = Cd into H−2 and let G∗ be its

adjoint operator from H+2 into H . The identity (1.2) still gives rise to a symmetric


operator S0 in H0. However, when ranG ⊂ H−2 the operator '01 in (4.2) is not well

defined anymore, and it has to be regularized. Let A0 be the [H0,H−2]-continuationof A0 to all of H0. The resolvent Rλ = (A0−λ)−1 of A0 is an [H−2,H0]-continuousoperator for λ ∈ ρ(A0), see [44]. Then the expression for the γ -field γ0 in (4.3) iswell defined, after A0 is replaced by A0, but a regularization of the Weyl functionM0 is needed.

An operator R ∈ [H−2,H0] is said to be a regularizing operator of Rλ if Rλ −R ∈ [H−2,H+2], and (Rλ −R)∗ = Rλ −R for λ ∈ ρ(A0). For example, one cantake R = 1

2 (Ri + R−i ) as a regularizing operator of Rλ, cf. [44]. If R1 and R2 aretwo regularizing operators of Rλ, then clearly R2 −R1 ∈ [H−2,H+2].THEOREM 4.3. Let A0 be a self-adjoint operator in the Hilbert space H0, letA0 be the [H0,H−2]-continuation of A0, and let R be a regularizing operator ofRλ = (A0− λ)−1. Let G be a linear injective mapping from H = C

d into H−2 anddefine the restriction S0 of A0 by (1.2). Then:

(i) The operator S0 is closed and symmetric in H0 and has defect numbers (d, d).(ii) The adjoint linear relation S∗0 of S0 is given by

S∗0 = {f = {f, A0f −Gh} : f ∈ H0, A0f −Gh ∈ H0, h ∈ H}. (4.13)

(iii) A boundary triplet for S∗0 can be defined by

H = Cd, '0

0 f = h,

'01 f = G∗(f −RGh

)+ Bh, f ∈ S∗0 , (4.14)

where B is a self-adjoint operator in H .(iv) The corresponding γ -field and the Weyl function are given by

γ0(λ) = (A0 − λ)−1G,

M0(λ) = G∗((A0 − λ)−1 −R)G+ B. (4.15)

(v) The resolvent of the extension Aτ = ker('0+ τ'1), τ = τ ∗ ∈ C(H), is givenby

(Aτ − λ)−1

= (A0 − λ)−1 − γ0(λ)(τ−1 +G∗((A0 − λ)−1 −R)G+ B)−1γ0(λ)

∗,λ ∈ ρ(Aτ ) ∩ ρ(A0).

Proof. (i) Observe, that kerG∗ is closed in H+2. This implies that S0 is a closedsymmetric operator in H0 with equal defect numbers which cannot be greater than(d, d), since kerG∗ has co-dimension d in H+2. The continuation A0 admits thefollowing symmetry property

(f, A0g) = (A0f, g)H0, f ∈ domA0, g ∈ H0,

where (·, ·) stands for the duality between H+2 and H−2. Hence, the equality (4.9)follows from (4.8). In particular, the defect numbers of S0 are (d, d).


(ii) It follows from (4.9) and (4.10) that every {f, f ′} ∈ S∗0 admits the represen-tation

{f, f ′} = {f0 + RλGh,A0f0 + λRλGh}, f0 ∈ domA0, h ∈ H ,

λ ∈ ρ(A0), (4.16)

which, due to the relation RλGh−RGh ∈ domA0, can be rewritten as

{f, f ′} = {f ′0 +RGh, A0(f′0 +RGh)−Gh}, f ′0 ∈ domA0.

Hence, S∗0 belongs to the left side of (4.13). Conversely, if A0f − Gh ∈ H0 forsome f ∈ H0 and h ∈ H , then equivalently f − RλGh ∈ domA0. In this case,f = f0 + RλGh for some f0 ∈ domA0 and f ′ = A0f −Gh = A0f0 + λRλGh,so that {f, f ′} ∈ S∗0 by (4.16).

(iii) It is clear from (4.13) that the mapping '0: S∗0 → H ⊕H determined by(4.14) is surjective. Moreover, for every {f, f ′} ∈ S∗0 of the form (4.16) and

{g, g′} = {g0 + RλGk,A0g0 + λRλGk} ∈ S∗0 , g0 ∈ domA0, k ∈ H ,

one obtains

(f ′, g)− (f, g′)= ((A0 − λ)f0, RλGk)− (RλGh, (A0 − λ)g0)

= (f0,Gk)− (Gh, g0)

= (G∗f0 + Bh, k)H − (h,G∗g0 + Bk)H

= (G∗(f −RGh)+ Bh, k)H − (h,G∗(g −RGk)+ Bk)H .

(iv) Decompose the defect vectors fλ ∈ Nλ(S∗0 ) as follows:

{fλ, λfλ} = {RλGh, λRλGh} = {RλGh, A0RλGh −Gh}.Then according to (4.14),

'00 fλ = h, '0

1 fλ = G∗(Rλ −R)Gh+ Bh,

which in view of (2.4) leads to (4.15).(v) The statement follows from Proposition 2.1. ✷The boundary operator '0

1 in Theorem 4.3 depends on a free parameter B ∈[H ]. When this parameter B is fixed, the family of perturbations A(α) of A0 can bedefined, in analogy with Theorem 4.1, as the family of self-adjoint extensions

Aτ = ker('00 + τ'0

1), τ = τ ∗ ∈ C(H). (4.17)

Some other, but equivalent, forms for defining these perturbations have been givenin [2, 22, 29, 33]. Observe that the resulting family A(α) is not additive in the senseof (4.11).


Remark 4.4. Comparing the statement (v) in Theorems 4.1 and 4.3 in the casewhere ranG ⊂ H−1 it is seen that A(α) = Aτ if and only if

τ−1 −G∗RG+ B = α−1. (4.18)

It follows from (4.18) that the operator α should be equal to 0 for G satisfyingranG ⊂ H−2 \ H−1 and τ = τ ∗ invertible. Such ‘perturbations’ Aτ of A0 werecalled in [33] infinitesimal; they can be interpreted as self-adjoint extensions of thesymmetric operator S0 in Theorem 4.3 and, hence, they can be parametrized by theresolvent formula (v) in Theorem 4.3 with τ = τ ∗ ∈ C(H).

Remark 4.5. If 0 ∈ ρ(A0), one can take A−10 ∈ [H−2,H0] as a regularizing

operator for the resolvent Rλ. Then the corresponding Weyl function M0 takes theform

M0(λ) = G∗((A0 − λ)−1 − A−10

)G+ B. (4.19)

The case of perturbations in H−2 is general in the sense that every closed sym-metric operator S with defect numbers (d, d) can be obtained as a restriction of aself-adjoint operator A0 via (1.2) with some linear injective mapping from H = C

d

into H−2, cf., e.g., [27].The following lemma gives some formulas which will be useful in Subsec-

tion 4.3 in order to describe the renormalization procedure for the resolvent Rλ =(A0 − λ)−1 and the Weyl function generated by singular perturbations of A0 withranG ⊂ H−k, k > 2.

LEMMA 4.6. Let G be a linear injective mapping from H = Cd into H−1, let

q be a scalar polynomial of degree n ∈ N, such that σ (q) ∩ σ (A0) = ∅, and letG0 = q(A0)

−1G. Define the block matrix T by

T = col(t2n−1, . . . , t1, t0),

tj = G∗0A

2n−1−j0 G0, j = 0, 1, . . . , 2n − 1. (4.20)

Then the following identities hold

G∗(A0 − λ)−1G = r(λ)+ q�(λ)G∗0(A0 − λ)−1G0q(λ), (4.21)

G∗(A0 − λ)−2G = d

dλ{r(λ)+ q�(λ)G∗

0(A0 − λ)−1G0q(λ)}, (4.22)

where

r(λ) = !(2n)(Bq�q ⊗ I )T , !(2n) = (IH , λIH , . . . , λ2n−1IH ) (4.23)

and Bq�q is the matrix associated with the polynomial q�q via (2.11).


Proof. It follows from

p(λ, z) = q(λ)�q(λ)− q(z)�q(z)

λ− z(4.24)

that the corresponding matrix polynomial admits the representation

p(λ, z)IH = !(2n)(Bq�q ⊗ I )Z�(2n) =2n−1∑j,k=0

bjkλjzkIH , (4.25)

where Z(2n) is defined similar to !(2n). Moreover, (4.24) implies that

G∗(A0 − λ)−1G = G∗0p(λ, A0)G0 + q�(λ)G∗

0(A0 − λ)−1G0q(λ).

