Mathematical Physics, Analysis and Geometry - Volume 3

Mathematical Physics, Analysis and Geometry3: 1–31, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

1

Asymptotic Distribution of Eigenvalues of WeaklyDilute Wishart Matrices

A. KHORUNZHY1 and G. J. RODGERS21Institute for Low Temperature Physics, Kharkov 310164, Ukraine.e-mail: [email protected] of Mathematics and Statistics, Brunel University, Uxbridge, Middlesex UB8 3PH,U.K. e-mail: [email protected]

(Received: 30 September 1998; in final form: 18 October 1999)

Abstract. We study the eigenvalue distribution of large random matrices that are randomly diluted.We consider two random matrix ensembles that in the pure (nondilute) case have a limiting eigen-value distribution with a singular component at the origin. These include the Wishart random matrixensemble and Gaussian random matrices with correlated entries. Our results show that the singularityin the eigenvalue distribution is rather unstable under dilution and that even weak dilution destroysit.

Mathematics Subject Classifications (2000):15A52, 82B44.

Key words: matrices, random, dilute, Wishart.

1. Introduction

Random matrices of large dimensions play a central role in a number of theoreticalphysics applications, such as statistical nuclear physics, solid state physics, statis-tical mechanics, including neural network theory and quantum field theory (see,e.g., the monographs and reviews [8 – 10, 13, 24, 27]). In this work, most interestis attached to the various ensembles of random matrices whose entries are all of thesame order of magnitude. This corresponds to the situation when the elements of asystem all strongly interact with one another. In the last decade, however, systemsin which some of the links between the different elements are broken have beenstudied in a variety of applications. This effect is particularly important in neuralnetwork theory where the total number of neurons is several orders of magnitudegreater than the average number of connections per neuron [2, 5, 13].

Such matrices known as dilute (or sparse) ones are also important in otherapplications, such as the theory of random graphs and linear programming [21].

2 A. KHORUNZHY AND G. J. RODGERS

1.1. STRONG DILUTION AND SEMICIRCLE LAW

In papers [17, 19] we have studied the limiting eigenvalue distribution of large ran-dom matrices that are strongly diluted. These are determined as theN-dimensionalmatrices that have, on average,pN nonzero elements per row and 1� pN � N

asN → ∞. We proved that under natural conditions the limiting eigenvalue dis-tribution of strongly dilute random matrices exists and coincides with the Wigner’sfamous semicircle law [32]. The semicircle law is also valid when the entries ofthe dilute random matrix are statistically dependent random variables [16, 20].This case is of special interest in applications (see, for instance, [1, 7, 10, 13]).In the pure (nondilute) regime these matrices have singularities in the eigenvaluedistribution. The strong dilution removes this singularity because the density of thesemicircle distribution is bounded. It should be noted that the Wigner’s semicircledistribution is typical for large random matrices with jointly independent entries.Therefore we have conjectured that the semicircle law arises in the ensembles of[16, 20] because the strong dilution eliminates the statistical dependence betweenrandom matrix entries. The same reasoning can explain the disappearance of thesingularity in the eigenvalue distribution. However, the last conjecture is not true.

1.2. WEAKLY DILUTE WISHART MATRICES

To study the transition to the semicircle law under dilution, we pass to the caseof weakly dilute random matrices. This means that we are now interested in theasymptotic regime whenpN = qN , q > 0 asN →∞. In this case the statisticaldependence between random matrix entries persists, provided it exists in the pure(nondilute) ensemble. We consider two random matrix ensembles with differenttypes of statistical dependence between the entries. These are the Wishart randommatricesHN and Gaussian random matricesAN with correlated entries. The firstensemble is widely known in multivariate statistical analysis (see, e.q., [1]). Recentapplications of these matrices are related with the the theory of disordered spinsystems of statistical mechanics [14, 26, 24] and learning algorithms of memorymodels of neural network theory [2, 13, 14]. The entries{HN(x, y)} are statisticallydependent (but uncorrelated) random variables. The degree of the dependence be-tweenHN(x, y) andHN(s, t) does not relate to the ‘distance’|x − s| + |y − t|. Incontrast, correlations between matrix elementsAN(x, y) andAN(s, t) in the sec-ond ensemble we consider decay when the distance between them increases. Dueto this property,{AN } can be regarded as the ensemble intermediate betweem ran-dom matrix models with strongly correlated entries (see, e.g., [7, 10]) and randommatrices with independent entries.

1.3. MAIN RESULTS AND STRUCTURE OF ARTICLE

We study the limit of the normalized eigenvalue counting function of weakly dilutereal symmetric matricesHN andAN asN →∞. We derive explicit equations for

EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES 3

the Stieltjes transform of the limiting distribution functions and determine recurrentequalities for their moments. Basing on these relations, we study the properties ofthe limiting eigenvalue distributions.

We show that both these distributions are different from the semicircle law. Weprove that, nevertheless, the singularity disappears from the limiting eigenvaluedensity and that this happens for arbitrary values ofq < 1.

Thus, our principal conclusion is that the singularity of the eigenvalue distrib-ution is rather unstable under dilution and is destroyed even when this dilution isweak.

To complete this introductory section, let us note that our results can be regardedas generalizations of the statements proved for strongly dilute random matrices in[19] and [20]. In this paper we use the technique developed in [20]. However, thecase of weak dilution studied here is more complicated and requires more accurateanalysis.

The paper is organised as follows. In Section 2 we present our main resultsfor the Gaussian random matricesAN with correlated entries. In Section 3 weconsider the weak dilution of the Wishart matricesHN . We prove the existence ofthe limiting eigenvalue distributionsσ (i), i = 1,2 in the limitN →∞ (Theorems2.1 and 3.1). At the second part of each of these sections we formulate Theorems2.2 and 3.2 concerning the properties of the respectiveσ (i). In Sections 4 and 5we derive the main equations that determine the eigenvalue distributions ofANandHN , respectively, asN → ∞. Section 6 contains the proofs of Theorems 2.2and 3.2. We also present there the recurrent relations for the moments ofσ (i) andmake conclusions about the support of the measure dσ (i). In Section 7 we give asummary of our results.

2. Gaussian Random Matrices with Correlated Entries

Let us considerN ×N symmetric random matrices

AN(x, y) = 1√Na(x, y), x, y = 1, . . . , N, (2.1)

where random variables{a(x, y), x 6 y, x, y ∈ N} have a joint Gaussian distrib-ution. We assume that{a(x, y)} satisfy the following conditions:

Ea(x, y) = 0, (2.2a)

Ea(x, y)a(s, t) = V (x − s)V (y − t)+ V (x − t)V (y − s), (2.2b)

where the signE represents the mathematical expectation with respect to the mea-sure generated by thea(x, y)’s andV (x) is a nonrandom function such thatV (−x)= V (x) andV is nonnegatively defined. Then the right-hand side satisfies condi-tions for the covariance of random variables (see Lemma 4.5 at the end of Sec-


tion 4). The eigenvalue distribution of the random matrix ensemble (2.1), (2.2),whereV (x) satisfies the condition∑

x

|V (x)| ≡ Vm <∞, (2.3)

was studied in [18]. This case is known as weakly correlated random variables.Indeed, condition (2.3) implies decay of the correlations between random matrixentriesAN(x, y) andAN(s, t) that are situated far enough from each other in thematrix.

2.1. ENSEMBLE AND MAIN EQUATIONS

In this section we consider the ensemble of real symmetric random matrices

A(q)

N =1√Na(x, y)πxy, x, y = 1, . . . , N, (2.4)

wherea(x, y) are the same as in (2.1) and the random variables{πxy, x 6 y} areboth independent between themselves and independent from{a(x, y)}. We assumethatπyx = πxy and the random variables have the common probability distribution

πxy = 1√q

{1, with probabilityq,0, with probability 1− q.

(2.5)

Our main result concerns the normalized eigenvalue counting function (NCF) ofA(q)

N given by the formula

σ(λ;A(q)N

) = #{λ(N)j 6 λ

}N−1, (2.6)

whereλ(N)1 6 · · · 6 λ(N)N are eigenvalues ofA(q)N .

THEOREM 2.1. Assume that(2.3)holds. Then

(i) givenq ∈ (0,1) the NCFσ(λ;A(q)N ) weakly converges in probability asN →∞ to a nonrandom functionσq(λ);

(ii) the Stieltjes transformfq(z) =∫(λ − z)−1dσq(λ), Imz 6= 0 can be found

from the the system of equations

fq(z) =∫ 1

0gq(p; z)dp, (2.7a)

gq (p; z) =[− z− (1− q)v2fq(z)− qV (p)

∫ 1

0V (r)gq(r; z)dr

]−1

, (2.7b)

where

V (p) =∑x∈Z

V (x)E{2πipx},


and

v ≡ V (0) =∫ 1

0V (p)dp;

(iii) system(2.7) is uniquely solvable in the class0 of functionsg(p; z), p ∈(0,1), z ∈ C± analytical in this region and such that

Im g(p; z)Im z > 0, z ∈ C±. (2.8)

Remarks.(1) Here and below we mean by the weak convergence of nonnega-tive nondecreasing functionsσ(λ;AN) the weak convergence of the correspondingmeasures

limN→∞

∫ ∞−∞

ϕ(λ)dσ(λ;AN) =∫ ∞−∞

ϕ(λ)dσ(λ), ϕ ∈ C∞0 (R).

Generally the convergence of integrals can be regarded as convergence in average,in probability or with probability 1.

(2) Each functiong ∈ 0 determines a nonnegative nondecreasing functionσ (λ)

such that [11]g(z) = ∫ (λ− z)−1 dσ (λ) and

σ (a)− σ (b) = limη↓0

1

π

∫ b

a

Img(µ+ iη)dµ. (2.9)

Relation (2.9), known as the inversion formula for the Stieltjes transform, is validfor all a, b such thatσ is continuous at these points.

Theorem 2.1 is proved in Section 4. Basing on (2.7), one can study the proper-ties ofσq(λ).

2.2. LIMITING EIGENVALUE DISTRIBUTION

To discuss the consequences of Theorem 2.1, let us note first that relations (2.7)considered withq = 0 can be reduced to the equation

f0(z) = 1

−z− v2f0(z). (2.10)

This equation is uniquely solvable and determines the Wigner semicircle distribu-tion σ0(λ) [32] with the density

%0(λ) ≡ σ ′0(λ) =1

2πv

{√4v2− λ2, if |λ| 6 2v,

0, otherwise.(2.11)

This observation shows that Theorem 2.1 generalizes the results of paper [20],where the eigenvalue distribution of the ensemble (2.4) has been studied under con-dition that random variablesπxy are replaced byπ(N)xy that take valuesN1/2p−1/2

with probabilityp/N and 0 with probability 1− p/N . Then the limitp,N →∞


considered in [20] corresponds to subsequent limiting transitionsN → ∞ andq → 0. Another limiting transitionq → 1 in (2.7) leads to equations

f1(z) =∫ 1

0g1(p; z)dp,

g1(p; z) =[− z− V (p)

∫V (r)g1(r; z)dr

]−1

.

This system has been derived in [18] the the limitf1(z) of the Stieltjes transformsof the NCFσ(λ;AN) of the ensemble determined by (2.1), (2.2), and (2.3) (seealso [3, 4] for others and more general ensembles).

It should be noted that if∫ 1

0

dp

V (p)= ∞, (2.12)

then corresponding tof1(z), measureσ1(dλ) has an atom at the origin. Indeed, onecan easily derive from (2.7) and (2.9) that if (2.12) holds, then

limε↓0

Imf1(iε) = ∞(see also [4]).

Relation (2.12) is known in stochastic analysis as the interpolation condition forthe infinite sequence of Gaussian random variables{γ (x), x ∈ Z} with zero aver-ageEγ (x) = 0 and covarianceEγ (x)γ (y) = V (x − y) [22]. If (2.12) holds, thenthe sequence{γ (x)} can be regarded as insufficiently random. Apparently, randomdilution makes the sequence of such random variables ‘more disordered’. Thisobservation can be regarded as a heuristic explanation of the following proposition.

THEOREM 2.2. If q < 1, then the density%q(λ) = σ ′q(λ) exists and is boundedeverywhere by1/(πv

√1− q).

Theorem 2.2 is proved in Section 6. There we also derive and analyse recurrentrelations for the momentsLk =

∫λkdσq(λ). Basing on these relations, we study

the support of the measure dσq(λ).

3. Weak Dilution of Wishart Matrices

In this section we study the eigenvalue distribution of symmetric matrices

H(q)

N (x, y) = 1

N

N∑µ=1

ξµ(x)ξµ(y)πxy, x, y = 1, . . . , N, (3.1)

where{ξµ(x), x, µ ∈ N} are independent random variables having joint Gaussiandistribution. We assume that these random variables satisfy conditions

Eξµ(x) = 0, Eξµ(x)ξν(y) = δxyδµν, (3.2)


whereδxy is the Kroneckerδ-symbol;

δxy ={

1, if x = y,0, if x 6= y.

We assume also thatπxy = πyx and{πxy, x 6 y}

are i.i.d. random variables (in-dependent of{Hµ}) that have probability distribution (2.5). Thus, (3.1) representsthe weak dilution of random matrices

HN,m(x, y) = 1

N

m∑µ=1

ξµ(x)ξµ(y), x, y = 1, . . . , N (3.3)

known since 30s in the multivariate statistical analysis as the Wishart matrices [1].Being at present of considerable importance in this field, the ensemble (3.3) isextensively studied in the statistical mechanics of the disordered spin systems (see,e.g., [6, 26, 31] for rigorous results). Another important application of (3.3) isrelated with the neural network theory, whereHN are used as the interation ma-trix of learning algorithms modelling auto-associative memory. In this approach,N-dimensional vectorsξµ are regarded as the patterns to be memorised by thesystem [13]. Dilution versions of (3.3) are important in this field of applications asthe models that can be tuned to give more precise correspondence with real systems(see, e.g., [2]). These models are mostly studied in the regime of strong dilution[5, 30]. The following statement concerns the normalized eigenvalue countingfunction (2.6) of of weakly dilute random matrices (3.1), (3.2).

THEOREM 3.1. For each fixedq ∈ (0,1) the NCFσ(λ;H(q)

N ) converges in thelimit N,m → ∞,m/N → c > 0 to a nonrandom functionσq,c. The Stieltjestransformfq,c(z) of σq,c(λ) satisfies equation

fq,c(z) =[− z− cu4(1− q)fq,c + cu2√q

1+ u2√qfq,c(z)]−1

. (3.4)

This equation is uniquely solvable in the class of functions0 determined in Theo-rem2.1and satisfying(2.8).