To prove (4.21), it remains to notice that due to (4.25),

G∗0p(λ, A0)G0 =

2n−1∑j,k=0

bjkλjG∗

0Ak0G0 = !(2n)(Bq�q ⊗ IH)T .

The identity (4.22) is obtained from (4.21) by differentiation. ✷Remark 4.7. Statements similar to those in Lemma 4.6 are still valid if ranG ⊂

H−2. In this case t0 = G∗0A

2n−10 G0 is not well defined and the resolvent needs

a regularizing term. Let R be a regularizing operator of Rλ commuting with Rλ.Then the difference

t0 −G∗RG = G∗(q�(A0)−1A2n−1

0 q(A0)−1 −R)G (4.26)

makes sense and by incorporating the regularizing terms G∗RG and q�G∗0RG0q

in the formula (4.21) one arrives at the following identity

G∗(Rλ −R)G = r(λ)+ q�(λ)q(λ)[G∗0(Rλ −R)G0 + B], (4.27)

where B = G∗0RG0 and the matrix polynomial r(λ) is now given by

r(λ) = !(2n)(Bq�q ⊗ I )T −G∗RG. (4.28)

In (4.28) r(λ) is well defined in view of (4.26). Differentiation of (4.27) gives againan expression for G∗(A0−λ)−2G analogous to (4.22). Notice also that the selectionof a regularizing operator R or even two different regularizing operators R1, R2

in (4.27) results in a difference for r and B only by some constant well-definedself-adjoint operators in H , since R −Rj ∈ [H−2,H+2], j = 1, 2.


4.3. PERTURBATIONS IN H−2n−1 AND H−2n−2

In a number of papers singular rank one perturbations of A0 generated by ω ∈H−2n−2 have been studied by means of exit space extensions of a symmetric op-erator S connected with A0, see [17, 18, 37, 38, 41]. In this subsection a modelfor finite rank singular perturbations of A0 generated by G with ranG ⊂ H−2n−j ,j = 1, 2, is established also in terms of exit space extensions of A0. Here the modelfor such perturbations is derived from a basic assumption that in an extending innerproduct space H ⊃ H0 the resolvents associated with the perturbations of A0 shouldbe finite rank perturbations of the resolvent generated in H by (A0 − λ)−1 (seeTheorem 4.8).

First consider the case, where G is a linear mapping from H = Cd into H−2n−1

and let A0 be the [H−2n+1,H−2n−1]-continuation of A0. The adjoint operator G∗maps H2n+1 into H . Observe, that if ranG∩H−2 = {0}, then the identity (1.2) givesrise to an essentially self-adjoint operator whose closure is equal to A0. Moreover,the vector RλGh = (A0 − λ)−1Gh, h ∈ H , λ ∈ ρ(A0), does not belong to thespace H0. To give a sense to the vector RλGh and to the resolvent formula (2.7)one needs to extend the space H0 by adding the subspaces

A−10 ranG, . . . , A−n0 ranG, (4.29)

assuming, for simplicity, that 0 ∈ ρ(A0). Then the vector

γ (λ)h := RλGh = A−10 Gh+ · · · + λn−1A−n0 Gh+ λnRλA

−n0 Gh (4.30)

can be considered as a vector from an extended inner product space H whichcontains both H0 and the subspaces (4.29):

H ⊃ span{H0, A−j0 ranG : j = 1, . . . , n}. (4.31)

In this space the continuation A0 of A0 generates an operator, say H0, for whichthe operator function γ (λ), λ ∈ ρ(A0), can be interpreted to form its γ -field in thesense that

γ (λ)− γ (µ)

λ− µ= (H0 − λ)−1γ (µ), λ, µ ∈ ρ(A0). (4.32)

This identity implies that

d

dλγ (λ) = (H0 − λ)−1γ (λ), λ ∈ ρ(A0). (4.33)

The inner product 〈u, ϕ〉H in H should coincide with the form (u, ϕ) generatedby the inner product in H0 if the vectors u, ϕ are in duality, say, u ∈ H2(n−j)+1,ϕ ∈ A

−j0 ranG. Now, for the other vectors in (4.31) it will be supposed that the

conditions⟨A−j0 Gh, A−k0 Gf

⟩H= (tj+k−1h, f )H , j, k = 1, . . . , n; h, f ∈ H , (4.34)


are satisfied for some operators tj = t∗j ∈ [H ], j = 1, . . . , 2n − 1. The nextresult shows that under such weak conditions on the extending space the structureof perturbed resolvents becomes already completely fixed even under some mildassumptions on H0. This fact yields an interpretation and a model for singular finiterank perturbations of A0.

THEOREM 4.8. Assume that 0 ∈ ρ(A0) and let ranG ⊂ H−2n−1\H−2n, let G0 =A−n0 G, let H ⊃ H0 be (an isometric image of ) an inner product space satisfying(4.31), (4.34), and let H and H0 be self-adjoint linear relations in H such that

(i) ρ(H0) = ρ(A0);(ii) γ (λ)′ = (H0 − λ)−1γ (λ) holds for (an isometric image of ) γ (λ) =

(A0 − λ)−1G, λ ∈ ρ(A0);(iii) (H − λ)−1 − (H0 − λ)−1 = −γ (λ)σ (λ)γ (λ)∗, λ ∈ ρ(H) ∩ ρ(H0);

for some matrix function σ (λ) holomorphic and invertible for λ ∈ ρ(H0) ∩ ρ(H).Then σ (λ)−1 can be represented in the form

σ−1(λ) = β + t (λ)+ λ2nM0(λ), (4.35)

where

β = β∗ ∈ [H ], t (λ) = t1λ+ · · · + t2n−1λ2n−1,

and M0(λ) = G∗0RλG0 is a Nevanlinna function in H .

Proof. Denote Rλ = (H0 − λ)−1, Rλ = (A0 − λ)−1. By assumption (ii)

γ (λ)′ = Rλγ (λ),[γ (λ)∗

]′ = γ (λ)∗Rλ. (4.36)

Now, differentiation of (iii) yields

(H − λ)−2 − R2λ = −γ (λ)σ (λ)′γ (λ)∗ − Rλγ (λ)σ (λ)γ (λ)

∗ −− γ (λ)σ (λ)γ (λ)∗Rλ, (4.37)

which together with (iii) implies that

γ (λ)σ (λ)′γ (λ)∗ = −γ (λ)σ (λ)γ (λ)∗γ (λ)σ (λ)γ (λ)∗. (4.38)

The identity (4.38) can be rewritten (by the assumption of isometry in (ii)) as

dσ−1

dλ= γ (λ)∗γ (λ) = (RλG)∗(RλG). (4.39)

It follows from (4.30) and (4.34) that for every h, f ∈ H , j = 1, . . . , n,⟨A−j0 RλGh, A−1

0 Gf⟩H= ⟨RλGh, A

−j−10 Gf

⟩H,⟨

RλGh, A−10 RλGf

⟩H= ⟨R2

λGh, A−10 Gf

⟩H.


Therefore,⟨RλGh, RλGf

⟩H= ⟨

RλGh, (I + λRλ)A−10 Gf

⟩H

= d

dλ

⟨λRλGh, A−1

0 Gf⟩H, (4.40)

and by applying (4.30), with 2n− 1 instead of n, and (4.34) one obtains⟨λRλGh, A−1

0 Gf⟩H

= λ⟨A−1

0 Gh, A−10 Gf

⟩H+ · · · + λ2n−1

⟨A−2n+1

0 Gh, A−10 Gf

⟩H+

+ λ2n⟨RλA

−2n+10 Gh, A−1

0 Gf⟩H

= (t (λ)h, f )H + λ2n(G∗0RλG0h, f )H .

(4.41)

It follows from (4.39), (4.40), and (4.41) that σ−1(λ) in (iii) takes the form (4.35),where β is a self-adjoint operator in H and M0(λ) = G∗

0RλG0 is a Nevanlinnafunction since ranG0 ⊂ H−1. ✷

Remark 4.9. Observe that one arrives at the same formula (4.35) for σ−1(λ) bycomparing (4.39) with (4.22) in Lemma 4.6. Similarly, in the case σ (q)∩σ (A0) =∅ one can derive from Lemma 4.6 the following representation of σ−1(λ):

σ−1(λ) = r(λ)+ q�(λ)q(λ)M0(λ), (4.42)

where r is given by (4.23). One can extend Theorem 4.8 also to the case whereranG ⊂ H−2n−2. Then the function σ−1(λ) in (iii) still has the same form (4.42),but the function M0(λ) in (4.42) takes the form G∗

0(Rλ−R)G0 +B, where R is aregularizing operator of Rλ, B = B∗, and r with deg r � 2n− 1 is given by (4.28);see Remark 4.7.

Remark 4.10. The function r(λ)+ q�(λ)q(λ)M0(λ) is the Weyl function of themodel Pontryagin space symmetric operator S considered in [13, Theorem 4.2]and does not belong to the class of Nevanlinna functions. In fact, substituting theformula (4.42) with β = τ−1, τ = τ ∗ ∈ C(H), for σ in (iii) one obtains theresolvent formula (2.7) in Proposition 2.1 with the Weyl function

M = r + q�M0q. (4.43)

The formulas (4.21) and (4.27) can now be seen as a renormalization procedurefor the Weyl function associated with the γ -field γ (λ) = (A0 − λ)−1G. In view ofTheorem 4.8 it is natural to identify the family of finite rank singular perturbationsA0+GαG∗ with the family of self-adjoint extensions Hτ of the symmetric operatorS in a Pontryagin space; see Theorem 4.12. The formula (4.42) implies also thatone cannot find a Hilbert space self-adjoint family satisfying the properties (ii) and(iii) in Theorem 4.8. Of course, one can still give a description in purely Hilbertspace terminology, but then the extensions will not be self-adjoint anymore (cf.[37, 38]).


A model space H and a self-adjoint relation H0 in H which satisfy the assump-tions in Theorem 4.8 can be constructed directly from A0 and G without usingriggings of the Hilbert space H0 as follows. Let G be an injective linear mappingfrom H = C

d into H−2n−j , j = 1, 2. Let q be an nth order monic d × d matrixpolynomial such that σ (q) ∩ σ (A0) = ∅ and define G0 = q(A0)

−1G, so that G0

maps H = Cd into H−1(A0) or H−2(A0), respectively. The restriction S0 of A0

to domA0 ∩ kerG∗0, is a closed symmetric operator in H0 with defect numbers

(d, d). The corresponding Weyl function M0 is as in (4.6), (4.15), or as in (4.19)if in addition 0 ∈ ρ(A0), with G replaced by G0. Let t0, . . . , t2n−1 be arbitraryself-adjoint d × d matrices and define the matrix polynomial r, deg r � 2n − 1,by (4.23) or (4.28). Parallel to (4.21) or (4.27) depending on ran G ⊂ H−2n−1 orranG ⊂ H−2n−2, respectively, the generalized Nevanlinna function M is definedby (4.43). The matrix polynomial Q of the form (2.10) gives rise to a modelinvolving a reproducing kernel Pontryagin space HQ and a corresponding multi-plication operator SQ in it, see Theorem 2.2. The model for M in (4.43) is nowobtained by applying Theorem 3.1. For simplicity the result is formulated for thecase 0 ∈ ρ(A0) and ranG ⊂ H−2n−1\H−2n.

THEOREM 4.11. Let A0 be a self-adjoint operator in the Hilbert space H0 suchthat 0 ∈ ρ(A0) and let G: H = C

d → H−2n−1\H−2n be injective. Let t0, . . . , t2n−1

be arbitrary self-adjoint d × d matrices and let the matrix polynomial r, deg r �2n−1, be given by (4.23), let G0 = (A0)

−nG, and let S0 be a symmetric restrictionof A0 defined by

dom S0 = {f ∈ domA0 : G∗0f = 0}.

Let {H , '00, '

01} be a boundary triplet for S∗0 with the Weyl function M0 and, more-

over, let the symmetric operator SQ in HQ and the boundary triplet for S∗Q be asin Theorem 2.2. Then the operator S0 is densely defined in H0 and, moreover, thefollowing statements hold:

(i) The linear relation S defined in Theorem 3.1 is a closed simple symmetricoperator in H = H0 ⊕ HQ with defect numbers (d, d).

(ii) The adjoint linear relation S∗ and the boundary triplet for S∗ are as given inparts (ii) and (iii) of Theorem 3.1.

(iii) The corresponding Weyl function has the form (4.43) where q(λ) = λn andM0 is given by (4.6).

(iv) The linear relations H0 = ker'0 and Hτ = ker('0 + τ'1), τ ∈ [H ], areself-adjoint extensions of S in H0 ⊕ HQ.

(v) The resolvent set ρ(Hτ) of Hτ , τ ∈ [H ], is nonempty, the spectrum σ (Hτ) inρ(A0) coincides with

{λ ∈ ρ(A0) : det(I + (r(λ)+ λ2nM0(λ))τ) = 0},and the compressed resolvent of H0 and Hτ are of the form (3.7) and (3.20).


(vi) The corresponding Štraus extensions Tτ , τ ∈ [H ], are given by the ‘interfaceconditions’ in (3.21).

The statements of the above theorem follow easily from Theorems 3.1, 3.5,Proposition 3.2, and [13, Theorem 5.3]. One can state a similar result for the casewhen ranG ⊂ H−2n−2 \ H−2n and with q an arbitrary polynomial by using theformula (4.28) for the polynomial r and the formula (4.15) for M0.

4.4. COMPLETION THEOREM

Another construction of a model for singular rank one perturbations generatedby ω ∈ H−2n, was given in [18] via a completion procedure which resulted ina symmetric operator in a Pontryagin space whose Q-function was of the form(4.43). Although the model in [18] differs from the one given in Theorem 3.1,the corresponding Weyl functions coincide and, hence, the underlying self-adjointextensions are unitarily equivalent. In the next theorem the spaces H0, H, as wellas A0, and its lifting H0 in H are connected to each others after such a completionprocedure, when applied to the model in the present paper.

THEOREM 4.12. Let ranG ⊂ H−2n−1\H−2n and assume that 0 ∈ ρ(A0). Let{H , '0, '1} be a boundary triplet defined by the equality (4.2). Let G0 = A−n0 G,let S0 be defined by dom S0 = domA0 ∩ kerG∗

0, let tj be self-adjoint operatorsin [H ], j = 1, 2, . . . , 2n − 1, let Q be as in (2.10), where q(λ) = λn and r isof the form (4.23), and let H = H0 ⊕ HQ and H0 = ker'0 be constructed as inTheorem 3.1. Consider the linear space

Pn = span{H2n, A−10 (ranG), . . . , A−2n

0 (ranG)},

and define the inner product of the vectors A−j0 Ghj , hj ∈ H , j = 1, . . . , 2n, by

the identities (4.34) with j, k = 1, . . . , 2n, where tj = G∗0A

2n−j−10 G0 if j � 2n.

Then

(i) the mapping V from Pn to H0 ⊕ HQ defined by

V : ϕ +2n∑j=1

A−j0 Ghj "→

(ϕ +

2n∑j=n+1

A−j0 Ghj

)⊕( ∑n

k=1 fk ⊗ ek∑nk=1 hk ⊗ ek

),

where

fk = G∗0A

k−10 ϕ +

2n∑i=n+1

tn−k+ihi, 1 � k � n,

is isometric and ranV is dense in H0 ⊕ HQ;


(ii) the closure of the graph of the operator

A0: ϕ +2n∑j=2

A−j0 Ghj "→ A0ϕ +

2n∑j=2

A−(j−1)0 Ghj, ϕ ∈ H2n+2,

coincides with the linear relation H0 in H0 ⊕ HQ under the isometry V .(iii) V maps (A0 − λ)−1G, λ ∈ ρ(A0), to the γ -field γ (λ) in (3.2).

Proof. (i) Let εj = A−j0 Ghj , hj ∈ H , j = 1, . . . , 2n. Since for i � n the vector

V εi takes the form

V εi = 0⊕(

0hi ⊗ ei

), (4.44)

one obtains for i, j � n

[V εi, V εj ] = (Brhi ⊗ ei , hj ⊗ ej )H(Q) = (ti+j−1hi, hj ) = 〈εi, εj 〉Pn.