We prove Theorem 3.1 in Section 5. Regarding (3.4), one can easily observethat, in complete analogy with (2.7), this equation determines a ‘mixture’ of twoequations: the one for the semicircle distribution withq = 0 (cf. with (2.10))

f0,c(z) = 1

−z− cu4f0,c(z)

and the equation (q = 1)

f1,c(z) =[− z+ cu2

1+ u2f1,c(z)

]−1

(3.5)


derived in [23] for the Stieltjes transform ofσ(λ;HN,m) in the limit N,m → ∞,m/N → c > 0.

Corresponding to (3.5) eigenvalue distribution has the density given by theformula [23]

dσ1,c(λ)

dλ= [1− c]+δ(λ)+ 1

2πu2λ

√4cu2 − (λ− (c + 1)u2

)2, (3.6)

where[x]+ = max(0, x) andδ(x) is the Dirac delta function. Let us stress thatif c < 1, then the density ofσ1,c(λ) has the singular component at the origin.The following statement shows that this singularity disappears in the weak dilutionregime.

THEOREM 3.2. If q < 1 then the density ofσq,c(λ) determined byfq,c(z) (3.4)is bounded from above by1/u2√c(1− q) for all c > 0.

We prove Theorem 3.2 in Section 6. In this section we also derive recurrentrelations for the momentsLk =

∫λkdσq,c(λ).

4. Proof of Theorem 2.1

In this section we use the resolvent method developed in a series of papers (see,e.g., [18]) and improved in [15]. This method is based on the derivation of relationsfor the moments of the normalized trace of the resolvent of random matrixAN

fN(z) ≡ 1

NTrGN(z) = 1

N

∑x=1

GN(x, x; z), GN(z) = (AN − z)−1. (4.1)

The important and in certain sense characteristic property of random matrices isthatEfN(z) converges asN →∞ to the variablef (z) and one can derive closedequations for it. The variance offN(z) vanishes asN → ∞ that means that thenormalized trace (4.1) is the self-averaging random variable.

The case of weakly diluted random matricesAN,q (2.7) is more complicated.The averageEGN(x, y) is expressed in terms of the generalized ‘trace’

TN(x, y) = 1

N

N∑r,s=1

πxrG(r, s)Vrsπsy, (4.2)

where we denoted

G(x, y) ≡ G(q)

N (z) = (AN,q − z)−1

andVxy ≡ V (x − y). Relations for the limit ofEGN(x, y) involve the limit of theaverageETN(x, y).

Our main observation is that this pseudo-traceTN(x, y) is also the selfaveragingvariable asN →∞. We obtain expression fort (x, y) = limN→∞ ETN(x, y).


In contrast with the strong dilution, matrixt (x, y) is nonzero and its entries taketwo different values depending on whetherx = y or not. These two limitst1 andt2 involve explicitly parameterq andt1 − t2 vanishes forq → 1. This determinesthe difference between equations that we obtain for weakly dilute random matricesand those derived for the corresponding pure (nondilute) ensemble.

To make the derivation self-consistent, we recall the basic elements of the methoddeveloped in [15, 18].

Given two symmetric (or Hermitian) operatorsH and H acting in the samespace, the resolvent identity holds forG = (H − z)−1 andG = (H − z)−1

G− G = −G(H − H )G. (4.3)

This identity leads to two important observations; that the dependence of the re-solvent on the random matrix can found explicitly and that this dependence isexpressed in terms of the resolvent (see formulas (4.7) and (4.9)). These two prop-erties make the resolvent approach a fairly powerful tool in the spectral theory ofrandom matrices.

We prove Theorem 2.1 by showing the convergence of the tracefN,q(z) =N−1TrG(q)

N (z);

limN→∞

E〈fN,q(z)〉 = fq,V (z), (4.4)

and

limN→∞

E⟨|fN,q(z)− E〈fN,q(z)〉|2

⟩(4.5)

for all

z ∈ 3q ={z ∈ C, |Imz| > 2Vmq

−1 + 1}, (4.6)

where we denoted by〈·〉 the mathematical expectation with respect to the measuregenerated by random variables{πxy}.

On this way, we derive equation forfq,V (z) that determines the limiting eigen-value distribution. Taking into account that the normalized trace of the resolvent(4.1) is the Stieltjes transform of the NCF, we conclude that relations (4.4) and (4.5)imply the weak convergence in probability ofσ(λ;AN,q), in the limitN →∞, toa nonrandom functionσq(λ; V ). This can be proved by the usual arguments of thetheory of Herglotz functions. The reasonong is based on weak compactness of thefamily σ(λ;AN,q) and the Helly theorem [11] (see, e.g., [18] for more details).

Thus, relations (4.4) and (4.5) prove items (i) and (ii) of Theorem 2.1. Item (iii)is proved in Lemma 4.4 (see the end of this section).

We split the remaining part of this section into three subsections. In the firstone we derive main relations that lead to the equation forfq,V (z). In the secondsubsection we derive relations leading to the proof of (4.5). The third subsectioncontains the proofs of the auxiliary facts and estimates.


4.1. DERIVATION OF MAIN RELATIONS

Let us consider identity (4.3) withH = AN,q andH = 0;

E〈G(x, y)〉 = ζδxy − ζN−1/2

⟨∑s

EG(x, s)a(s, y)π(s, y)⟩, (4.7)

whereζ = −z−1. In (4.7) and below, we omit the varablez and subscriptsq,Nand do not indicate limits of summations, if no confusion can arise.

To compute the averageEG(x, s)a(s, y), we use the following elementary facts(see also [18]). It is related to the Gaussian random vectorγ = (γ1, . . . , γk) withzero average:

EγjF (γ1, . . . , γk) =k∑l=1

EγjγlE{∂F

∂γl

}, (4.8)

whereF is a nonrandom function such that all integrals in (4.8) exist. This formulacan be proved by using the integration by parts technique.

We will also use the formula that is a direct consequence of identity (4.3);

∂G(x, s)

∂a(p, r)= − 1√

NG(x, p)G(r, s)πpr . (4.9)

Now we can write that

EG(x, s)a(s, y) = − 1√N

∑x,p

[VpsVry + VpyVrs

]EG(x, p)G(r, s)πpr .

Substituting this relation into (4.7), we obtain equality

E〈G(x, y)〉 = ζδxy + ζ∑p

E〈G(x, p)T (p, y)〉Vpy + ζE⟨ψ(1)N (x, y)

⟩, (4.10)

whereT is determined in (4.2) and we denoted

ψ(1)N (x, y) =

1

N

∑p,r,s

G(x, p)πprG(r, s)VpsπsyVry. (4.11)

In the last part of this section we prove thatψ(1)N vanishes in the limitN →∞ (see

Lemma 4.1).Turning back to (4.7), we can write for the averageE〈G(x, y)〉 ≡ g(x, y) the

following relation

g(x, y) = ζδxy + ζ∑p

g(x, p)t (x, y)Vpy + ζE⟨ψ(1)N (x, y)

⟩ ++ ζψ(2)

N (x, y), (4.12)


wheret (x, y) ≡ E〈T (x, y)〉 and

ψ(2)N (x, y) =

∑p

{E〈G(x, y)T (x, y)〉 − g(x, y)t (x, y)}Vpy.

Given a random variableγ with finite mathematical expectation, let us introducethe centered random variable

γ 0 ≡ γ − Eγ.

We will also use denotation[γ ]0 for more complex expressions. In what follows(see Subsection 4.2), we prove that

limN→∞ sup

x,y

E⟨|G0(x, y)|2⟩ = 0 for all z ∈ 3q. (4.13)

Let us note that the last inequality of (4.11) implies that supx,y |T (x, y)| 6Vmq

−1η−1. This estimate together with (4.13) leads to relation

limN→∞ sup

x,y

∣∣ψ(2)N (x, y)

∣∣ = 0. (4.14)

It should be noted also that (4.13) implies (4.5).To derive the final equation (2.7b) from (4.12), it remains to compute the limit

of the averaget (x, y) = E〈T (x, y)〉. Basing on (4.2), one can write that

E〈T (x, y)〉 = 1

N

∑r,s

〈πxrπsy〉g(r, s)Vrs + ψ(3)N (x, y), (4.15)

where the term

ψ(3)N (x, y) =

1

N

∑r,s

E⟨πxrG

0(r, s)πsy⟩Vrs

vanishes due to the self-averaging property (4.13).Examining the first term on the right-hand side of (4.15), we observe that it

depends only on whetherx = y:

1

N

∑r,s

〈πxrπsy〉g(r, s)Vrs ={t1, if x = y,t2, if x 6= y,

where

t1 = (1− q)gV0+ qgV , t2 = qgV ,and

g = 1

N

N∑x=1

g(x, x), gV = 1

N

N∑x,y=1

g(x, y)Vxy .


Let us consider the equation

g(x, y) = ζδxy + ζ(t1− t2

)g(x, y)V (x, y) + ζ t2

N∑p=1

g(x, p)Vpy, (4.16)

wherex, y = 1, . . . , N and ti are determined by the same formulas asti withg(x, y) replaced byg(x, y).

It is not hard to prove that (4.16) has a unique solution forz ∈ 3q (4.6) (seeLemma 4.3 at the end of this section). Also it is easy to see that

limN→∞

1

N

N∑x=1

[g(x, x) − g(x, x)] = 0. (4.17)

Some elementary calculations based on the finite-difference form ofVxy =V (x − y) show that Equation (4.16) leads in the limitN →∞ to Equation (2.7b)and therefore

limN→∞

1

N

N∑x=1

g(x, x) = fq,V (z),

wherefq,V (z) is given by (2.7a). Relation (4.4) is proved.

4.2. SELFAVERAGING PROPERTY

Let us prove relation (4.13) that obviously implies (4.5). We consider, at the sametime withG(x, y) ≡ G(x, y; z) the resolventG′(x, y) ≡ G(x, y; z′), and studythe average

SN(x, y) = E⟨G0(x, y)G′(x, y)

⟩. (4.18)

Loosely speaking, the main idea is to derive relation of the formSN(x, y) =B(N)S (x, y) + 8N(x, y) whereB is certain expression involvingSN and term8N

vanishes asN → ∞. The crucial observation used in this approach is thatB(N)S

can be estimated bySN itself multiplied by coefficients depending onη = |Imz|−1

(see, e.g., [15]). This lead to relations (4.13) providedη is large enough.

4.2.1. Selfaveraging ofS(x, y)

Applying (4.3) to the last factor of (4.18), we obtain equality

S(x, y) = −ζ ′N−1/2∑t

E⟨G0(x, y)G′(x, t)a(t, y)πty

⟩,

whereζ ′ = −1/z′. Using (4.8) and (4.9), we derive relation

S(x, y) = ζ ′∑p

E⟨G0(x, y)G′(x, p)T ′(p, y)

⟩Vpy + 0N(x, y). (4.19)


Here we denoted

0N(x, y) = ζ ′E⟨G0(x, y)φ′N (x, y)

⟩ ++ 1

N

∑p,r,t

{G(x, p)πprG(r, y)G

′(x, t)πt,y}(VptVry + VpyVrt ),

whereφ′N(x, y) is given by (4.11) withG replaced byG′.Using identity

E⟨ξ0

1ξ2ξ3⟩ = E

⟨ξ0

1ξ2⟩E〈ξ3〉 + E

⟨ξ0

1ξ2ξ03

⟩,

we can rewrite (4.19) in the form

S(x, y) = ζ ′∑p

E⟨[G(x, y)]0T ′(p, y)⟩E⟨G′(x, p)⟩Vpy +

+ ζ ′∑p

E⟨[G(x, y)]0[G′(x, y)]0T ′(p, y)⟩Vpy + ζ ′0N(x, y). (4.20)

Repeating the reasoning used to estimate (4.11) (see Lemma 4.1), one can easilyshow that forz, z′ ∈ 3q (4.6)

|0N | ≡ supx,y

|0N(x, y)| = O(N−1/2+ N−1/2|SN |

),

where

|SN | ≡ supx,y

|SN(x, y)|.

Substituting this inequality into (4.20), we obtain that forz′ = z|SN |2 6 |SN ||UN |η−2Vm + q−1η−2V 2

m|SN |2+ η−1|0N |, (4.21)

where

|UN | ≡ supx,y

(E⟨|T 0(x, y)|2⟩)1/2.

Now it is clear that (4.21) implies (4.13) provided the estimate

|UN | = o(1) asN →∞ (4.22)

is true.

4.2.2. Selfaveraging ofT (x, y)

To prove (4.23), we treat the average

UN(x, y) ≡ E⟨T 0(x, y)T ′(x, y)

⟩ = E⟨T 0(x, y)

1

N

∑r,s

πxrG′(r, s)Vrsπsy

⟩


by the same procedure as is used to studySN(x, y). We apply toG′(r, s) theresolvent identity (4.3) and use (4.8) and (4.9) to compute the mathematical ex-pectations. As a result, we obtain relations

UN(x, y) = ζ ′E⟨T 0(x, y)ρ(x, y)

⟩V0+

+ ζ ′N−1E⟨T 0(x, y)

∑p,r,s

πxrVrsG′(r, p)T ′(p, s)Vpsπs,y

⟩+

+ ζ ′θ(1)N (x, y), (4.23)

where we have denoted

ρ(x, y) = 1

N

∑r

πxrπry

and

θ(1)N (x, y) = 1

N2

∑α,β,p,r,s

πxαG(α, l)πljG(j, β)πβy ×

× [VplVsj + VpjVsl]πxrG′(r, p)πpsVrsπsy.Standard computations (see, e.g., Lemma 4.1) show that∣∣θ(1)N ∣∣ ≡ sup

x,y

∣∣θ(1)N (x, y)∣∣ 6 2V 4

m

q3η3.

Let us rewrite (4.23) in the following form

UN(x, y) = ζ ′E⟨T 0(x, y)ρ(x, y)

⟩V0+

+ ζ ′N−1E⟨T 0(x, y)

∑p,r,s

πxrVrsG′(r, p)[T ′(p, s)]0Vpsπs,y

⟩+

+ ζ ′N−1E⟨T 0(x, y)

∑p,r,s

πxrVrs[G′]0(r, p)Vpsπs,y〉E〈T ′(p, s)⟩+

+ ζ ′N−1E⟨T 0(x, y)

∑p,r,s

πxrVrsVpsπs,y

⟩E〈G′(r, p)〉E〈T ′(p, s)〉 +

+ ζ ′θ(1)N (x, y).