For j > n the vector V εj takes the form

V εj = εj ⊕( ∑n

k=1 tn−k+j hj ⊗ ek0

), (4.45)

and hence one obtains for i � n < j � 2n

[V εi, V εj ] =((

hi ⊗ en−i+1∑nl=1 ti+l−1hi ⊗ ei

),

( ∑nk=1 tn−k+j hj ⊗ ek

0

))= (ti+j−1hi, hj ) = 〈εi, εj 〉Pn

.

It follows from (4.45) that

[V εi, V εj ] = 〈εi, εj 〉Pn, i, j > n.

Finally, the equality

V ϕ = ϕ ⊕( ∑n

k=1 G∗0A

k−10 ϕ ⊗ ek

0

)(4.46)

yields

[V ϕ, V εj ] = (ϕ, εj )H0 = 〈ϕ, εj 〉Pn, j > n,

and

[V ϕ, V εj ] = (G∗0A

n−j0 ϕ, hj)H = (ϕ, A

−j0 G∗hj) = 〈ϕ, εj 〉Pn

, j � n.

Thus, the mapping V is isometric. It follows from (4.44), (4.45), (4.46) also thatranV is dense in H0 ⊕ HQ, since H2n is dense in H.


(ii) It follows from (3.9)–(3.13) that

H−10 VA0f0 = f0 ⊕

( ∑nj=1 G

∗0A

j

0f0 ⊗ ej0

)= Vf0, f0 ∈ H2n+2.

Therefore, to prove the second statement, it remains to show that

V A−(j+1)0 Gh = H−1

0 V A−j0 Gh, j = 1, . . . , 2n− 1. (4.47)

For j < n it follows from (4.44) and (3.9)–(3.13) that

H−10 V A

−j0 Gh = 0⊕

(0

hj ⊗ ej+1

)= V A

−(j+1)0 Gh.

If j = n one obtains from (3.9)–(3.13)

H−10 V A−n0 Gh = (A−1

0 G0h)⊕( ∑n

i=1 t2n−i+1h⊗ ei0

)= V A

−(n+1)0 Gh.

Similarly, for j > n one obtains from (3.9)–(3.13) that

H−10 V A

−j0 Gh = (A

−(j+1)0 Gh)⊕

( ∑ni=1 tn+j−i+1h⊗ ei

0

)= V A

−(j+1)0 Gh.

To see (iii) decompose RλG = (A0−λ)−1G as in (4.30) with n replaced by 2n.Then

(A0 − λ)−1Gh = ε1 + · · · + λ2n−1ε2n + λ2nRλε2n.

It is easy to see that V maps the sum u1 = ε1 + · · · + λn−1εn to the vector(0, 0,!�h). Moreover, the image of u2 = λn(εn+1 + · · · + λn−1ε2n) + λnRλε2n

is of the form V (u2) = (u2, v, 0). In view of (4.30)

u2 = λnRλA−n0 Gh = q(λ)(A0 − λ)−1G0h = q(λ)γ0(λ)h.

The expression for the components of v is obtained after several applications of(4.30) with different values of n and by taking into account the definition of !1 in(2.14). The results is

v = λn!�(G∗0(A0 − λ)−1G0)h+!�1 h = q(λ)!�M0(λ)h+!�1 h.

Therefore, V (RλG) = γ (λ), λ ∈ ρ(A0). ✷Remark 4.13. An analog of Theorem 4.12 is still true when ranG ⊂

H−2n−2\H−2n. Then t2n, the resolvent Rλ on the left side of (4.30) and the constantterm A−1

0 on the right side of (4.30) should be regularized. If t2n is replaced byt2n = G∗

0(A−10 − R)G0 + B then V (RλG) = γ (λ) is still given by (3.2) with

M0(λ) = G∗0(Rλ −R)G0 + B; here also the selection R = A−1

0 is allowed since0 ∈ ρ(A0).


Finally, it is emphasized that the model constructed above for singular pertur-bations admits all the properties in Theorem 4.8. The property (i) in Theorem 4.8was shown in Proposition 3.4. Part (iii) of Theorem 4.12 shows that the isometricimage γ (λ) of (A0 − λ)−1G is the γ -field associated with H0, so that it satisfies(4.32) and hence also (4.33): γ (λ)′ = (H0 − λ)−1γ (λ). (One can check this lastidentity also directly by applying the formulas given for (H0− λ)−1 in the proof ofProposition 3.4 with G = γ (λ).) Moreover, the property (iii) in Theorem 4.8 wasproved in Proposition 3.2.

5. The Dirac Operator

As an application of the model constructed in Section 3 some singular perturbationsof the Dirac operator are studied.


Let A0 be the free Dirac operator in H0 = L2(R) ⊕ L2(R) given on the domainW 1

2 (R)⊕W 12 (R) by the expression

D = −ic d

dx⊗ σ1 + (c2/2)⊗ σ3 =

c2

2−ic d

dx

−ic d

dx−c2

2

, (5.1)

where

σ1 =(

0 11 0

), σ3 =

(1 00 −1

),

are Pauli matrices in C2 and c > 0 is the velocity of light. The spectrum of A0

coincides with the set (−∞,−c2/2]∪ [c2/2,∞), and the resolvent operator (A0−λ)−1 is given by the integral kernel

Rk(λ, x − x′) = i

2c

(ζ(λ) sgn(x − x′)

sgn(x − x′) ζ(λ)−1

)eik(λ)|x−x

′|, (5.2)

where

k(λ) = 1

c

√λ2 − c4/4, Im k(λ) � 0, ζ(λ) = λ+ c2/2

ck(λ).

Define γ0: H = C2 → H0 by γ0(λ) = Rk(λ, x), so that in particular

γ0(0) = 1

2c

(1 i sgn x

i sgn x −1

)e−c/2|x|. (5.3)


Consider the two-dimensional perturbations of A0,

A(α) = A0 +G0αG∗0, α ∈ [H ], (5.4)

with the operator G0: H → H−2 given by G0h = δ ⊗ h, h ∈ H . Then G∗0:

H+2 (= domD) → H is given by G∗0f = f (0). Let S0 be the domain restriction

of A0 given by

dom S0 = {y = (y1, y2)� ∈ W 1

2 (R)⊕W 12 (R) : y(0) = 0},

and let A0 be the [H0,H−2]-continuation of A0. Then γ0(λ) = (A0 − λ)−1G0 andaccording to Theorem 4.3,

S∗0 = {{y0 + γ0(0)h, A0y − δ ⊗ h} : y0 ∈ W 12 (R)⊕W 1

2 (R), h ∈ H}. (5.5)

The boundary operators '00 and '0

1 for S∗0 can be given by (4.14). It follows from(5.3) and (5.5) that

'00y = h = −icσ1(y(0+)− y(0−)), (5.6)

and, for the special choice of B = (1/2c)σ3, that

'01y = y0(0)+ 1

2cσ3h = y(0+)+ y(0−)

2. (5.7)

Due to (4.17) the perturbations A(α) are determined by the self-adjoint extensionsAτ = ker('0 + τ'1) with τ a self-adjoint 2× 2 matrix in H . Now f ∈ Aτ can berewritten as

y(0+) = !y(0−), (5.8)

where ! is a σ1-unitary matrix given by

! =(iσ1 − 1

2cτ

)−1(iσ1 + 1

2cτ

). (5.9)

In view of (5.7) the corresponding Weyl function is given by

M0(λ) = γ0+(λ)+ γ0−(λ)2

= i

2c

(ζ(λ) 0

0 ζ(λ)−1

).

Clearly, limy→∞M0(iy) = (i/2c)IH is not self-adjoint and therefore, by [26,Section 2] or [10, Theorem 4.4], ranG0 ⊂ H−2\H−1. The description (5.8), (5.9)of self-adjoint extensions Aτ of the operator S0 was given in [5]. In [3] it wasshown that the extensions Aτ can be considered as perturbations of A0. In fact, thedefinition of A(α) depends on the choice of a free parameter B. In the present paperthe choice B = (1/2c)σ3 is made in order to obtain the same family Aτ as in [5]and [3], cf. also [30]. For the special cases

τ =(a 00 0

)or τ =

(0 00 b

), a, b ∈ R,


one obtains the boundary conditions

y2(0+)− y2(0−) = − i

cay1(0) or y1(0+)− y1(0−) = − i

cby2(0), (5.10)

which characterize the one-parameter families Da and Tb of perturbations of A0,

Da = A0 + aδ ⊗ e1(· , δ ⊗ e1) or Tb = A0 + bδ ⊗ e2(· , δ ⊗ e2),

respectively, cf. [1] and [21]. Here (f, δ⊗ e1) = f1(0) and (f, δ⊗ e2) = f2(0) forall f ∈ domD.


Now assume that G maps H = C2 into H−4. For the sake of simplicity let

Gh = A0G0h = −icδ′ ⊗ σ1h+ (c2/2)δ ⊗ σ3h, h ∈ H = C2.

Then G∗: H+4 (= domD2) → H is given by G∗f = (Df )(0). Setting q(λ) =IHλ one obtains in the model in Theorem 2.2:

Cq = Cq� = 0, B = σ1 ⊗ IH .

According to Theorem 3.1 and [13, Theorem 5.3], the operator

S = y

'01y

0

,

S∗0y0

'00y

: y ∈ dom S∗0

, (5.11)

is a simple symmetric operator in the Pontryagin space H0 ⊕ C4 whose inner

product is determined by IH0 ⊕ B. The finite rank perturbations A0 + GαG∗are identified with the self-adjoint extensions Hτ , τ ∈ [H ], of the operator S,as specified in the following theorem. It is obtained by applying Theorem 3.1 withthe data given above.

THEOREM 5.1. Let the operator S in H0 ⊕ C4 be defined by (5.11). Then:

(i) The adjoint linear relation S∗ takes the form

S∗ =F =

y

'01y

f

,

S∗0yϕ

'00y

: y ∈ dom S∗0 , f , ϕ ∈ H

.

(ii) The boundary triplet {C2, '0, '1} for S∗ and the corresponding Weyl functionM, which has two negative squares, are given by

'0F = f , '1F = ϕ, M(λ) = iλ2

2c

(ζ(λ) 0

0 ζ(λ)−1

),

|λ| > c2/2. (5.12)


(iii) The self-adjoint extensions Hτ = ker('0 + τ'1) are given by

Hτ = y

'01y

f

,

S∗0yϕ

'00y

: y ∈ dom S∗0 , f + τ ϕ = 0

. (5.13)

(iv) The spectrum of Hτ in C\((−∞,−c2/2]∪[c2/2,∞)) is characterized by theequivalence

λ ∈ σp(Hτ )⇔ det

(I + iλ2

2c

(ζ(λ) 0

0 ζ(λ)−1

)τ

)= 0. (5.14)

(v) The compression of the resolvent (Hτ − λ)−1 to H0 takes the form

PH0(Hτ − λ)−1

= (A0 − λ)−1 − λ2γ0(λ)τ

(I + iλ2

2c

(ζ(λ) 0

0 ζ(λ)−1

)τ

)−1

γ0(λ)∗. (5.15)

(vi) The function y = PH0(Hτ−λ)−1z is a solution of the boundary value problemwith the λ-depending interface condition

(S∗0 − λ)y = z, y(0+) = !(λ)y(0−), (5.16)

where !(λ) is given by the formula

!(λ) = (2icσ1 − λ2τ)−1(2icσ1 + λ2τ). (5.17)

By special choices of τ ∈ [H ] it is possible to generate frequently occurringcases. For instance, if

τ =(a 00 0

)or τ =

(0 00 b

), a, b ∈ R,

then one obtains the one-parameter families of ‘perturbations’ D(1)a or T (1)

b of A0,

D(1)a = A0 + aω1(·, ω1), ω1 =

((c2/2)δ−icδ′

), (5.18)

T(1)b = A0 + bω2(·, ω2), ω2 =

(icδ′

(c2/2)δ

), (5.19)

respectively. Here (f, ω1) = (c2/2)f1(0) − icf ′2(0) and (f, ω2) = icf ′1(0) +(c2/2)f2(0) for all f ∈ domD2. Their compressed resolvents are characterizedby the following interface conditions

y2(0+)− y2(0−) = − i

cλ2ay1(0)

or y1(0+)− y1(0−) = − i

cλ2by2(0), (5.20)


respectively. As is known [1, 5, 21], the perturbations A(τ) in (5.4) are relatedto the corresponding nonrelativistic interactions of the Schrödinger operator viathe nonrelativistic limit. For perturbations in H−4 the situation is different. Thenonrelativistic limit does not distinguish the perturbations D(1)

a . Namely,

limc→∞PH0

(D(1)

a −(λ+ c2

2

))−1

�H0 = (−Aa,∞ − λ)−1 ⊗(

1 00 0

),

where Aa,∞ stands for

Aa,∞ = {{y,−D2y} : y ∈ W 22 (R\{0}), y(0) = 0}.

Acknowledgements

We would like to thank Yury Arlinskiı for discussions and valuable commentswhich led to some improvements in the paper.

The first author (V.D.) was partially supported by the Academy of Finland(project 52528) and the Dutch Association for Mathematical Physics (MF00/34).The second author (S.H.) was supported by the Academy of Finland (project 40362).

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385

Spectral Analysis of One-Dimensional DiracOperators with Slowly Decreasing Potentials

MATHIEU MARTINUniversité Paris 7, Mathématiques, case 7012, 2 place Jussieu, 75251 Paris Cedex 05, France.e-mail: [email protected]

(Received: 11 October 2001; in final form: 22 August 2002)

Abstract. We prove that the absolutely continuous spectrum of Dirac operators on the half-line withsquare integrable potentials fills the whole real axis. We also establish an estimate on the number ofeigenvalues for Coulomb-like potentials.

Mathematics Subject Classifications (2000): 47E05, 34L40, 34B20, 34L15, 34L25.

Key words: Dirac operator, spectral analysis, scattering theory, Weyl function, Jost solution, Coulombpotential.

1. Introduction

Our goal is to study the spectral properties of Dirac operators

[H(α)u](x) =(

0 1−1 0

)u′(x) +

(p1(x) q(x)

q(x) p2(x)

)u(x) (1.1)

acting on L2((0,∞); C2) with boundary condition

u1(0) cos(α)+ u2(0) sin(α) = 0, (1.2)

where α ∈ [0, π) and u(x) = (u1(x), u2(x))t . Here we assume that p1, p2 and

q are three real-valued functions which belong to L1loc([0,∞)). These conditions

ensure the self-adjointness of the operator H(α) (see [15, 16] for the proof of thisresult and for the exact definition of the domain). Let V denotes the followingcomplex-valued function:

V (x) = q(x) + i

2(p1(x) − p2(x)).

The function V is often called the potential of the operator H(α).Recall that a Borelian S of R is a minimal support of a Borel measure dµ on R

if µ(R \ S) = 0 and µ(B) > 0 for any Borelian B ⊂ S such that |B| > 0, where|B| denotes the Lebesgue measure of B. Our main result is:

THEOREM 1.1. If V ∈ L2(0,∞) then the whole real axis is a minimal supportof the absolutely continuous part of the spectral measure dρα of H(α).

386 MATHIEU MARTIN

The analog of this result has been proved recently by Deift and Killip [3] for theSchrödinger case (see also [7, 11] for more general potentials). Then Theorem 1.1has the following consequence:

THEOREM 1.2. If V ∈ L2(0,∞) then the absolutely continuous part of H(α) isunitarily equivalent to Mid in L2(R) where Mid is the multiplication operator bythe function id(x) = x. In particular, σac(H(α)) = R.

If P,Q ∈ L1loc([0,∞); R) then we will denote by H the Dirac operator (1.1)

with p1 = P , p2 = −P , q = Q and α = 0 (i.e. with a Dirichlet boundarycondition). In Section 2 we will prove that Theorem 1.1 is a consequence of thefollowing theorem:

THEOREM 1.3. If P,Q ∈ L2(0,∞) then the whole real axis is a minimal supportof the absolutely continuous part of the spectral measure of H .

As in [3], the proof is based on an estimate of the Jost function but in our case wehad to use a different method in order to obtain it. More precisely, they obtain theirestimate by performing an integration in the complex plane whereas our method isbased on harmonic analysis.