Then using a-priori estimate|T (x, y)| 6 Vm/(qη) and takingz′ = z, we deriveinequality

|UN |2 6 vVm

qη2supx,y

(E[ρ0(x, y)

]2)1/2+ V 2m

qη2|UN |2+

+ V 3m

q2η2|UN | |SN | + |UN |

⟨N−1

∑r,s

[πxrπsy]0Vrs⟩1/2

+

+ η−1∣∣θ(1)N ∣∣. (4.24)


Now we use elementary estimates

supx,y

(E⟨∣∣ρ0(x, y)

∣∣2⟩)1/2 = O(N−1/2

)and

E

∣∣∣∣( 1

N

∑r,s

πxrVrsπsy

)0∣∣∣∣2 = O(N−1

)and derive from (4.24) that

|UN |2 6 V 2m

qη2|UN |2+ V 3

m

q2η2|UN ||SN | +O

(N−1/2).

This inequality regarded jointly with (4.21) forz ∈ 3q (4.6) implies (4.22). Rela-tion (4.5) is derived.

We complete this section with the outline of the proofs of the following auxiliarystatements.

4.3. PROOF OF THE AUXILIARY FACTS

LEMMA 4.1. Relation

supx,y

|φN(x, y)| = O(V 2mη−2q−1N−1/2

). (4.25)

holds with probability1.Proof.Taking into account that|πxy| < q−1/2 with probability 1, one can write

the inequality∣∣ψ(1)N (x, y)

∣∣ 6 1

qN

∑p,r,s

|G(x, p)|VpsG(r, s)Vry |.

Let us look at|G(x, p)| ≡ Gx(p) and|G(s, r)| ≡ Gr(s) for given fixedx andr asvectors inl2(N) and observe that the norm of the linear operatorV in l2(N) withthe kernelV (p − s) = |Vps| is bounded byVm (2.3). Then using the estimate∑

p

|G(x, p)|2 = |Gex |2 6 |G|2 6 η−2, (4.26)

where we introduced the unit vectorsex with componentsex(j) = δxj , we obtainthat

|φN(x, y)| 6 Vm

qηN

[∑s

(∑r

|G(s, r)| · |Vry|)2]1/2

6 V 2m

qηN

[∑s

∑r

|G(s, r)|2]1/2

.


This estimate, together with inequalities (4.26) leads us to (4.25). 2LEMMA 4.2. Equation(4.16) has a unique solution in the classG of matrices{g(x, y; z), x, y = 1, . . . , N} such that

supx,y

∣∣g(x, y; z)∣∣ 6 2η−1, η = |Imz| > 2Vm. (4.27)

Proof. The proof is based on the use of the method of subsequent approxima-tions. We introduce the sequence of matrices{g(k)(x, y)}, k = 1,2, . . . by therelations

g(1)(x, y) = ζδxy, ζ = −z−1,

g(k+1)(x, y) = ζδxy + ζ t(k)2

[g(k)V

](x, y) +

+ ζ (t (k)1 − t (k)2

)g(k)(x, y)V (x, y),

where

t(k)1 = V0N

−1Tr g(k) + q[N−1Tr(g(k)V

)− V0N−1Tr g(k)

]and

t(k)

2 = qN−1Tr g(k)V .

It is easy to see that ifg(k) satisfies (4.27), theng(k+1) also satisfies (4.27). Thensimple computation shows that there exists suchα(z) that

1k+1 ≡ supx,y

∣∣g(k+1)(x, y) − g(k)(x, y)∣∣ < α(z)1k

and|α(z)| < 1 for z ∈ 1q . This completes the proof of the Lemma 4.2. 2LEMMA 4.3. Relation(4.17)holds forz ∈ 3q (4.6).

Proof. Subtracting (4.17) from (4.15), we obtain for the differenceδg(x, y) =g(x, y) − g(x, y) relation

δg(x, y) = ζδ(2)t

[gV](x, y) + ζ t2

[δgV

](x, y) +

+ ζ (δ(1)t − δ(2)t )g(x, y)V (x, y) + ζ (t1− t2)δg(x, y)V (x, y) ++ ζψ ′N(x, y), (4.28)

where

δg(x, y) = g(x, y) − g(x, y),δ(1)t = N−1Trδg + q

[N−1Tr

(δgV

)− V0N−1Trδg

],

δ(2)t = qN−1TrδgV


and

supx,y∣∣ψ ′N(x, y)∣∣ = O(1) asN →∞. (4.29)

Now we regard (4.28) as the matrix relation

δg = Lζ,V δg + ζδ(2)t gV + ζ(δ(1)t − δ(2)t

)g(x, y)V (x, y) + ζψ ′N,

whereL is a linear operator

(Lζ,V δ)(x, y) = ζ t2∑p

δ(x, p)V (p, y)+ ζ(t1− t2)δ(x, y)V (x, y).

It is easy to show that for large enough values of|Imz| there existsβ(z) < 1 suchthat

‖Lζ,V ‖1 < β(z), (4.30)

where the norm‖ · ∥∥1 is determined as

‖δ‖1 = supx,y

|δ(x, y)|.

A priori estimates forg(x, y) andti

|g(x, y)| 6 η−1, |ti | 6 Vmη−1

and estimates (4.27) and (4.29) allow one to deduce from (4.30) that

‖δg‖1 = o(1) asN →∞.This proves Lemma 4.3. 2LEMMA 4.4. Item(iii) of Theorem2.1 is true.

Proof.We introduce a sequence{g(k), k = 0,1,2, . . .} of functionsg(k)(p; z),p ∈ (0,1), z ∈ C by the formulasg(0)(p; z) = −1/z and

g(k+1)(p; z) =[−z− (1− q)v2

∫g(k)(r; z)dr − qV (p)×

×∫V (r)g(k)(r; z)dr

]−1

.

It is easy to verify that ifg(k)(p; z) satisfies conditions (2.8), theng(k+1)(p; z)also does. Next, it is not hard to deduce that inequality

supp

∣∣g(k)(p; z)∣∣ 6 2η−1, z ∈ 3q (4.31)

implies the same forg(k+1)(p; z).


The next step is to show that the sequence{g(k)} is the Cauchy one that deter-mines, in the limitk → ∞, the functionγ (p; z) ∈ 0 that satisfies (2.7b). Thisfunction satisfies (4.31) that proves uniqueness of the solution of (2.7b). 2LEMMA 4.5. Due to positivity ofVxy = V (x − y) the matrix

M(x, y; s, t) = VxsVyt + VxtVysis also positively determined on the vectorsψ with complex componentsψ(x, y),x, y = 1,2, . . . , n;

n∑x,y,s,t=1

M(x, y; s, t)ψ(s, t)ψ(x, y) > 0. (4.32)

Proof.Let us consider the Gaussian random variables{γ (x), x ∈ N} such that

Eγ (x) = 0, Eγ (x)γ (y) = V (x − y).Then one can derive with the help of (4.8) that

Eγ (x)γ (y)γ (s)γ (t) = M(x, y; s, t).Therefore the right-hand side of (4.32) can be rewritten in the follpwing form

n∑x,y,s,t=1

Eγ (x)γ (y)γ (s)γ (t)ψ(s, t)ψ(x, y) = E∣∣∣∣ n∑x,y=1

γ (x)γ (y)ψ(x, y)

∣∣∣∣2.This proves the lemma. 2

5. Proof of Theorem 3.1

To prove Theorem 3.1, we again use the resolvent approach of Section 3. Howeverthe present case ofH(q)

N is more complicated because of the bilinear structure ofH(x, y) (3.3) with respect to the random variables{ξµ(x)}. We modify the generalscheme and regard the resolventG

(q)

N (x, y; z) of matricesH(q)

N (3.1) first as a func-tion of the random variables{ξµ(x)} and derive equations for the mathematical

expectationEG(q)

N . Then we prove the selfaveragenessin the limitN → ∞ of therandom variableG(q)

N (x, y; z) as a function of random variables{πxy}.As before, our tools are the resolvent identity (4.3) and the version of (4.8)

applied for jointly independent Gaussian random variables{γj }

E{γjF (γ1, . . . , γk)

} = Eγ 2j E

∂F

∂γj. (5.1)


We will also use the consequence of (4.3) applied forG(q)

N (we omit the super- andsubscripts inG(q)

N ):

∂G(x, s)

∂ξρ(t)= − 1

N

∑r

[G(x, r)G(t, s) +G(x, t)G(r, s)]ξρ(r)πrt . (5.2)

5.1. DERIVATION OF MAIN RELATIONS

We start with the resolvent identity (4.3) written in the form

E〈G(x, y; z)〉 = ζδxy − ζE〈M(x, y; z)〉, ζ = −z−1,

where we denoted

M(x, y) = 1

N

∑s,µ

G(x, s)ξµ(s)ξµ(y)πsy.

Here and below we omit the variablez when it is not important.To compute the mathematical expectation ofM(x, y), we can use (5.1) Taking

into account (5.2), we obtain the following relation

E〈M(x, y)〉 = mu2

NE⟨G(x, y)πyy

⟩−− u

2

N

∑s,t

E⟨G(x, y)πytH(t, s)G(t, s)πsy

⟩+ ϑ(1)N , (5.3)

whereH is the same as in (3.3) and

ϑ(1)N =

u2

N

∑s,t

E⟨G(x, t)πtyH(t, s)G(y, s)πsy

⟩. (5.4)

In Lemma 5.1 (see the second part of this section) we prove that the variableϑ(1)N

vanishes in the limitN → ∞. The proof is based on the observation that all themoments of the random variables

√NH(t, s) andH(t, t) are bounded for allN .

We also use elementary inequality (4.26).Let us introduce variable

L(x, y) = 1

N

∑s,t

πxsH(s, t)G(s, t)πty (5.5)

that resembles the generalized traceT (x, y) (4.2). Taking into account that〈πyy⟩ =

q and denotingcN = mN−1, we rewrite (5.3) in the form

E〈M(x, y)〉 = cNqu2E〈G(x, y)〉 − u2E〈G(x, y)〉E〈L(y, y)〉 −

− ϑ(1)N (x, y) + ϑ(2)N (x, y), (5.6)


where

ϑ(2)N (x, y) = cNu2E

⟨G0(x, y)πyy

⟩+ u2E⟨G0(x, y)L0(y, y)

⟩(5.7)

and the subscripts 0 denote the centered random variables. In Lemma 5.2 we provethatϑ(2)N vanishes in the limitN →∞.

Let us note that Equation (5.6) is similar to Equation (4.12). Following the sameideas as of Section 4,we derive relations for the variableE〈L(x, y)⟩. To do this, weemploy once more (5.1) and obtain relation (cf. (4.15))

E〈L(x, y)〉 = cNu2∑s

E⟨πxsG(s, s)πsy

⟩−− u

2

N

∑t

E⟨L(x, t)G(t, t)πty

⟩+ ϑ(2)N , (5.8)

where

ϑ(3)N = −

u2

N2

∑s,t,r

E⟨πxsH(s, t)G(s, t)πsrG(r, s)πty

⟩. (5.9)

Denoting lN(x, y) = E〈L(x, y)〉 and gN(x, y) = E〈G(x, y)〉 and taking intoaccount definition (2.8), we can rewrite (5.6) and (5.7) as the system of relations

gN(x, y) = ζδxy − ζcu2√qgN(x, y) + ζu2gN(x, y)lN (y, y) ++ ϑ(1)N (x, y) + ϑ(2)N (x, y), (5.10a)

lN(x, y) = cu2gN[δxy + (1− δxy)q

]− u2√qN

∑t

lN (x, t)gN(t, t) +

+ ϑ(3)N + ϑ(4)N , (5.10b)

wheregN = N−1∑x gN(x, x) and

ϑ(4)N (x, y) = cN

u2

N

∑s

E⟨πxsG

0(s, s)πsy⟩− u2

N2

∑t

E⟨L0(x, t)G(t, t)πty

⟩−− u2

N2

∑t

E⟨L(x, t)

⟩E⟨G0(t, t)πty

⟩.

In Lemma 5.2 we prove that

supx,y

∣∣ϑ(i)N (x, y)∣∣ = o(1) asN →∞, i = 3,4. (5.11)

Let us consider the system of matrix equations

RN(x, y) = ζδxy − ζcu2√qRN(x, y) + ζu2RN(x, y)SN(y, y), (5.12a)

SN(x, y) = cu2RN(x, y)[δxy + (1− δxy)q

]−− u

2√qN

N∑t=1

SN(x, t)RN(t, t). (5.12b)


The subscriptN indicates that we considerN-dimensional matrices. It is not hardto show that this system is uniquely solvable providedz ∈ 3q (4.6) withVm =2u√c. See the proof in Lemma 5.3 at the end of this section. Under this condition

it is not hard to show that vanishing of termsϑ(i)N , i = 1,2,3,4 leads to relation

limN→∞

1

N

N∑x=1

[E〈gN(x, x)〉 − RN(x, x)

] = 0. (5.13)

This can be proved by the same reasoning as used Lemma 4.3 of the previoussection.

The next observation is that the system (5.12) admits the solution such that thediagonal elementsRN = RN(x, x) do not depend onx andSN(x, y) = δxyS(1)N +(1− δxy

)S(2)N Therefore we conclude that (5.10) is equivalent to the scalar system

RN = ζ − ζcu2√qRN + ζu2RNS(1)N , (5.14a)

S(1)N = cu2RN − u2√qRN

(1

NS(1)N +

N − 1

NS(2)N

), (5.14b)

S(2)N = cqu2RN − u2√qRN

(1

NS(1)N +

N − 1

NS(2)N

)(5.14c)

that naturally has the unique solution (Lemma 5.3). Now elementary computationsshow thatRN satisfies the same Equation (3.4) that determinesfq,c(z). Then wecome to the conclusion that

limN→∞

∣∣gN − fq(z)∣∣ = 0. (5.15)

Since the normalized tracegN = N−1∑x gN(x, x) represents the Stieltjes trans-

form of certain measure and limN→∞gN is unique, then the Helly theorem and(5.15) imply thatfq,c(z) also represents the Stieltjes transform of certain measureσq,c(λ).

Thus, to complete the proof of Theorem 5.1, it remains to show that

lim N→∞E⟨∣∣∣∣ 1

N

N∑x=1

G0N(x, x)

∣∣∣∣2⟩ = 0. (5.16)

According to the general scheme, relations (5.15) and (5.16) imply convergence inprobability of the measureσ(λ;H(π)

N ) to the limitσq,c(λ). Relation (5.16) is provedin Lemma 5.2.

5.2. PROOF OF THE AUXILIARY FACTS

LEMMA 5.1. Variables(5.7)and(5.9)admit the following estimate for allz ∈ 3q

(4.6)

supx,y

∣∣ϑ(i)N (x, y)∣∣ = O(N−1/2

)asN →∞

for i = 1,3.


Proof.We can estimate|ϑ(1)N (x, y)| by the sum

u2

qN

∑s 6=t

E〈|G(x, t)H(t, s)G(y, s)|〉 + u2

qN

∑s

E〈|G(x, s)H(s, s)G(y, s)|〉.(5.17)

For the first term of this sum we have inequality

u2

qN

∑s 6=t

E〈|G(x, t)H(t, s)G(y, s)|〉

6 u2

qN3/2

∑s 6=t

E⟨|G(x, t)|2|G(y, s)|2⟩1/2⟨∣∣√NH(t, s)∣∣2⟩1/2.

Let us note that all the moments of the random variables√NH(t, s) andH(t, t)

are bounded for allN and do not depend onx andy. Taking into account that

1

N

∑s 6=t

E⟨|G(x, t)|2|G(y, s)|2⟩1/2 6 E

⟨∑s 6=t|G(x, t)|2|G(y, s)|2

⟩1/2

and a priori estimate (4.26), we deduce that the first term of (5.17) is of the orderO(N−1/2) asN →∞. The second one is also of the same order of magnitude.

Let us consider the case ofi = 3. As in the previous case, we can writeinequality∣∣ϑ(3)N (x, y)∣∣ 6 u2

q3/2N5/2

∑r 6=t

∑s

E⟨|G(t, s)G(s, r)| · ∣∣√NH(r, t)∣∣⟩+

+ u2η

q3/2N2

∑s,t

E〈|G(t, s)| · |H(t, t)|〉.

Now it is easy to see that estimates (5.4) and (5.5) complete the proof of thelemma. 2LEMMA 5.2. Regarding the centered random variables

G0(x, y) = G(x, y) − EG(x, y) and L0(x, y) = L(x, y) − EL(x, y),

the following relations hold in the limitN →∞supx

E⟨∣∣G0(x, x)

∣∣2⟩ = o(1), (5.18)

and

supx,y

E⟨∣∣L0(x, y)

∣∣2⟩ = o(1). (5.19)


Remark.Relations (5.18) reflect the selfaveraging property of the random vari-ablesG(x, x) andL(x, y). It is easy to see that (5.18) and (5.19) imply estimates(5.11). Let us also note that (5.18) implies relation (5.16).

Proof. We prove (5.18) using again the resolvent identity (4.3) and relations(5.1) and (5.2). According to the definition of the centered random variable, wecan write relation

E⟨L0(x, y)L(x, y)

⟩ = E⟨L0(x, y)

1

N

∑r,t

πxrH(r, t)G(r, t)πty

⟩. (5.20)

Let us apply relation (4.4) to the last average in (5.20). With the help of (5.2) weobtain equality

E⟨L0(x, y)L(x, y)

⟩ = mu2

NE⟨L0(x, y)πxtG(t, t)πty

⟩−− u2E

⟨L0(x, y)

1

N

∑t

L(x, t)G(t, t)L′(t, y)⟩+

+ φ(1)N (x, y) + φ(2)N (x, y), (5.21)

where

φ(1)N (x, y) = −E

⟨L0(x, y)

1

N2

∑r,s,t

πxtG(t, s)πsrH(t, r)G(r, s)πsy

⟩

andφ(2)N (x, y) can be estimated by inequality

∣∣φ(2)N (x, y)∣∣ 6 1

q2N2

∑l,p,t

E〈|G(l, p)[H(x, t) +H(l, t)]G(t, p)|〉 +

+ 1

q3N3

∑k,l,p,r,t

E〈|G(k, l)H(r, t)H(k, p)G(r, p)G(t, l)|〉 +

+ 1

q3N3

∑k,l,p,r,t

E〈|G(k, r)H(r, t)H(k, p)G(l, p)G(t, l)|〉.

It is not hard to find, with the help of the estimates (5.4) and (5.5), that

supx,y

∣∣φ(1)N (x, y)∣∣ 6 ⟨∣∣L0∣∣2⟩1/2 u4η2

q√N. (5.22)

A little more cumbersome computations lead to the estimate

supx,y

∣∣φ(2)N (x, y)∣∣ = O

(1

q3N3/2

). (5.23)


Taking into account the identity⟨L0(x, y)

1

N

∑t

L(x, t)G(t, t)πty

⟩=⟨L0(x, y)

1

N

∑t

L(x, t)0G(t, t)πty

⟩+

+⟨L0(x, y)

1

N

∑t

G(t, t)πty

⟩ ⟨L(x, t)

⟩,

we derive from (5.21) inequality⟨∣∣L0(x, y)L(x, y)∣∣⟩

6 mu2

N|PN(x, y)| + η√

q

⟨∣∣L0(x, y)∣∣2⟩1/2 sup

t

⟨∣∣G(x, y)0∣∣2⟩1/2++ η√q sup

t

|QN(x, t)| + u4η2

q√N

⟨∣∣L0(x, y)∣∣2⟩1/2+

+ ∣∣φ(3)N (x, y)∣∣|ψN(x, y)|, (5.24)

where

PN(x, y) =⟨L0 1

N

∑t

πxtG(t, t)πty

⟩,

and

QN(x, y, t) =⟨L0(x, y)G(t, t)

⟩6⟨∣∣L0(x, y)

∣∣2⟩1/2⟨∣∣G(t, t)0∣∣2⟩1/2.The termψN contains the terms arising when one passes fromL0G to L0G that areindependent from random variables{πxy} and back toL0G, after averaging overπty.

In what follows, we derive inequalitites∣∣PN(x, y)∣∣ 6 η√N

⟨∣∣L0(x, y)∣∣2⟩1/2+ η√

q

⟨∣∣L0(x, y)∣∣2⟩1/2 sup

t

⟨∣∣L0(x, t)∣∣2⟩1/2+

+ η supt

∣∣QN(x, t)∣∣+ φ(4)N (x, y), (5.25)

and ⟨∣∣G0(t, t)∣∣2⟩ 6 mηu2√q

N

⟨∣∣G0(t, t)∣∣2⟩+ η⟨∣∣G0(t, t)

∣∣2⟩〈|L(x, x)|〉 ++ η2

⟨G0(t, t)L0(t, t)

⟩+ 8ηqu2N−1/2, (5.26)

where the variable|φ(4)N (x, y)| is of the order O(N−1/2) asN →∞.


Then elementary calculations show that (5.24), (5.25) and (5.26) imply (5.18)and therefore (5.19).

Let us considerPN(x, y). Using the resolvent identity (2.10), we can write that

PN(x, y) = ζE⟨L0 1

N

∑t

πxtπty

⟩−

− ζE⟨L0 1

N

∑t,k

πxtG(t, k)H(k, t)πktπty

⟩, (5.27)

where we denotedζ = −1/z andL0 ≡ L0(x, y). It is easy to derive that∣∣∣∣E⟨L0 1

N

∑t

πxtπty

⟩∣∣∣∣ 6 ⟨∣∣L0∣∣2⟩1/2⟨∣∣∣∣[ 1

N

∑t

πxtπty

]0∣∣∣∣2⟩1/2

6 1√N

⟨∣∣L0∣∣2⟩1/2.

Turning to the second term in the right-hand side of (5.27), we use (5.2) to computethe average. Then we obtain

E⟨L0 1

N

∑t,k

πxtG(t, k)H(k, t)πktπty

⟩= E

⟨L0 m

N2

∑t,k

πxtG(t, t)πktπty

⟩−

− E⟨L0 1

N2

∑t,k

πxtG(t, t)H(t, t)πktπty

⟩+ φ(5)N (x, y),

whereφ(5)N (x, y), in common with (5.24) and (5.25), includes the terms of theorderN−1/2〈|L0|2〉1/2 and those that are of the order O(N−3/2). Now, repeatingarguments used to derive expressions (5.22) and (5.23) we arrive at (5.19).

Let us derive (5.18). Using the resolvent identity (4.3) and formula (5.2), weobtain the equality

E⟨G0(x, x)G(x, x)

⟩ = −ζ u2m

NE⟨G0(x, x)G(x, x)πxx

⟩++ ζE

⟨G0(x, x)G(x, x)L(x, x)

⟩ ++ ζ

N

∑s,t

E⟨G0(x, x)G(x, t)H(s, t)πtxG(s, x)πsx

⟩++ 2ζ

N

∑s,t

E⟨G(x, t)H(s, t)πtxG(x, x)G(x, s)πsx

⟩.

Then (5.18) easily follows from (5.19). Lemma is proved. 2


LEMMA 5.3. The system of equations(5.12)is uniquely solvable and(5.15)holds.Proof.Let us introduce recurrence relations

R(k+1)N (x, y) = ζδxy − ζcu2√qR(k)N (x, y) + ζu2R

(k)N (x, y)S

(k)N (y, y),

S(k+1)N (x, y) = cu2R

(k)N (x, y)

[δxy + q

(1− δxy

)]− u2

N

√q∑t

S(k)N (x, t)R

(k)N (t, t)

for k > 0 andR(0)N (x, y) = S(0)N (x, y) = 0. Then one can easily show that for allz ∈ 31 (2.9b)∥∥R(k)N ∥∥ ≡ sup

x,y

∣∣R(k)N (x, y)∣∣ 6 2η−1 (5.28)

and the same relation holds forR(k)N . Then elementary computations show that thedifferences

1k+1(x, y) ≡ R(k+1)N (x, y) − R(k)N (x, y)

and

δk+1(x, y) ≡ S(k+1)N (x, y) − S(k)N (x, y)

are such that

‖1k+1‖ 6[4(c + 1)u2]kη−k−1‖10‖

and

‖δk+1‖ 6[4(c + 1)u2

]kη−k−1‖δ0‖.

This proves the first statement of the lemma. Using the estimates (5.28), it is easyto prove (5.15). Lemma is proved. 2

6. Properties of the Eigenvalue Distribution

In this section we prove Theorems 2.2 and 3.2. The reasonings based on the in-version formula (2.9) are similar in both cases and use Equations (2.7) and (3.4),respectively.

Proof of Theorem 3.2.Let us denoteI (z) = Im fq,c(z) andR(z) = Refq,c(z).It is easy to derive from (3.6) that

I

I 2+ R2= Im z+ cu4(1− q)I + cu2√q u2√qI

(1+ u2√qR)2+ u4qI 2.

Taking into account inequalityI (λ+ iε) > 0, we derive the following estimate

1

I (λ+ iε) > ε + cu4(1− q)I (λ+ iε) (6.1)


for all values ofε > 0 andλ. Using (2.9), one can easily deduce from (6.1) theestimate mentioned in Theorem 3.2. 2

Proof of Theorem 2.2.Taking into account thatgq(p; z) posesses property (2.8),one can easily derive from (2.7b) inequality

Im gq(p;λ+ iε) 6 1

ε + (1− q)v2Im fq(λ+ iε) .

Integrating both parts of this relation overp, one obtains inequality

[Im fq(λ+ iε)]2 6 1

(1− q)v2.

This estimate added by formula (2.9) completes the proof. 2We continue the study of the distributionsσq(λ) andσq,c(λ) and derive from

Equations (2.7) and (3.4), respectively, relations for the moments

Mk =∫λk dσq(λ) and Lk =

∫λk dσq,c(λ).

Let us note first that item (iii) of Theorem 2.1 implies thatgq (p; z) can be repre-sented in the form

gq(p; z) = −∞∑k=0

mk(p)

zk+1. (6.2)

Then (2.7a) implies thatMk =∫mk(p)dp.

Writing down similar to (6.2) expressions forfq(z),∫gq (r; z)V (r)dr and their

products, one can easily derive from (2.7b) the system

m0(p) = 1, m1(p) = 0,

mk+1(p) = (1− q)v2∑

i+j=k−1

m(0)i mj (p)+ q

∑i+j=k−1

m(1)i mj (p)V (p),

wherem(l)i =∫mi(r)V

l(r)dr, l = 0,1 and the sum is taken overi, j > 0. Thissystem can be reduced to the system

m(l)

0 = Vl ≡∫V l(r)dr, m

(l)

1 = 0, (6.3a)

m(l)

k+1 = (1− q)v2∑

i+j=k−1

m(0)i m

(l)j + q

∑i+j=k−1

m(1)i m

(l+1)j . (6.3b)

THEOREM 6.1. (i) Distribution σq(λ) is even in the sence thatM2k+1 = 0,∀k ∈ N;


(ii) the support of the measuredσq(λ) is bounded;

supp dσq(λ) ⊂ (−2l,2l), l = √v[(1− q)v + qVm]. (6.4)

Proof. Item (i) obviously follows from (6.3).We start the proof of item (ii) with the following observation. Since matrixV

with enriesV (x − y) is positively determined, then obviouslym(l)0 > 0. Now it iseasy to show that ifm(l)s > 0, thenm(l)s+2 > 0.

The second step is the following statement.

PROPOSITION 6.1.For all s > 0 and l > 0

m(l+1)s 6 Vmm(l)s . (6.5)

Proof. Relation (6.5) obviously holds fors = 0. It is easy to deduce from theform of (6.3b) that if (6.5) is valid form(l)s with s 6 t then (6.5) is true also form(l)

t+1. 2Using Proposition 6.1, we can derive from (6.3b) inequality

m(l)k+1 6

∑i+j=k−1

((1− q)v2m

(0)i + qVmm(1)i

)m(l)j . (6.6)

Now let us introduce the numbersm(l)k by the following recurrent relations

m(l)

0 = Vl ≡∫V l(r)dr, m

(l)

1 = 0, (6.7a)

m(l)

k+1 =∑

i+j=k−1

((1− q)v2m

(0)i + qVmm(1)i

)m(l)j . (6.7b)

It follows from Proposition 6.1 and (6.6) that

m(l)k 6 m

(l)k . (6.8)

Let us determine the support of the measure dσq(λ) given by its momentsm(0)k . Todo this, we introduce the function

g(l)(z) = −∞∑k=0

m(l)

zk+1.

It is not hard to derive from (6.7) that it satisfies the system of equations

g(l)(z) = m(l)0

−z− (1− q)v2g(0)(z)− qVmg(1)(z), l = 0,1,2, . . . . (6.9)

It follows from (6.9) thatg(1)(z) = m(1)0 g(0)(z),m(1)0 ≡ v and then (cf. (2.10))

g(0)(z) = 1

−z− [(1− q)v2+ qvVm]g(0)(z) . (6.10)


It is clear that dσq(λ) determined by its Stieltjes transformg(0)(z) (6.10) is thesemicircle distribution (2.11) withv2 replaced by(1 − q)v2 + qvVm. Thus, itssupport is given by the interval from the right-hand side of (6.4). Relation (6.8)implies that

supp dσq(λ) ⊂ supp dσq(λ).