For the Schrödinger operators L = −d2/dx2 + q acting on L2(0,∞) (with anyself-adjoint boundary condition at the origin), it is proved in [8] that iflim supx→∞ x|q(x)| = C then the possible positive eigenvalues λn > 0 of Lsatisfy

∑n λn � C2/2. In particular, L has no positive bound states as long as

q(x) = o(|x|−1). Following their method, for Dirac systems we obtain the follow-ing theorem:

THEOREM 1.4. Assume that

C ≡ lim supx→∞

x|V (x)| < ∞. (1.3)

Then H(α) has at most 4C2 distinct eigenvalues. In particular, if C < 1/2 thenσpp(H(α)) is empty.

This result is quite interesting because Naboko proved in [12] that for eachε, C > 0 there exists a potential V satisfying |V (x)| � C/|x|1−ε as x → +∞ andsuch that the point spectrum of the corresponding Dirac operator is dense in R. Seealso in [12] for further references on this topic.

The paper is organized as follows. In Section 2 we show that it is enough toprove Theorem 1.1 in the special case where α = 0 (i.e. for the operator with aDirichlet boundary condition) and p1 + p2 = 0 (which we have reformulated asTheorem 1.3). Section 3 is devoted to recall what we need about Weyl functions. InSection 4 we study some properties of the Jost function associated to Dirac systemswith compactly supported potentials. In Section 5 we prove Theorem 1.3, using theresults from Section 4. Finally, we give the proof of Theorem 1.4 in Section 6.

SPECTRAL ANALYSIS OF ONE-DIMENSIONAL DIRAC OPERATORS 387

2. Reduction of the Problem

It is well known (see Lemma 1 of [1, 2, 5]) that there exists a minimal support ofthe absolutely continuous part of the spectral measure dρα of H(α) which is inde-pendant of α. Using the result: if two measures are both absolutely continuous w.r.t.the Lebesgue measure and have a common minimal support then all their minimalsupports coincide, we see that it is enough to prove Theorem 1.1 in the specialcase of the Dirichlet boundary condition (i.e. α = 0). Consider the multiplicationoperator A in L2((0,∞); C

2) by the matrix-valued function

A(x) =(

cos(χ(x)) sin(χ(x))− sin(χ(x)) cos(χ(x))

),

where

χ(x) = −1

2

∫ x

0[p1(t) + p2(t)] dt.

Then A is a unitary operator in L2((0,∞); C2). Let P and Q denote the following

real-valued functions:

P(x) = Im[V (x)e2iχ(x)],Q(x) = Re[V (x)e2iχ(x)].

Then a simple computation shows that

D(H) = AD(H(0)) and H = AH(0)A−1,

where H is the Dirac operator corresponding to P and Q. Thus H and H(0)are unitarily equivalent. Since H (resp. H(0)) is unitarily equivalent to Mid inL2(R, dρ) (resp. L2(R, dρ0)) where dρ (resp. dρ0) denotes the spectral measureof H (resp. H(0)), the operators Mid in L2(R, dρac) and Mid in L2(R, dρ0

ac) areunitarily equivalent, which implies that the measures dρac and dρ0

ac are equivalent.As a consequence the minimal supports of dρac and dρ0

ac coincide. On the otherhand, if V ∈ L2(0,∞) then P ∈ L2(0,∞) and Q ∈ L2(0,∞) thus Theorem 1.1is a consequence of Theorem 1.3.

3. Basic Results about the Weyl Function m(z)

Let us fix the following notations:

C± = {z ∈ C : ±Im z > 0} and C± = {z ∈ C : ±Im z � 0}.

For each z ∈ C, consider the solutions η(x; z) and θ(x; z) of the equation(0 1

−1 0

)u′(x)+

(P(x) Q(x)

Q(x) −P(x))u(x) = zu(x) (3.1)

388 MATHIEU MARTIN

such that

η1(0; z) = −1, θ1(0; z) = 0,

η2(0; z) = 0, θ2(0; z) = 1.

For each z ∈ C\R let φ(x; z) the unique solution of (3.1) wich is square integrableat infinity and such that φ1(0; z) = −1. Recall that if f and g are solutions of thesame Equation (3.1) their Wronskian

W [f (x), g(x)] =∣∣∣∣f1(x) g1(x)

f2(x) g2(x)

∣∣∣∣is independent of x. Thus φ(x; z) has the form

φ(x; z) = η(x; z) +m(z)θ(x; z). (3.2)

m is called the Weyl function (sometimes called Weyl–Titchmarsh function). Inparticular, we have

m(z) = −φ2(0; z)φ1(0; z) . (3.3)

Note that it implies m(z) = m(z) for each z ∈ C \ R. Moreover, using (3.2), asimple computation shows that

Imm(z)

Im z=

∫ ∞

0‖φ(x; z)‖2

C2 dx (3.4)

and it is known (see [6]) that m is analytic on C+. The spectral measure dρ of the

operator H is connected to m(z) via its representation as a Herglotz function, i.e.

m(z) = A+ Bz +∫

R

[1

λ− z− λ

λ2 + 1

]dρ(λ) (3.5)

for some A ∈ R and B � 0. It follows that [6, 13, 14]

dρ(λ) = w- limε↓0

1

πIm[m(λ+ iε)] dλ

and the absolutely continuous part of dρ satisfies

dρac(λ) = 1

πImm(λ+ i0) dλ.

As we already mentioned, the most important fact concerning the spectral measureis that the operator H is unitarily equivalent to the operator Mid acting inL2(R, dρ)(see [9, 16]).


4. The Jost Function a(z) in the Case where the Potential is CompactlySupported

In this section we assume that P and Q are two real-valued functions belongingto L2

0([0,∞); R). Let us extend P and Q to functions defined on R by P(x) =Q(x) = 0 for x < 0. Put V (x) = Q(x) + iP (x) for each x ∈ R. Assume thatsupp(V ) ⊂ [0,D]. For each z ∈ C let denote by ψ(x; z) the unique solution of theequation(

0 1−1 0

)u′(x)+

(P(x) Q(x)

Q(x) −P(x))u(x) = zu(x), x ∈ R, (4.1)

such that

ψ(x; z) =(

1−i

)eizx for x � D. (4.2)

By analogy with the Schrödinger case, ψ(x; z) will be called the Jost solution. Onthe left of the support of V , ψ(x; z) takes the form

ψ(x; z) = a(z)

(1−i

)eizx + b(z)

(1i

)e−izx for x � 0. (4.3)

The coefficient a(z) is called the Jost function. Moreover, since ψ(x; z) is a solu-tion of Equation (4.1) its components ψ1(x; z) and ψ2(x; z) are entire functions ofz for each x ∈ R fixed. Thus a(z) and b(z) are entire. Since ψ(x; z) and ψ(x; z)are both solutions of Equation (4.1), their Wronskian is constant:

a(z)a(z)− b(z)b(z) = 1. (4.4)

In particular, for each λ ∈ R

|a(λ)|2 = 1 + |b(λ)|2. (4.5)

In the following lemma we give some properties of the function log |a(z)|:

LEMMA 4.1.

(i) The function log |a(z)| is harmonic on C+, bounded and continuous on C+.

(ii) For each z ∈ C+ we have

log |a(z)| � 1

2 Im(z)

∫ ∞

0|V (x)|2 dx.

(iii)∫ ∞

−∞log |a(λ)|λ2 + 1

dλ � π

2

∫|V (x)|2 dx.

390 MATHIEU MARTIN

Proof. (i) Since a(z) is analytic, in order to prove that log |a(z)| is harmonic onC

+ and continuous on C+ we only have to show that a(z) does not vanish on C+.According to (4.4), for each λ ∈ R we have |a(λ)|2 = 1 + |b(λ)|2 � 1. Now fixz ∈ C

+ and assume that a(z) = 0. From (4.2) and (4.3) we see that ψ is L2 at +∞and −∞. Thus ψ ∈ L2(R; C

2) which is in contradiction with the fact that ψ(x; z)is a solution of Equation (4.1). It remains to show that log |a(z)| is bounded on C+.For each x ∈ R and each z ∈ C define(

a(x; z)b(x; z)

)= 1

2i

(ie−izx −e−izxieizx eizx

)ψ(x; z),

then

ψ(x; z) = a(x; z)(

1−i

)eizx + b(x; z)

(1i

)e−izx, x ∈ R (4.6)

and

a′(x; z) = V (x)e−2izxb(x; z),b′(x; z) = V (x)e2izxa(x; z).

Since a(x; z) = 1 and b(x; z) = 0 if x � D and a(x; z) = a(z) and b(x; z) = b(z)

if x � 0, we obtain

a(x; z) = 1 −∫ D

x

V (y)e−2izyb(y; z) dy,

b(x; z) = −∫ D

x

V (y)e2izya(y; z) dy.

Consequently, for each x ∈ R and each z ∈ C we have

|a(x; z)| � 1 +∫x<x1<x2

|V (x1)| |V (x2)| e2(Im z)x1e−2(Im z)x2|a(x2; z)| dx1 dx2.

By iterations, we obtain

|a(x; z)| � 1 +∞∑n=1

In(x; z), (4.7)

where

In(x; z) =∫x<x1<···<x2n

|V (x1)| |V (x2)| . . . |V (x2n−1)| |V (x2n)| × (4.8)

× e2(Im z)x1e−2(Im z)x2 . . . e2(Im z)x2n−1e−2(Im z)x2n dx1 . . . dx2n.


The following inequalities ensure that the series appearing in (4.7) converges:

In(x; z) �

1

(2n)!(∫ D

x

|V (t)| dt

)2n

if Im z � 0 and x ∈ R,

e2(Im z)xe−2(Im z)D1

(2n)!(∫ D

x

|V (t)| dt

)2n

if Im z < 0 and x � 0,

e−2(Im z)D1

(2n)!(∫ D

0|V (t)| dt

)2n

if Im z < 0 and x � 0.

In turn, for each x ∈ R and each z ∈ C+ this gives

|a(x; z)| � cosh

(∫ D

x

|V (t)| dt

),

which proves that log |a(z)| is bounded on C+.(ii) Let z ∈ C

+ and put β ≡ Im (z) > 0. From (4.8), we have

In(x; z) =∫x<x1<···<x2n

2n∏k=1

|V (xk)| exp

[2β

2n∑j=1

(−1)j+1xj

]dx1 . . . dx2n.

Using the Cauchy–Schwarz inequality, we get

In(x; z) � [fn(x; z)]1/2[gn(x; z)]1/2,

where

fn(x; z) ≡∫x<x1<···<x2n

n−1∏k=0

|V (x2k+1)|2 exp

[2β

2n∑j=1

(−1)j+1xj

]dx1 . . . dx2n,

gn(x; z) ≡∫x<x1<···<x2n

n∏k=1

|V (x2k)|2 exp

[2β

2n∑j=1

(−1)j+1xj

]dx1 . . . dx2n.

Define

f (x; z) = 1

2β

∫ ∞

x

|V (t)|2 dt.

Let us prove by induction that

fn(x, z) � 1

n! [f (x; z)]n,

gn(x, z) � 1

n! [f (x; z)]n

(4.9)

for each n � 1. Integrations by parts show that f1(x; z) = f (x; z) and

g1(x; z) = 1

2β

∫ ∞

x

|V (t)|2 dt − e2βx

2β

∫ ∞

x

|V (t)|2e−2βt dt � f (x; z),

392 MATHIEU MARTIN

which proves (4.9) for n = 1. Assume that both estimates (4.9) are true for n andlet us prove them for n+ 1. We have

fn+1(x; z) =∫x<x1<x2

|V (x1)|2e2β(x1−x2)fn(x2; z) dx1 dx2

thus by induction’s hypothesis,

fn+1(x; z) � 1

n!∫x<x1<x2

|V (x1)|2e2β(x1−x2)[f (x2; z)]n dx1 dx2.

But ∫ ∞

x1

e−2βx2[f (x2; z)]n dx2 � 1

2β[f (x1; z)]ne−2βx1,

which gives

fn+1(x; z) � 1

n!∫ ∞

x

1

2β|V (x1)|2[f (x1; z)]n dx1 = 1

(n+ 1)! [f (x; z)]n+1.

By the same kind of argument, it is easy to prove (4.9) for gn+1(x; z). Conse-quently, for each n � 1, we have

In(x; z) � 1

n! [f (x; z)]n.

Thus using (4.7) we see that |a(x; z)| � exp[f (x; z)] which implies (ii).(iii) From (i), for each z0 ∈ C

+ we have (see Lemma 3.4 of [4])

log |a(z0)| =∫ ∞

−∞log |a(λ)|Pz0(λ) dλ, (4.10)

where Pz0(λ) denotes the Poisson kernel for the upper half-plane, i.e.,

Pz0(λ) = 1

π

y0

(x0 − λ)2 + y20

, z0 = x0 + iy0.

In particular, (4.10) with z0 = i gives∫ ∞

−∞log |a(λ)|λ2 + 1

dλ = π log |a(i)|

and thus (iii) follows from (ii). ✷The next lemma makes the link between the Weyl function and the Jost function:

LEMMA 4.2. The Weyl function m(z) has an analytic continuation to R and foreach λ ∈ R we have

1

|a(λ)|2 � 4 Imm(λ). (4.11)


Proof. When z ∈ C+, from (4.2) we see that ψ(x; z) is square summable

on (0,∞). Consequently, the functions ψ(x; z) and φ(x; z) are dependent, sofrom (3.3) we have that

m(z) = −ψ2(0; z)ψ1(0; z) = i

a(z)− b(z)

a(z)+ b(z).

Thus, from (4.4), m(z) has an analytic continuation to R and using (4.5) we obtain

Imm(λ) = 1

|a(λ)+b(λ)|2 .

Now (4.11) follows from the fact |a(λ)| � |b(λ)|. ✷

5. Absolutely Continuous Spectrum of H

Before proving Theorem 1.3, let us recall the following lemma (see [3] for a proof):

LEMMA 5.1. Let w(x) ∈ L1loc(R) be a.e. positive. Assume that dµn is a sequence

of positive measures which converge weakly to dµ. Then for each compact I ⊂ R

of positive Lebesgue measure we have

lim infn

1

w(I)

∫I

− log

(dµn

dx

1

w(x)

)w(x) dx � log

(w(I)

µ(I )

), (5.1)

where w(I) ≡ ∫Iw(x) dx. ✷

Proof of Theorem 1.3. Let P,Q ∈ L2(0,∞). For each n � 1 define

Pn = Pχ[0,n] and Qn = Qχ[0,n].

Denote by Hn the Dirac operator

[Hnu](x) =(

0 1−1 0

)u′(x) +

(Pn(x) Qn(x)

Qn(x) −Pn(x))u(x)

on L2((0,∞); C2) with Dirichlet boundary condition u1(0) = 0. To each operator

Hn associate its Jost function an(z), its spectral measure dρn and its Weyl functionmn(z). Define Vn(x) ≡ Qn(x) + iPn(x).

In order to apply Lemma 5.1, we have to prove the weak convergence of thespectral measures dρn to dρ. Let denote by ηn(x; z) (resp. θn(x; z), φn(x; z)) thesolution η(x; z) (resp. θ(x; z), φ(x; z)) corresponding to Pn(x) and Qn(x) in Sec-tion 3. Since φn(x; z) satisfies the free Dirac equation on (n,∞) this solution takesthe form φn(x; z) = Kn(z)

( 1−i

)eizx for each x � n. But ηn(x; z) = η(x; z) and

θn(x; z) = θ(x; z) for 0 � x � n, so it follows from (3.2) that

mn(z) = − iη1(n; z)+ η2(n; z)iθ1(n; z)+ θ2(n; z) .

394 MATHIEU MARTIN

By standard arguments from the Weyl theory of the limit point case (see [9]) thisimplies that for each z ∈ C

+ and each n � 1 the point mn(z) in the complex planeis inside the circle

Cn(z) ={

− λη1(n; z)+ η2(n; z)λθ1(n; z)+ θ2(n; z) ;λ ∈ R

}

of radius

rn(z) = 1

2(Im z)∫ n

0 ‖θ(x; z)‖2 dx.