This observation completes the proof of Theorem 6.1. 2We finish this section with relation for the momentsLk. Using representation

fq,c(z) = −∞∑k=0

Lk

zk+1,

one can easily derive from (3.6) the system

L0 = 1, L1 = bc, L2 = ac + (1+ c)bL1,

Lk+1 = (c − 1)bLk + b∑i+j=k

LiLj + ac∑

i+j=k−1

LiLj −

− abc∑

i+j+l=k−2

LiLjLl, for k > 2, (6.6)

wherea = u4(1− q) andb = u2√q.Analysis of this system is more complicated than that of (6.3).

7. Summary

We have studied the eigenvalue distribution of weakly dilutedN × N randommatrices in the limitN → ∞. The basic ensemble represents dilution of theWishart random matricesHN which are widely known in multivariate statisticalanalysis. The second ensemble is the dilute version of the Gaussian random matri-cesAN with weakly correlated entries. In the pure (nondilute) case both of theseensembles,{HN} and{AN }, have limiting eigenvalue distributions that can have asingular component at the origin.

We derived explicit equations determining the limiting eigenvalue distributionsof the dilute versions of{HN} and {AN }. We showed that in the case of weakdilution, when each matrix row contains, on average,qN , 0 < q < 1 nonzeroentries, the density of the eigenvalue distribution is bounded by const.(1− q)−1.

Thus, we can conject that in general the singularities (if any) in the spectra ofrandom matrices are rather unstable and disappear when the dependence betweenmatrix entries is waived. Taking into account our results, one can say that thisperturbation can be fairly weak, as it is represented by the weak random dilution.


Acknowledgements

A.K. would like to thank the Department of Mathematical Sciences at Brunel Uni-versity for hospitality while this work was performed and both authors are gratefulto the Royal Society, London for the financial support.

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33

On Ground-Traveling Waves for the GeneralizedKadomtsev–Petviashvili Equations

A. PANKOV1 and K. PFLÜGER21Department of Mathematics, Vinnitsa State Pedagogical University, Ukraine.e-mail: [email protected]

2Institut für Mathematik I, Freie Universität Berlin, Germany. e-mail: [email protected]

(Received: 18 March 1999; in final form: 18 October 1999)

Abstract. As a continuation of our previous work, we improve some results on convergence ofperiodic KP traveling waves to solitary ones as the period goes to infinity. In addition, we presentsome qualitative properties of such waves, as well as nonexistence results, in the case of generalnonlinearities. We suggest an approach which does not use any scaling argument.

Mathematics Subject Classifications (2000):35Q53, 35B10, 35A35, 35A15.

Key words: generalized Kadomtsev–Petviashvili equation, traveling waves, variational methods.

1. Introduction

Kadomtsev–Petviashvili (KP) equations, both original and generalized, appear inthe theory of weakly nonlinear dispersive waves [7]. They read

ut + uξξξ + f (u)ξ + εvy = 0, vξ = uy (1)

or, eliminatingv,

(ut + uξξξ + f (u)ξ )ξ + εuyy = 0. (2)

More precisely, these are KP-I equations ifε = −1, and KP-II equations ifε = +1.The original KP equations correspond to the casef (u) = 1

2u2, form a completely

integrable Hamiltonian system, and were studied extensively by means of algebro-geometrical methods (see, e.g., [8]). There are also a number of papers dealing withmore general equations (1) or (2), mainly in the case of pure power nonlinearity:[1, 3 – 6, 10, 17, 20, 23], to mention a few. In particular, solitary traveling waveswere studied [1, 4 – 6, 10, 17, 23]. Here we consider mainly the case of KP-I equa-tions. Remark that KP-II equations do not possess traveling waves at all (see [5]for the case of solitary waves and pure power nonlinearity, and Section 4 for thegeneral case).

The present paper is a direct continuation of our previous work [17]. It concernsthe existence of ground-traveling waves, both periodic and solitary, and the limit

34 A. PANKOV AND K. PFLÜGER

behavior of periodic waves, as the period goes to infinity. Corresponding equationsfor traveling waves read

−cux + uxxx + f (u)x + εvy = 0, vx = uy (3)

and

(−cux + uxxx + f (u)x)x + εuyy = 0, (4)

respectively. Herex = ξ − ct , c > 0 is the wave speed. In [17], among otherresults we have proved thatk-periodic inx ground waves converge to a solitaryground wave in a very strong sense (Theorem 5 of that paper). Unfortunately, thatresult does not cover the case of the original KP equation, but includes the casef (u) = u3. The first aim of this paper is to extend the results of Theorems 4and 5, [17], in order to include nonlinearities likef (u) = |u|p−1, 2 < p < 6.This will be done in Section 2. Our second goal is to discuss, in Section 3, somequalitative properties of KP traveling waves: symmetry, continuity, and rate ofdecay. As for continuity and decay properties, we follow very closely the paper [6]and point out only the main differences. On the contrary, our proof of symmetrywith respect toy-variable relies on a quite different variational characterization ofground waves and permits us to treat the case of nonhomogeneous nonlinearity.Finally, in Section 4, we discuss the nonexistence of traveling waves, both solitaryand periodic, for general nonlinearities.

All the assumptions we impose here are satisfied for the nonlinearities

f (u) = c|u|p−1 and f (u) = c|u|p−2u+k∑i=1

ci |u|pi−2u,

with

c, ci > 0, 2< p < 6, 2< pi < p.

Unfortunately, Assumptions (N) and (N1) below are not satisfied for a very inter-esting nonlinearityf (u) = u2 − u3 which appears in some physical models [18].

In addition, let us remark that it is natural to look at(k, l)-periodic travelingwave solutions with respect to(x, y), as well asl-periodic waves with respect toy which are decaying inx. All the results below, except of Theorem 2, have theirstraightforward counterparts in both these cases. The case of double periodic wavesis even simpler, since the corresponding functional satisfies the Palais–Smale con-dition. However, as we already mentioned in our previous paper [17], our techniquedoes not work, at least directly, when we try to study the behavior of such travelingwaves as(k, l)→∞ (respectively,l→∞).

2. Ground Waves

Denote byF(u) = ∫ u0 f (t)dt the primitive function off . We make the followingassumptions:

GROUND-TRAVELING WAVES AND KP EQUATIONS 35

(1) f ∈ C(R), f (0) = 0;(2) |f (u)| 6 C(1+ |u|p−1), 2 2 such thatµF(u) 6 uf (u) for all u ∈ R.

Remark that ifF(u) > 0 for all u 6= 0, then Assumption (3) follows from (4).LetQk = (−k/2, k/2)× R, 0< k 6∞. We set

D−1x,ku(x, y) =

∫ x

−k/2u(s, y)ds, k ∈ (0,∞]. (5)

We shall simply writeD−1x in the casek = ∞. Define the Hilbert spaceXk as the

completion of{ϕx : ϕ ∈ C∞k }, whereC∞k is the space of smooth functions onR2

which arek-periodic inx and have finite support iny, with respect to the norm‖u‖k = (u, u)1/2k ,

(u, v)k =∫Qk

uxvx +D−1x,kuy ·D−1

x,kvy + cuv.

Similarly, X = X∞ is the completion of{ϕx : ϕ ∈ C∞0 (R2)} with respect to thenorm‖u‖ = ‖u‖∞ = (u, u)1/2 = (u, u)1/2∞ . The operatorD−1

x,k is well-defined onthe spaceXk, k ∈ (0,∞].

For k-periodic traveling waves,k ∈ (0,∞), Equation (4) may be written in theform [17]

(−uxx +D−2x,kuyy + cu− f (u))x = 0. (6)

Solitary waves are solutions of the same Equation (6), withk = ∞. The actionfunctional associated with (6) reads [17]

Jk(u) = 1

2‖u‖2k −

∫Qk

F (u); (7)

Jk is of the classC1 onXk. We consider weak solutions of (6), i.e. critical pointsof Jk in Xk.

Now let us consider the so-called Nehari functional

Ik(u) = 〈J ′k(u), u〉 = ‖u‖2k −∫Qk

uf (u), (8)

and the Nehari manifold

Sk = {u ∈ Xk : Ik(u) = 0, u 6= 0}.All traveling wave solutions lie in the corresponding Nehari manifold and wewill find ground waves, i.e. solutions with minimal action among all nontrivialsolutions, solving the following minimization problem:

mk = inf{Jk(u) : u ∈ Sk}. (9)


Remark that

Jk(u) =∫Qk

1

2uf (u)− F(u), u ∈ Sk. (10)

In what follows, we will omit the subscriptk if k = ∞ and write simplyJ, I, . . . .Throughout this section, in addition to Assumptions (1)–(4), we impose the

following one

(N) For anyu ∈ L2(R2) such that∫R2uf (u) > 0,

the function oft

t−1∫R2uf (tu)

is strictly increasing on(0,+∞).In the proof of Theorem 1, [17], we have considered the Mountain Pass Values

ck for Jk and proved that they are uniformly bounded from below and above bypositive constants. More precisely,

ck = infγ∈0k

maxt∈[0,1]

Jk(γ (t)),

where

0k = {γ ∈ C([0,1],Xk) : γ (0) = 0, Jk(γ (1)) < 0}.Here we have defined0k in a slightly different way than in [17], but it does noteffect on the value ofck . Consider also another minimax value

c′k = infv∈X+k

supt>0

Jk(tv),

where

X+k ={v ∈ Xk :

∫Qk

F (v) > 0}.

Due to Assumption (4),X+k 6= ∅.

LEMMA 1. For everyv ∈ X+k , there exists a uniquetk = tk(v) such thattkv ∈ Sk,Jk(tkv) = max

t>0Jk(tv),

and tk(v) depends continuously onv ∈ X+k .


Proof.Assumption (4) implies that∫Qk

vf (v) > 0

for anyv ∈ X+k . Therefore, due to Assumption (N), the function

d

dtJk(tv) = Ik(tv) = t2

(‖v‖2k − t−1

∫Qk

vf (tv)

)vanishes at only one pointtk = tk(v) > 0. Equation (10) and Assumption (4) implythatJk is positive onSk. SinceJk(0) = 0, we see thattk is a point of maximum forJk(tv). Continuity oftk(v) is easy to verify. 2LEMMA 2. ck = c′k = mk.

Proof. Sinceuf (u) is subquadratic at 0 and the quadratic part ofJk is positivedefined, we see thatIk(v) > 0 in a neighborhood of the origin, except of 0. Hence,Ik(γ (t)) > 0, γ ∈ 0k , for small t > 0. Due to Assumption (4), forv ∈ X+k wehave

2Jk(v) = ‖v‖2k − 2∫Qk

F (v) > ‖v‖2k − µ∫Qk

F (v)

> ‖v‖2k − 2∫Qk

vf (v) = Ik(v).

Hence,Ik(γ (1)) < 0. Therefore,γ (t) crossesSk and this implies thatck > mk.By Assumption (4), for anyv ∈ X+k we haveF(tv) > αtµ, α > 0, if t > 0 is

large enough. This implies thatJk(tv) < 0 for everyv ∈ X+k and sufficiently larget > 0. Hence, the half-axis{tv : t > 0} generates in a natural way an element of0k. This implies the inequalityck 6 c′k.

Now let v ∈ Sk. By the definition ofIk, σ =∫Qkvf (v) > 0, and (N) implies

that

d

dt

∫Qk

F (tv) =∫Qk

vf (tv) > t−1∫Qk

vf (tv) > σ > 0

providedt > 1. Hence, fort > 0 large enough∫Qk

F (tv) > 0.

By definitions ofc′k andmk, we see thatc′k = mk. 2THEOREM 1. Assume Assumptions(1)–(4) and (N) are fulfilled. Then, for anyk ∈ (0,∞), there exists a minimizeruk ∈ Sk of (9) which is a critical point ofJk. Moreover,Jk(uk) = mk is bounded from above and below by positive constantsindependent onk.


Proof. In the proof of Theorem 1, [17], it is shown that there exists a Palais–Smale sequenceuk,n ∈ Xk at the levelck, i.e.

J ′k(uk,n)→ 0, Jk(uk,n)→ ck

asn → ∞. Moreover,uk,n → uk weakly inXk and strongly inLploc(R2), whereuk ∈ Xk is a nontrivial solution of (6). Therefore,

Ik(uk,n) = 〈J ′k(uk,n), uk,n〉 → 0

and

Jk(uk,n)− 12Ik(uk,n) =

∫Qk

(12uk,nf (uk,n)− F(uk,n)

)→ ck.

Due to Assumption (4), the integrand here is nonnegative and, sinceuk,n → uk inL2

loc(R2), we have∫Qk

(12ukf (uk)− F(uk)

)6 ck.

However,uk is a nontrivial solution, hence,uk ∈ Sk. Therefore, we deduce from(10) that

Jk(uk) =∫Qk

(12ukf (uk)− F(uk)

)> mk.

Now Lemma 2 implies thatJk(uk) = mk anduk is a ground-wave solution.The last statement of the theorem follows immediately from Lemma 2 and

uniform estimates forck. 2Remark 1.The Nehari variational principle suggested in [13] was used success-

fully in many papers (see, e.g., [2, 9, 14 – 17, 22]). In all these papers, except [16],the geometry of Nehari manifold is simple enough: it is a bounded surface with-out boundary around the origin, like a sphere. In the case we consider here, thepicture is different:Sk may look like a sphere if, e.g.,f (u) = |u|p−2u, and maybe unbounded if, e.g.,f (u) = |u|p−1. Nevertheless, in any case,Sk separates theorigin and the domain of negative values ofJk, which is sufficient for our purpose.In [16], such a manifold is also unbounded in general, but there we have useddifferent arguments.

Now we are going to study the behavior ofuk, ask→∞. Recall the definitionof cut-off operatorsPk : Xk → X, [17]. Letχk ∈ C∞0 (R) be a nonnegative functionsuch thatχk(x) = 1 for x ∈ [−k/2, k/2], χk(x) = 0 for |x| > (k + 1)/2, and|χ ′k|, |χ ′′k | 6 C0, with some constantC0 > 0. We set

Pku(x, y) = [χk(x)D−1x,ku(x, y)]x .


THEOREM 2. Assume that Assumptions(1)–(4) and (N) are satisfied. Letuk ∈Xk be a sequence of ground-wave solutions. Then there exists a nontrivial groundwaveu ∈ X and a sequence of vectorsζk ∈ R2 such that, along a subsequence,Pkuk(· + ζk)→ u weakly inX. If, in addition,

|f (u+ v)− f (u)| 6 C(1+ |u|p−2+ |v|p−2)|v|, v ∈ R, (11)

then, along the same subsequence,

limk→∞‖uk(· + ζk)− u‖k = 0.

Proof. By Theorem 2, [17], there is a nontrivial solutionu ∈ X such thatPkuk(· + ζk) → u weakly inX for someζk ∈ R2 (along a subsequence). Letus prove thatu is a ground wave, i.e.

J (u) = inf{J (v) : v ∈ S} = m.First of all, for anyv ∈ S and anyε > 0, there existkε andvk ∈ Sk such that

Jk(vk) 6 J (v)+ ε, k > kε.

Indeed, sinceJ andI are continuous, we can findϕk ∈ C∞0 (Qk) such thatηk =Dxϕk → v in X and, hence,

J (ηk)→ J (v), I (ηk)→ I (v) = 0.

SinceI (v) = 0 andv 6= 0, we have∫Qk

vf (v) = ‖v‖2 > 0.

Hence,∫Qkηkf (ηk) > 0 for k large enough. Due to (N), there existsτk > 0

such thatI (τkηk) = 0 andτk → 1. Let vk be a uniquek-periodic function whichcoincides withτkηk onQk. ThenJk(vk) = J (τkηk) 6 J (v)+ ε providedk is largeenough.

In particular, we have lim supk→∞mk 6 m. Now, exactly as in the proof ofTheorem 5, [17], we see that lim infk→∞mk > J (u) > m. Hence,m = J (u) andu is a ground wave solution.

The second part of the theorem follows from Theorem 3, [17], exactly as at theend of the proof of Theorem 5, [17]. 2

3. Qualitative Properties of Traveling Waves

Now we are going to study such properties of KP traveling waves as symmetry,regularity, and decay. We start with the following:


LEMMA 3. Suppose that Assumptions(1) and (2) are satisfied. In the case2 <p 6 5, assume, in addition, thatf ∈ C2(R) and

|f (j)(u)| 6 C(1+ |u|p−1−j ), j = 1,2, u 6= 0. (12)

Then any traveling wave is continuous. Moreover, any solitary(resp. periodic)wave tends to zero as(x, y)→∞ (resp.y →∞).

Proof.For such a waveu ∈ Xk , we have

−cvxx − vyy + vxxxx = f (u)xx = gxx. (13)

Let

(Fk,xh)(ξ) =∫ k/2

−k/2h(x)exp(−iξx)dx

be the Fourier transform ifk = ∞ (then we simply writeFx), and the sequence ofFourier coefficients ifk <∞. In the last case,ξ ∈ (2π/k)Z. Now we get from (13)

Fk,xFyu = p(ξ1, ξ2)(Fk,xFyg), (14)

where

p(ξ) = p(ξ1, ξ2) = ξ21

cξ21 + ξ4

1 + ξ22

,

ξ1 andξ2 are dual variables tox andy, respectively. Ifk = ∞, there is nothing todo. In the case 5< p < 6, one needs only to repeat the proof of Theorem 1.1, [6],which does not use any particular property of power nonlinearity, except of itsgrowth rate. In the case 2< p 6 5 the arguments from the proof of Lemma 4.1,[5], work and just here assumption (12) is needed.

Now we explain how to cover the case of periodic waves. Recall the followingLizorkin theorem [11]. Letp(ξ), ξ ∈ Rn, be of the classCn for |ξj | > 0, j =1, . . . , n. Assume that∣∣∣∣ξk1

1 · · · ξknn∂kp

∂ξk11 · · · ∂ξknn

∣∣∣∣ 6 M,with kj = 0 or 1,k = k1+· · ·+kn = 0,1, . . . , n. Thenp(ξ) is a Fourier multiplieronLr(Rn), 1< r <∞.

We rewrite now (14) as follows:

Fk,xu = F −1y [p(ξ1, ξ2)FyFk,xg] = P(ξ1)g,

whereP(ξ1) is the operatorF −1y p(ξ1, ·)Fy for any fixedξ1. It is easy to verify that

P(ξ1) ∈ L(Lr(Ry)), the space of bounded linear operators inLr(Ry). Moreover,due to the Lizorkin theorem,p(ξ) is a multiplier inLr(R2). Hence, so is it forP(ξ1) in the spaceLr(Rx, Lr(Ry)) = Lr(R2). It is not difficult to verify that


P(ξ1) depends continuously onξ1 with respect to the norm inL(Lr(Ry)) at anypoint ξ1 6= 0. Therefore, by Theorem 3.8 of Ch. 7, [21], we see thatP(ξ1) isalso a multiplier in the spaceLr((−k/2, k/2), Lr(Ry))0 = Lr(Qk)0 considered asthe space ofk-periodic inx functions. The subscript 0 means that for functionsfrom this spaceFk,xu vanishes atξ1 = 0. Sincep(0, ξ2) = 0, the correspondingmultiplier vanishes on

{u ∈ Lr(Qk) : Fk,xu = 0 if ξ1 6= 0}and, hence, is a bounded operator on the entire spaceLr(Qk). In fact, we needhere an extension of that theorem for operator-valued multipliers which may bediscontinuous at the point 0. However, in this case, the proof presented in [21]works without any change.

To complete the proof in the case 5< p < 6, we can now use the samereiteration argument as in [6]. In the case 2< p 6 5, again one needs to invokethe arguments of the proof of Theorem 4.1, [5]. Here we have to apply the remarkon operator-valued multipliers top1(ξ) = ξ2

1p(ξ), p2(ξ) = ξ2p(ξ), as well as top(ξ) itself. 2

We also need the following additional assumption:

(N1) f ∈ C1(R) and, for anyv ∈ L2(R2) such that∫R2 f (v)v > 0, we have∫

R2f (v)v <

∫R2f ′(v)v2

and∫R2 f (tv)v > 0 ∀t > 0.

Calculating the derivative oft−1∫R2 f (tv)v, we see that (N1) implies (N).

Let us introduce the functional

Lk(v) =∫Qk

[12f (v)v − F(v)

], v ∈ Xk.

As we have seen,Lk = Jk onSk andLk(v) > 0, ∀v ∈ Xk.

LEMMA 4. Under Assumption(N1), Lk(tv) is a strictly increasing function oft > 0, provided

∫Qkf (v)v > 0.

Proof. It follows immediately from the following elementary identity

d

dtLk(tv) = 1

2t

[∫Qk

f ′(tv)t2v −∫Qk

f (tv)tv

]. 2

We also need the following dual characterization of ground-traveling waves:

LEMMA 5. Suppose Assumptions(1)–(4)and(N1) are satisfied. For nonzerou ∈Xk, k ∈ (0,∞], the following statements are equivalent:


(i) u is a ground wave,(ii) Ik(u) = 0 andLk(u) = mk = inf{Lk(v) : v ∈ Sk},

(iii) Ik(u) = 0= sup{Ik(v) : v ∈ Xk,Lk(v) = mk}.Proof. Implication (i)⇒ (ii) is proved in Section 2.To prove (ii)⇒ (i) assume thatu ∈ Xk satisfies (ii). SinceJk = Lk onSk, there

exists a Lagrange multiplierλ such thatλI ′k(u) = J ′k(u). Then

λ〈I ′k(u), u〉 = 〈J ′k(u), u〉 = Ik(u) = 0.

On the other hand,

〈I ′k(u), u〉 = 2‖u‖2k −∫Qk

f ′(u)u2 −∫Qk

f (u)u

= 2Ik(u)+∫Qk

f (u)u−∫Qk

f ′(u)u2

=∫Qk

f (u)u−∫Qk

f ′(u)u2.

However,∫Qkf (v)v > 0 onSk and, due to (N1),〈I ′k(u), u〉 < 0. Therefore,λ = 0

andu is a ground wave.Now let us prove (ii)⇒ (iii). For u as in (ii), Ik(u) = 0. Assume that there is

v ∈ Xk such thatLk(v) = mk andIk(v) < 0. Then∫Qkf (v)v > 0 and there

existst0 ∈ (0,1) such thatIk(t0v) = 0. By Lemma 4,Lk(t0v) < Lk(v) = mk,which is impossible.

Finally, we prove (iii)⇒ (ii). Let u ∈ Xk satisfies (iii). Then,Lk(u) > mk.Assume thatLk(u) > mk. Again we have

∫Qkf (u)u > 0. By Lemma 4, there

existst0 ∈ (0,1) such thatLk(t0u) = mk. However,Ik(t0u) > 0 and this contra-dicts (iii). 2

Now we are ready to prove the symmetry property for all kinds of ground waveswe consider. As in [6], we use the approach suggested in [12] (see also [22]).

THEOREM 3. In addition to Assumptions(1)–(4) and (N1), suppose thatf ∈C2(R). In the case2 < p 6 5 assume also that inequality(12) is fulfilled. Thenany ground waveu ∈ Xk, k ∈ (0,∞], is symmetric with respect to some line1 = {(x, y) ∈ R2 : y = b}.

Proof.Chooseb in such a way that∫1+∩Qk

[12f (v)v − F(v)

] = ∫1−∩Qk

[12f (v)v − F(v)

] = mk

2,

where1+ and1− are the corresponding upper and lower half-planes. Letu± be asymmetric (with respect to1) function such thatu± = u on1±. Thenu± ∈ Xkand

Lk(u±) = Lk(u) = mk.


By Lemma 5,Ik(u±) 6 0. On the other hand,

Ik(u+)+ Ik(u−) = 2Ik(u) = 0.

Using Lemma 5, we conclude thatu± is a ground wave.To conclude thatu± = u and, hence, complete the proof, it is sufficient to use

the same unique continuation result as in [6], and just here we need the assump-tion f ∈ C2(R) and Lemma 3. Remark that a periodic version (with5 = 1±)of unique continuation Theorem A.1, [6], can be proved exactly as that theoremitself. 2

In addition, we formulate the following direct generalization of results of [6]for decay of solitary waves.

THEOREM 4. Suppose Assumptions(1) and (2) to be satisfied. Letu ∈ Xk, k ∈(0,∞], be a traveling wave. Ifk = ∞, then

r2u ∈ L∞(R2), r2 = x2+ y2.

If 0< k <∞, theny2u ∈ L∞(Qk).

The proof is essentially the same as in [6]. In the casek < ∞, one needsonly to use the partially periodic Fourier transform as in Lemma 3. Let us remarkthat the classical rational KP-solitons decay exactly asr−2. On the other hand,in [24] a family of traveling waves is constructed for the original KP-I equationswhich are periodic inx and decay exponentially fast with respect to the transversevariable. Thus, it seems that the statement of Theorem 4 is not exact in the casek <∞, while it is so in the casek = ∞. Also unknown are Zaitsev’s ground-wavesolutions.

4. On Nonexistence of Traveling Waves

In this section we turn to general KP equations (3), withε = ±1, and discuss thenonexistence problem. We use the same approach as in [5]. However, the case ofperiodic waves is more involved (see the proof of Lemma 6). Here we considertraveling waves belonging to the space

Yk = {u ∈ Xk : u ∈ H 1(Qk), uxx,D−1x,kuyy ∈ L2(Qk), f (u)u ∈ L1(Qk)}

if k <∞, and

Y = Y∞ = {u ∈ X : u ∈ H 1(R2), uxx,D−1x,kuyy ∈ L2

loc(R2), f (u)u ∈ L1(R2)}.

First, we collect some useful identities.

LEMMA 6. Suppose thatf satisfies Assumptions(1) and (2). Let u ∈ Yk, k ∈(0,∞], be a solution of Equations(3). Then

44 A. PANKOV AND K. PFLÜGER∫Qk

[c

2u2+ 3

2u2x + ε

v2

2− uf (u)+ F(u)

]= 0, (15)∫

Qk

[c

2u2+ 1

2u2x + ε

v2

2+ F(u)

]= 0, (16)∫

Qk

[cu2+ u2x − εv2− f (u)u] = 0. (17)

Proof.First, we remark that, for anyk, (17) is an extention of the caseε = ±1of Ik(u) = 0 stated in Section 2. Therefore, we concentrate on (15) and (16) only.

In the case of solitary waves (k = ∞), the calculations carried out in the proofof Theorem 1.1, [5], work equally well for general nonlinearities. Therefore, welook at periodic waves (k <∞).

Fixedκ ∈ (0,1), let ϕT ∈ C∞0 (R) be a nonnegative function such thatϕT = 1on [−T/2, T /2], ϕT (x) = 0 if |x| > (T + T κ)/2, andϕ(j)(x) 6 Cj/|x|j , j =1,2, . . ., if T/2 6 |x| 6 (T + T κ)/2 (the construction of such a function will begiven later on).

Multiplying the first equation (3) byxϕT u and integrating overR2, after anumber of integrations by parts, we get

c

2

∫ϕT u

2−∫ϕT uf (u)+

∫ϕT F(u)+ 3

2

∫ϕT u

2x+

+ ε1

2

∫ϕT v

2+ 1

2

∫xϕ′T u

2−∫xϕ′T uf (u)+

∫xϕ′T F (u)+

+ 2∫ϕ′T uux +

∫xϕ′′T uux +

3

2

∫xϕ′T u

2x + ε

1

2

∫xϕ′T v

2 = 0.

Dividing the last identity byT , we are going to pass to the limit asT →∞. First,we point out that here the integrals containingϕT are taken over

QT ∪Q′T ∪Q′′T= QT ∪ {(T /2, (T + T κ)/2)×R} ∪ {(−(T + T κ)/2,−T/2)×R},

while those containingϕ′T andϕ′′T are overQ′T ∪Q′′T . Moreover,ϕT = 1 onQT .Now, letg ∈ L1

loc(R2) be a function which isk-periodic inx. Then, it is easy toverify that

limT→∞

1

T

∫QT

g =∫Qk

g.

Next, due to the properties ofϕT , all the integrals overQ′T can be estimated fromabove by

∫Q′Tg, with a nonnegativek-periodic inx functiong ∈ L1

loc(R2). Now

1

T

∫Q′Tg 6 T κ + 1

T

∫Qk

g.


This justifies the passage to the limit and gives rise to (15).Identity (16) can be proved exactly as (2.8), [5], with the only change: take the

cut-off functionsχj depending ony only.Now we construct the functionϕT . Fix ε > 0 and let

g(x) ={1 if x 6 T/2,

1− log(x/T ) if T/2< x 6 (T + T κ)/2+ ε,0 if x > (T + T κ)/2+ ε.

We choose a nonnegative functionh ∈ C∞0 (R) such that supph ⊂ (0, ε) and∫h = 1, and set

ϕT (x) =∫h(x − t)g(t)dt, ϕT (x) = ϕT (|x|).

For this function, it is easy to verify all the properties we need. 2THEOREM 5. Suppose thatf ∈ C(R) satisfies Assumption(4). Then there is nonontrivial traveling waveu ∈ Yk, k ∈ (0,∞], providedε = +1, or ε = −1 andµ > 6.