Moreover, if z ∈ C+, then {Cn(z)} form a sequence of concentric circles in the

complex plane which converges to the point m(z). In particular, mn(z) convergesto m(z) on C

+ and for each n � 1 and each z ∈ C+ we have

|mn(z)−m(z)| � 2rn(z) = 1

(Im z)∫ n

0 ‖θ(x; z)‖2 dx. (5.2)

Let K denotes a compact subset of C+ and fix A > 0 arbitrary. Then for each

z ∈ K there exists N(z) � 1 such that∫ N(z)

0 ‖θ(x; z)‖2 dx > A. For each n � 1the map z �→ ∫ n

0 ‖θ(x; z)‖2 dx is continuous on C so for each z ∈ K there exists

a neighbourhood V(z) of z in C such that∫ N(z)

0 ‖θ(x; z′)‖2 dx > A for each z′ ∈V(z). Now from the compactness of K it follows that there exists N � 1 suchthat for each z ∈ K we have

∫ N

0 ‖θ(x; z)‖2 dx > A. Since A > 0 is arbitrary, itfollows from (5.2) that mn(z) converges to m(z) uniformly on compact subset ofC

+. Consequently, from the representation (3.5) we have that for each z ∈ C \ R

and each integer k � 3,

limn→∞

∫R

dρn(λ)

(λ− z)k=

∫R

dρ(λ)

(λ− z)k.

Then, by the Stone–Weierstrass theorem, this implies that (dρn) converges weaklyto dρ.

From Lemma 4.1(iii), for each n � 1 we have∫ ∞

−∞log |an(λ)|λ2 + 1

dλ � π

2‖Vn‖2

2.

Put w(λ) = 1/(λ2 + 1). Since Vn → V in L2, there exists C > 0 such that foreach n � 1 we have∫ ∞

−∞log |an(λ)|w(λ) dλ � C. (5.3)

But Pn and Qn are compactly supported, so Hn has purely absolutely continuousspectrum, which implies that

dρndλ

= 1

πImmn(λ+ i0) a.e.


Then from Lemma 4.2, for almost every λ ∈ R, we get

dρndλ

1

w(λ)� 1

4π |an(λ)|2because w(λ) � 1. Using (5.3), we see that for each compact I ⊂ R of positiveLebesgue measure we have

1

w(I)

∫I

− log

(dρndx

1

w(x)

)w(x) dx � log(4π)+ 2C

w(I).

Now Lemma 5.1 allows us to conclude that

0 <1

4πexp

(− 2C

w(I)

)w(I) � ρ(I )

for each compact I ⊂ R such that |I | > 0. This implies that the Lebesgue measureis absolutely continouous w.r.t. dρ. As a consequence, R is a minimal support ofthe absolutely continuous part of dρ. Indeed, if B ⊂ R satisfies |B| > 0 and if 6denotes a support of the singular part of dρ such that |6| = 0, we have |B \6| > 0and thus ρac(B) � ρac(B \ 6) = ρ(B \ 6) > 0. This completes the proof of thetheorem. ✷

6. Number of Eigenvalues of H (α)

In order to prove Theorem 1.4, we have to study the solutions of the differentialsystem(

0 1−1 0

)u′(x)+

(p1(x) q(x)

q(x) p2(x)

)u(x) = λu(x). (6.1)

To this end, we will use the Prüfer variables R and ϕ given by

u(x) = R(x)

(cos ϕ(x)sin ϕ(x)

). (6.2)

Straightforward computations show that (6.1) and (6.2) are equivalent to

R′(x)R(x)

= q(x) cos 2ϕ(x) + 12 (p2(x) − p1(x)) sin 2ϕ(x),

ϕ′(x) = λ− q(x) sin 2ϕ(x) + 12 (p2(x) − p1(x)) cos 2ϕ(x)

− 12 (p2(x)+ p1(x)).

(6.3)

Before giving the proof of Theorem 1.4, we recall the following two elementarylemmas (see [8] for the proofs).

396 MATHIEU MARTIN

LEMMA 6.1. Let f, g ∈ C([1,∞)) such that g is real-valued and |g′f | + |f ′| ∈L1. Then for each λ ∈ R \ {0} the integral∫ B

1f (x)ei(λx+g(x)) dx

is bounded as B → ∞.

LEMMA 6.2. Let {ei}Ni=1 a family of unit vectors in a Hilbert space H such thatα ≡ N supi �=j 〈ei, ej 〉 < 1. Then for each g ∈ H we have

N∑i=1

|〈g, ei〉|2 � (1 + α)‖g‖2.

Proof of Theorem 1.4. Assume thatH(α) hasN distinct eigenvalues λ1, . . . , λN .We have to prove that N � 4C2. Denote by Rn (resp. ϕn) the R (resp. ϕ) corre-sponding to λ = λn in (6.2). We will normalize Rn by Rn(0) = 1. Accordingto (6.2), λn is an eigenvalue if and only if R2

n ∈ L1. In particular,

limx→∞

x ·N∑n=1

R2n(x) = 0.

Hence, there exists a sequence Bj → ∞ such that Rn(Bj) � B−1/2j . Then, us-

ing (6.3) we get∫ Bj

0{q(y) cos 2ϕn(y) + 1

2(p2(y) − p1(y)) sin 2ϕn(y)} dy

=∫ Bj

0

d

dy(lnRn(y)) dy � − 1

2 lnBj . (6.4)

Let us set

Hj = L2((0, Bj ), (1 + x) dx)2 and :(x) = (q(x), 12 (p2(x) − p1(x))).

Then using (1.3) we see that

‖:‖2Hj

�∫ Bj

0C2(1 + x)−2(1 + x) dx

� C2 lnBj + O(1).

Let

e(j)n (y) = 1√N(j)

(cos 2ϕn(y)

1 + y,

sin 2ϕn(y)

1 + y

). χ[0,Bj ](y),

where

N(j) =∫ Bj

0

dy

1 + y= ln(1 + Bj) = lnBj + O(1). (6.5)


The vectors {e(j)n }Nn=1 are unit vectors of Hj . On the other hand, for n �= m we have

〈e(j)n , e(j)m 〉Hj= O((lnBj)

−1). (6.6)

Indeed, for n �= m

N(j)〈e(j)n , e(j)m 〉Hj

=∫ Bj

0

cos(2ϕn(y)) cos(2ϕm(y)) + sin(2ϕn(y)) sin(2ϕm(y))

1 + ydy

=∫ Bj

0

cos{2(λn − λm)y + 2[ϕn(y)− ϕm(y) − (λn − λm)y]}1 + y

dy.

Then by setting

f (y) = 1

1 + y,

g(y) = 2[ϕn(y) − ϕm(y)− (λn − λm)y],and using (6.3) we see that Lemma 6.1 applies to f and g because λn − λm �= 0.This means that N(j)〈e(j)n , e

(j)m 〉Hj

= O(1) and (6.5) ensures (6.6). But N is fixed

and Bj → +∞, then one can apply Lemma 6.2 to {e(j)n }Nn=1 for sufficiently large j :

N∑n=1

|〈:, e(j)n 〉Hj|2 � (1 + O((ln Bj)

−1))‖:‖2Hj.

To be able to conclude the proof we need to estimate 〈:, e(j)n 〉Hj. According to (6.4),

〈:, e(j)n 〉Hj� − 1

2(N(j))1/2 ln Bj � − 12 (ln Bj)

1/2 + O(1).

Thus14N lnBj � C2 lnBj + O(1),

and so N � 4C2. ✷

Acknowledgements

This work is a part of my PhD thesis. I would like to thank my advisor, A. Boutetde Monvel, for her support, and J. Sahbani and L. Zielinski for useful discussionsduring the preparation of this work. I also express my gratitude to D. Gilbert for herhospitality at the Dublin Institute of Technology and to S. Naboko for his invitationat the Banach center in Warsaw where this work was completed.

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Mathematical Physics, Analysis and Geometry 6: 399–400, 2003. 399

Contents of Volume 6 (2003)

Volume 6 No. 1 2003

MARÍA LAURA BARBERIS / Hyper-Kähler Metrics Conformal toLeft Invariant Metrics on Four-Dimensional Lie Groups 1–8

HSUNGROW CHAN / Embedding Misner and Brill–Lindquist Ini-tial Data for Black-Hole Collisions 9–27

S. V. BREIMESSER and D. B. PEARSON / Geometrical Aspects ofSpectral Theory and Value Distribution for Herglotz Functions 29–57

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GREGORIO FALQUI and MARCO PEDRONI / Separation of Vari-ables for Bi-Hamiltonian Systems 139–179

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L. BERTINI, A. DE SOLE, D. GABRIELLI, G. JONA-LASINIO andC. LANDIM / Large Deviations for the Boundary DrivenSymmetric Simple Exclusion Process 231–267

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