Proof.Adding (15), (16) and subtracting (17), we get∫Qk

u2x = −2ε

∫Qk

v2.

This rules out the caseε = +1. In the caseε = −1 (KP-I equations) the lastidentity together with (15) and (17), respectively, implies∫

Qk

[c

2u2+ 5

2v2− f (u)u+ F(u)

]= 0

and ∫Qk

[cu2+ 3v2− f (u)u] = 0.

Eliminatingv, we get

2c∫Qk

u2 =∫Qk

[6F(u)− f (u)u].

If µ > 6, we have

2c∫Qk

u2 6∫Qk

[µF(u)− f (u)u] 6 0.

Hence,u = 0 and we conclude. 2


Acknowledgements

This work was carried out during the visit of the first author (A.P.) to the Institut fürMathematik, Humboldt Universität Berlin (September–December, 1998) under thesupport of Deutsche Forschungsgemeinschaft. A.P. is very grateful to K. Grögerfor his kind hospitality and a lot of stimulating discussions. The authors thank theanonimous referee for information on paper [24].

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49

Pole Dynamics for Elliptic Solutions of theKorteweg–deVries Equation

BERNARD DECONINCK1 and HARVEY SEGUR21Department of Applied Mathematics, Box 352420, University of Washington, Seattle, Washington,98195, U.S.A.2Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, U.S.A.

(Received: 21 April 1999; in final form: 24 December 1999)

Abstract. The real, nonsingular elliptic solutions of the Korteweg–de Vries equation are studiedthrough the time dynamics of their poles in the complex plane. The dynamics of these poles isgoverned by a dynamical system with a constraint. This constraint is solvable for any finite numberof poles located in the fundamental domain of the elliptic function, often in many different ways.Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit.

Mathematics Subject Classifications (2000):34M05, 35A20, 35Q53, 37K10, 37K20.

Key words: KdV equation, elliptic, finite gap solutions, pole dynamics, Calogero–Moser.

1. Introduction

In 1974, Kruskal [15] considered the interaction of solitons governed by the Korte-weg–de Vries equation (KdV),

ut = 6uux + uxxx. (1.1)

Each KdV soliton is defined by a meromorphic function in the complexx-plane(i.e., sech2k(x − x0)), so Kruskal [15] suggested that the interaction of two ormore solitons could be understood in terms of the dynamics of the poles of thesemeromorphic functions in the complexx-plane, where the poles move according toa force law deduced from (1.1). This was followed by the work of Thickstun [17]who considered the case of two solitons in great detail.

Following a different line of thought, Airault, McKean and Moser [2] studied ra-tional and elliptic solutions of the KdV equation. An elliptic solution of KdV is bydefinition a solution of the KdV equation that is doubly periodic and meromorphicin the complexx-plane, for all time. Note that the soliton case is an intermediarycase between the elliptic and the rational case. It was treated as such in [2].

Airault, McKean and Moser [2] approached these elliptic solutions and theirdegenerate limits through the motion of their polesxi(t) in the complexx-plane.In particular, they looked for elliptic KdV solutions of the form

u(x, t) = −2N∑i=1

℘(x − xi(t)). (1.2)

50 BERNARD DECONINCK AND HARVEY SEGUR

Here℘(z) denotes the Weierstrass elliptic function. It can be defined by its mero-morphic expansion

℘(z) = 1

z2+

∑(m,n) 6=(0,0)

(1

(z+ 2mω1+ 2nω2)2− 1

(2mω1+ 2nω2)2

), (1.3)

with ω1/ω2 not real.? More properties of the Weierstrass function will be given asthey are needed.

It is shown in [2] that the dynamics of the polesxi(t) is governed by thedynamical system

xi = 12N∑

j=1,j 6=i℘ (xi − xj ), i = 1,2, . . . , N, (1.4a)

(the dot denotes differentiation with respect to time) with the invariant constraint

N∑j=1,j 6=i

℘′(xi − xj ) = 0, i = 1,2, . . . , N. (1.4b)

Here the prime denotes differentiation with respect to the argument. The solutions(1.2) generalize an elliptic solution given earlier by Dubrovin and Novikov [7], cor-responding to the caseN = 3. These authors also recall the Lamé–Ince potentials[13]

u(x) = −g(g + 1)℘ (x), (1.5)

which are the simplestg-gap potentials of the stationary Schrödinger equation

∂2ψ

∂x2+ u(x)ψ = λψ. (1.6)

The remarkable connection between the KdV equation and the stationary Schrödin-ger equation has been known since the work of Gardner, Greene, Kruskal andMiura [9]. Dubrovin and Novikov show [7] that the(N = 3)-solution discussed in[7] is a 2-gap solution of the KdV equation with a 2-gap Lamé–Ince potential asinitial condition.

If one considers the rational limit of the solution (1.2) (i.e., the limit in which theWeierstrass function℘(z) reduces to 1/z2), then the constraint (1.4b) is solvableonly for a triangular number of poles,

N = n(n+ 1)

2, (1.7)

? In this paper it is always assumed thatω1 is real andω2 is imaginary. This is necessary toensure reality of the KdV solution (1.2) whenx is restricted to the realx-axis. Other considerationsfor reality of the elliptic KdV solutions will be discussed in Section 4.

POLE DYNAMICS FOR ELLIPTIC SOLUTIONS OF THE KdV EQUATION 51

for any positive integern [2]. Notice that the Lamé–Ince potentials are given byg(g + 1)/2 times anN = 1 potential. Based on these observations, it was con-jectured in [2] that also in the elliptic case given by (1.2), the constraint (1.4b) issolvable only for a triangular numberN , ‘or very nearly so’. From the momentit appeared this conjecture was known not to hold, because it already fails in thesoliton case, where the Weierstrass function degenerates to hyperbolic functions.This failure of the conjecture easily follows from the work of Thickstun [17].

A further understanding of the elliptic case had to wait until 1988, when Verdierprovided more explicit examples of elliptic potentials of the Schrödinger equation[21]. Subsequently, Treibich and Verdier demonstrated that

u(x) = −24∑i=1

gi(gi + 1)

2℘(x − x0− ωi) (1.8)

(ω3 = ω1+ω2, ω4 = 0, thegi are positive integers) are finite-gap potentials of thestationary Schrödinger equation (1.6) and, hence, result in elliptic solutions of theKdV equation [19, 18, 20].

The potentials of Treibich and Verdier were generalized by Gesztesy and Wei-kard [10, 11]. They showed that any elliptic finite-gap potential of the stationarySchrödinger equation (1.6) can be represented in the form

u(x) = −2M∑i=1

gi(gi + 1)

2℘(x − αi), (1.9)

for someM and positive integersgi . Notice that this formula coincides with (1.2)if all the gi are 1.

The focus of this paper is the constrained dynamical system (1.4a–b). We re-turn to the ideas put forth by Kruskal [15] and Thickstun [17]. This allows us toderive the system (1.4a–b) in a context which is more general than [2]: there it wasobtained as a system describing a class of special solutions of the KdV equation.Here, it is shown that any meromorphic solution of the KdV equation which isdoubly periodic inx is of the form (1.2). Hence the consideration of solutions ofthe form (1.2) and the system of equations (1.4a–b) leads toall elliptic solutions ofthe KdV equation.

Simultaneously, some of the results of Gesztesy and Weikard [10, 11] are recov-ered here. Because of the connection between the KdV equation and the Schrödin-ger equation, any potential of Gesztesy and Weikard can be used as an initialcondition for the KdV equation, which determines any time dependence of theparameters of the elliptic KdV solution with that initial condition. Our approachdemonstrates which parameters in the solutions (1.9) are time independent andwhich are time dependent.

The following conclusions are obtained in this paper:


• All finite-gap? elliptic solutions of the KdV equation are of the form (1.2),with all xi(t) distinct for almost all times (see below). In other words, ifu(x, t)

is a finite-gap KdV solution that is doubly periodic in the complexx-plane,thenu necessarily has the form (1.2) except at isolated instants of time.

• Any numberN 6= 2 of xj is allowed in (1.2). This is trivially true, since itfollows from the work of Thickstun [17] that it is true in the soliton limit,which is a special case of the elliptic case. As a consequence, the constraint(1.4b) is solvable for any positive integer?? N 6= 2. In our numerical method,the elliptic case is viewed as a deformation of the soliton limit of the system(1.4a–b). This viewpoint is useful because it provides good initial guessesfor many of the numerical solutions of (1.4b) in Section 5. This deformationconcept does not lead to all elliptic solutions of the KdV equation, but onlyto those that have nonsingular soliton limits. In particular, we are unable tofind the solutions corresponding to the Treibich–Verdier potentials (1.8) inthis way; for these solutions it is necessary to find an initial guess by someother means.In Section (5.2), a solution corresponding to one particular Treibich–Verdierpotential is discussed. It is the only solution discussed in this paper that doesnot have a nonsingular soliton limit. Its inclusion allows us to point out somedifferences with the other examples in a very concrete way.

• If |ω1/ω2| � 1, then for a givenN > 4, nonequivalent configurations satis-fying the constraint exist that donot flow into each other under the KdV flowand which cannot be translated into each other. To the best of our knowledgethis is a new result.

• Thexi are allowed to coincide, but only in triangular numbers: if some of thexi coincide at a certain timetc, thengi(gi +1)/2 of them coincide at that timetc. At this time tc, the solution can be represented in the form (1.9) with notall gi = 1.Such timestc are referred to as collision times and the poles are said to collideat the collision time. Before and after each collision time allxi are distinct,hence pole collisions are isolated events. At the collision times, the dynamicalsystem (1.4a) is not valid. The dynamics of the poles at the collision times iseasily determined directly from the KdV equation.Gesztesy and Weikard [10, 11] demonstrate that (1.9) are elliptic finite-gappotentials of the Schrödinger equation. These potentials generalize to solu-tions of the KdV equation, but this requires thegi to be nonsmooth functionsof time. Only at the collision timestc are thegi not all one. Furthermore, atall times but the collision times, the number of parametersαi (which are timedependent) isN =∑M

i=1 gi(gi+1)/2. This shows that the generalization from[10, 11] to solutions of the KdV equation is nontrivial.

? See Section 3.?? The constraint (1.4b) is solvable forN = 2 [2]. As noted there, the corresponding solution

reduces to a solution forN = 1, with smaller periods. This case is therefore trivial and is disregarded.


In the terminology of Chapter 7 of [3], the poles are ingeneral positionifall gi are equal to one. Otherwise, if not allgi = 1, the poles are said to bein special position. We conclude that at the collision times the poles are inspecial position. Otherwise they are in general position.

• The solutions discussed here are finite-gap potentials of the stationary Schrö-dinger equation (1.6), witht treated as a parameter. In other words, eachsolution specifies a one-parameter family of finite-gap potentials of the sta-tionary Schrödinger equation. It follows from our methods that to obtain ag-gap potential that corresponds to a nonsingular soliton potential, one needsat leastN = g(g + 1)/2 polesxj . The Lamé–Ince potentials show that thislower bound is sharp. If we consider potentials that do not have soliton lim-its (such as the Treibich–Verdier potentials (1.8)) then it may be possible toviolate this lower bound.

• In Section 5, we present an explicit solution of the form (1.2) withN = 4.Notice that if in (1.8) allgi are one, the solution is reducible to a solutionwith N = 1 and smaller periods. To see this it is convenient to draw the poleconfiguration corresponding to (1.8) in the complex plane. This is actuallytrue, even ifgi is not one, but allgi are equal. In that case, (1.8) reduces to aLamé–Ince potential (1.5). Unlike any of the Treibich–Verdier solutions, the(N = 4)-solution, presented in Section 5, has a nonsingular soliton limit.

The first five conclusions are all discussed in Section 4. In Section 2, the resultsof Kruskal [15] and Thickstun [17] for the soliton solutions of the KdV equation arereviewed, but they are obtained from a point of view that is closer to the approachwe present in Section 3 for the elliptic solutions of the KdV equation. Finally, inSection 5, some explicit examples are given, including illustrations of the motionof the poles in the complexx-plane.

2. The Soliton Case: Hyperbolic Functions

In this section, the results of Kruskal [15] and Thickstun [17] for the dynamics ofpoles of soliton solutions are discussed from a point of view that will allow us togeneralize to the periodic case.

Consider the one-soliton solution of the KdV equation

u(x, t) = 2k2sech2k(x + 4k2t − ϕ). (2.1)

Herek is a positive parameter (the wave number of the soliton) determining thespeed and the amplitude of the one-soliton solution. Using the meromorphic ex-pansion [12]

1

T 2cosech2

x

T=

∞∑n=−∞

1

(x + inπT )2 (2.2)


(uniformly convergent except at the pointsx = inπT ), one easily obtains the fol-lowing meromorphic expansion for the one-soliton solution of the KdV equation:

u(x, t) = −2k2∞∑

n=−∞

1

(k(x + 4k2t − ϕ)+ i π2 + inπ)2. (2.3)

From this expression, one easily finds that the locations of the poles of the one-soliton solution of the KdV equation for all time are given by

xn = ϕ − 4k2t − iπk

(n+ 1

2

). (2.4)

This motion is illustrated in Figure 1a. Notice that the locations of the poles aresymmetric with respect to the realx-axis. This is a consequence of the reality ofthe solution (2.1). In order for a solution to be real it is necessary and sufficient thatif xn(t) is a pole, then so isx∗n(t), where ∗ denotes the complex conjugate.? Theclosest distance between any two poles isd = π/k and is constant both along thevertical line Re(x) = ϕ − 4k2t and in time. Note that the poles are moving to theleft. This is a consequence of the form of the KdV equation (1.1), which has timereversed, compared to the version Kruskal [15] and Thickstun [17] used.

Since two solitons of the KdV equation cannot move with the same speed,a two-soliton solution of the KdV equation asymtotically appears as the sum oftwo one-soliton solutions which are well-separated: the higher-amplitude soliton,which is faster, is to the right of the smaller-amplitude soliton ast → −∞. Ast →∞, the higher-amplitude soliton is to the left of the smaller-amplitude soliton.Hence ast → −∞, the pole configuration of a two-soliton solution with wavenumbersk1 and k2 is as in Figure 1b. In this limit, the two-soliton solution isa sum of two one-soliton solutions. Each results in a vertical line of equispacedpoles, with interpolar distance respectivelyd1 = π/k1 andd2 = π/k2. As long asthe solitons are well-separated, these poles move in approximately straight lines,parallel to the real axis, with respective velocitiesv1 = −4k2

1 andv2 = −4k22. Since

|v1| > |v2|, the solitons interact eventually. This interacting results in nonstraightline motion of the poles. After the interaction, the situation is as in Figure 1b, butwith the two lines of poles interchanged.

Thickstun [17] considered the case wherek1 andk2 are rationally related, sok1/k2 = p/q, wherep andq are positive integers. In this case, one can defineD = pd1 = qd2. The complexx-plane is now divided into an infinite number ofequal strips, parallel to the realx-axis, each of heightD. The realx-axis is usuallytaken to be the base of such a strip. It is easy to show [17] that the motion of thepoles in one strip is repeated in every strip. Hence, one is left studying the motionof a finite numberN (= p+q) of poles in the fundamental strip, whose base is the

? The vertical line of poles can be rotated arbitrarily. The expression (2.3) still results in a solutionof the KdV equation, but it is no longer real. Again, we will only consider real, nonsingular solutions,when restricted to the realx-axis.


(a)

(b)Figure 1. (a) The motion of the poles of a one-soliton solution in the complex plane. (b) Theasymptotic motion of the poles of a two-soliton solution.


realx-axis. Thickstun examined this motion by analyzing the exact expression fora two-soliton solution of the KdV equation.

Any two-soliton solution is expressible as [1]

u(x, t) = 2∂2x ln τ(x, t). (2.5)

It follows from this formula that the poles ofu(x, t) are the zeros ofτ(x, t) ifτ(x, t) is entire in x. Then the Weierstrass Factorization Theorem [5] gives afactorization forτ(x, t):

τ(x, t) = C∞∏k=1

(1− x

xk

)ex/xk . (2.6)

Since only the second logarithmic derivative of this function is relevant, the con-stantC is not important. If the solution is periodic in the imaginaryx-direction,this is rewritten as

τ(x, t) = CN∏n=1

∞∏l=−∞

(1− x

xn + ilD)

ex/(xn+ilD), (2.7)

where the first product runs over the poles in the fundamental strip. The secondproduct runs over all strips. Using the uniform convergence of (2.7),

u(x, t) = −2N∑n=1

∞∑l=−∞

1

(x − xn − ilD)2 , (2.8)

which, using (2.2), is rewritten as

u(x, t) = −2π2

D2

N∑n=1

cosech2π(x − xn)

D

= −2k2

2

q2

p+q∑n=1

cosech2k2(x − xn)

q, (2.9)

where the pole locationsxn depend on time:xn = xn(t). One recovers the one-soliton solution (2.1) easily, by equatingk1 = 0, p = 0, q = 1. Equation (2.9)essentially expresses a two-soliton solution as a linear superposition ofN one-soliton solutions with nonlinearly interacting phases. Note that the first equality in(2.9) is valid for arbitrary soliton solutions that are periodic inx with period iD.This is the case for ag-soliton solution if its wavenumberski , i = 1,2, . . . , g areall commensurable:(k1 : k2 : . . . : kg) = (p1 : p2 : . . . : pg), for positive distinctintegerspi , i = 1,2, . . . , g which have no overall common integer factor. Thetotal number of poles in a strip is thenN = p1+ p2+ · · · + pg. In obtaining (2.8)and (2.9), we have deviated from Thickstun’s approach [17] to an approach that isgeneralized to the elliptic case of the next section in a straightforward way.


Next, we derive the dynamics imposed on the polesxn(t) by the KdV equation.This is conveniently done by substituting (2.8) into (1.1) and examining the behav-iour near one of the poles:x = xn + ε. This results in several singular terms asε → 0, corresponding to negative powers ofε. The dynamics of the poles is thendetermined by the vanishing of the coefficients of these negative powers and thezeroth power. This results in only two nontrivial equations, obtained at orderε−3

andε−2 respectively:

xn = 12N∑

k=1,k 6=n

∞∑l=−∞

1

(xk − xn − ilD)2 + 12∞∑

l 6=0,l=−∞

1

(−ilD)2 , (2.10a)

0=N∑

k=1,k 6=n

∞∑l=−∞

1

(xk − xn − ilD)3 , (2.10b)

for n = 1,2, . . . , N . Using (2.2) and its derivative,

xn = −4π2

D2+ 12

π2

D2

N∑k=1,k 6=n

cosech2π(xk − xn)

D, (2.11a)

0=N∑

k=1,k 6=ncosech2

π(xk − xn)D

cothπ(xk − xn)

D, (2.11b)

for n = 1,2, . . . , N . Hence the dynamics of the polesxn(t) is determined by(2.11a). This dynamics is constrained by the equations (2.11b). These constraintequations (2.11b) are invariant under the flow of (2.11a). This follows from a directcalculation.

Remarks

• Since the KdV equation has two-soliton solutions for any ratio of the wave-numbersk1/k2 6= 1, the constraint (2.11b) is solvable for any value ofN ,excludingN = 2, which can only be obtained byp = q = 1, resulting inequal wavenumbersk1 andk2.

• In particular, it follows that for almost all times (i.e., all times except collisiontimes) the minimum number of poles in a fundamental strip required to obtainag-soliton solution isN = 1+ 2+ 3+ · · · + g = g(g+ 1)/2, correspondingto ag-soliton solution with wavenumbers which are related as(k1 : k2 : . . . :kg−1 : kg) = (g : g − 1 : . . . : 2 : 1).

• Equatingk1 = 0, p = 0, q = 1, one obtains from (2.11a)x1 = −4k22,

corresponding to the dynamics of the one-soliton solution. The asymptoticbehavior of the poles of a two-soliton solution also follows from (2.11a): fromthe separation of the poles into distinct vertical lines, it follows from (1.4a)that the velocity of these vertical lines is given by the one-soliton velocity for


each line, as expected. This result follows from easy algebraic manipulationand the identity

p2− 1

3=

p−1∑n=1

cosec2(nπ

p

), (2.12)

valid for any integerp > 1.• A full analysis of the interaction of the poles for the case of any two-soliton

solution withk1/k2 = p/q is given in [17].

3. The Elliptic Case

Consider the quasiperiodic finite-gap solutions of the KdV equation withg phases[14]

u(x, t) = 2∂2x ln θg(kx + ωt + φ|B), (3.1)

where

θg(z|B) =∑m∈Zg

exp(

12m · B ·m+ im · z

), (3.2)

a hyperelliptic Riemann theta function of genusg. Theg × g real Riemann matrix(i.e., symmetric and negative definite)B originates from a hyperelliptic Riemannsurface with only one point at infinity. Furthermore,k, ω andφ areg-dimensionalvectors.

The derivation of equations (2.9), (2.11a) and (2.11b) is easily generalized tothe case where the solution is periodic not only in the imaginaryx-direction, butalso in the realx-direction:

u(x + L1, t) = u(x, t) = u(x + iL2, t). (3.3)

This divides the complexx-plane into an array of rectangular domains, each ofsizeL1 × L2. One of these domains, called the fundamental domainS, is conve-niently placed in the lower left corner of the first quadrant of thex-plane. The thetafunction has the property [6]

θg(z + iB ·M + 2πN |B) = θg(z|B)exp(− 1

2M · B ·M + iM · z), (3.4)

for any pair ofg-component integer vectorsM,N . This expression is useful todetermine conditions on the wavevectork and on the Riemann matrixB in orderfor u(x, t), given by (3.1), to satisfy (3.3):

∃N0,M0 ∈ Zg : kL1 = 2πN0, kL2 = B ·M0. (3.5)


These results are now used to determine the number of polesN of u(x, t) in thefundamental domain. The poles of (3.1) are given by the zeros ofϑ(x, t) = θg(kx+ωt + φ|B), regarded as a function ofx:

N = 1

2πi

∮∂S

d ln ϑ(x, t)

= 1

2πi

(∫ L1

0d ln ϑ(x, t) +

∫ iL2

0d ln ϑ(x + L1, t)−

−∫ L1

0d ln ϑ(x + iL2, t)−

∫ iL2

0d ln ϑ(x, t)

)using(3.4)= 1

2πi

∫ L1

0d ln

ϑ(x, t)

ϑ(x + iL2, t)

using(3.4)= 1

2πi

∫ L1

0d(−ixM0 · k)

= −M0 ·N0 = − L1

2πL2M0 ·B ·M0. (3.6)

The first equality of (3.6) confirms thatN is an integer. The second equality showsthatN is positive, by the negative-definiteness ofB.

We now proceed to determine the dynamical system satisfied by the motion oftheN poles ofu(x, t) in the fundamental domainS. Again, the poles ofu(x, t) arethe zeros ofϑ(x, t). Furthermore, zeros ofϑ(x, t) result in double poles ofu(x, t),as in the hyperbolic case.

The Weierstrass Factorization theorem [5] gives the following form forϑ(x, t):

ϑ(x, t) = ecx2/2∏k

(1− x

xk

)exxk+ x2

2x2k , (3.7)

where the product runs over all polesxk. The additional exponential factors, ascompared to (2.6), are required because the poles now appear in a bi-infinite se-quence: both in the vertical and horizontal directions. These exponential factorsensure uniform convergence of the product. The parameterc is allowed to de-pend on time. It determines the behavior ofϑ(x, t) as x approaches infinity inthe complexx-plane [6]. Using (3.3), this is rewritten as

ϑ(x, t) = exp(cx2/2)N∏n=1

∞∏m=−∞

∞∏l=−∞

(1− x

xn +mL1+ ilL2

)×

×exp

(x

xn +mL1+ ilL2+ x2

2(xn +mL1+ ilL2)2

). (3.8)

The first product runs over the number of poles (N) in the fundamental domain,the second and third products result in all translations of the fundamental domain.


From the uniform convergence of (3.8),

u(x, t) = 2c − 2N∑n=1

∞∑m=−∞

∞∑l=−∞

(1

(x − xn −mL1− ilL2)2−

− 1

(xn +mL1+ ilL2)2

). (3.9)

Using the definition of the Weierstrass function (1.3), this is rewritten as

u(x, t) = 2c − 2N∑n=1

℘(x − xn)+ 2N∑j=1

℘(xn), (3.10)

where the periods of the Weierstrass function are given by 2ω1 = L1,2ω2 = iL2.Define

c = 2c + 2N∑n=1

℘(xn). (3.11)

The dynamics of the polesxn = xn(t) is determined by substitution of (3.10)or (3.9) into the KdV equation and expanding in powers ofε for x near a pole:x = xk + ε. Equating the coefficients ofε−3, ε−2 andε0 to zero result in

xn = 12N∑

j=1,j 6=n℘ (xj − xn), (3.12a)

0=N∑

j=1,j 6=n℘′(xj − xn), (3.12b)

˙c = 0 ⇐⇒ c(t) = α = 0, (3.12c)

for n = 1,2, . . . , N . (The constantα can always be removed by a Galilean shift, soit is equated to zero, without loss of generality.) The constraints (3.12b) are invari-ant under the flow, as can be checked by direct calculation. Notice that (3.12a–b)are identical to the equations obtained by Airault, McKean and Moser [2]. Theseequations are obtained here in greater generality: any solution (3.1) that is doublyperiodic in thex-plane gives rise to a system (3.12a–b). This allows us to reach theconclusions stated in the next section.

Remarks

• In the limit L1 → ∞, Equations (3.12a–b) reduce to (2.11a–b). This limitis most conveniently obtained from the Poisson representation of the


Weierstrass function:

℘(x) =(π

L2

)2(

1

3+ cosech2

πx

L2+

+∞∑

n=−∞,n6=0

{cosech2

π

L2(x + nL1)− cosech2

nπL1

L2

}). (3.13)

This representation is obtained from (1.3) by working out the summation inthe vertical direction. It gives the Weierstrass function as a sum of expo-nentially localized terms, hence few terms have important contributions inthe fundamental domain. A Poisson expansion for℘′(x) is obtained fromdifferentiating (3.13) term by term with respect tox.

• Define the one-phase theta functionθ1(z, q) [12]:

θ1(z, q) = 2∞∑n=0

(−1)nq(n+1/2)2 sin(2n+ 1)z, (3.14)

with |q| < 1. If L2 < L1, then the relationship

℘(z) = a − ∂2x ln θ1(πz/L1, iL2/L1),

with a a constant [12], allows us to rewrite (1.2) as

u(x, t) = a + 2∂2x ln

N∏j=1

θ1

(π

L1(x − xj (t)), i L2

L1

), (3.15)

with a = −2aN . Hence, for the doubly-periodic solutions of the KdV equa-tion of the form (3.1), it is possible to rewrite theg-phase theta function asa product ofN 1-phase theta functions, with nonlinearly interacting phases.Note that this does not imply that theg-phase theta function appearing in (3.1)is reducible. Reducible theta functions do not give rise to solutions of the KdVequation [6].

• By taking another time derivative of (3.12a) and using (3.12b), one obtains

xn = −(12)2N∑

j=1,j 6=n℘′(xj − xn)℘ (xj − xn). (3.16)

It is known that this system of differential equations is Hamiltonian [4], withHamiltonian


H = 1

2

N∑k=1

p2k +

(12)2

2

N∑k=1

N∑j=1,j 6=k

℘2(xk − xj ), (3.17)

and canonical variables{xk, pk = xk}. A second Hamiltonian structure for theequations (3.16) is given in [4]. A Lax representation for the system (3.12a–c)is also given there. This Lax representation is a direct consequence of the lawof addition of the Weierstrass function [12]. It is unknown to us whether aHamiltonian structure exists for the constrained first-order dynamical system(3.12a–b).The Hamiltonian structure (3.17) shows that the system (3.12a–b) is a (con-strained) member of the elliptic Calogero–Moser hierarchy [3].

4. Discussion of the Dynamics

In this section, the constrained dynamical system (3.12a–c) is discussed. In partic-ular, the assertions made in the introduction are validated here.

For reality of the KdV solution (3.1) whenx is restricted to the real line, itis necessary and sufficient that ifxj (t) appears, then so doesx∗j (t). Because theWeierstrass function is a meromorphic function of its argument that is real valuedon the real line, this reality constraint is invariant under the dynamics (3.12a).

As a consequence, the distribution of the poles in the fundamental domainS issymmetric with respect to the horizontal centerline ofS. Poles are allowed on thecenterline. Most of what follows is valid for both real KdV solutions? and KdVsolutions that are not real, but we restrict our attention to real KdV solutions.

4.1. ALL FINITE -GAP ELLIPTIC SOLUTIONS OF THE KDV EQUATION ARE OF

THE FORM (1.2), UP TO A CONSTANT

A straightforward singularity analysis of the KdV equation [2] shows that anyalgebraic singularity of a solution of the KdV equation is of the typeu(x, t) =−2/(x−α(t))2+O(x−α), for almost all timest . At isolated timestc, the leadingorder coefficient is not necessarily−2. It can be of the form−g(g+1) (see below),but the exponent of the leading term is always−2.

Hence, an elliptic function ansatz foru(x, t) can only have second-order polesand with the substitutionu(x, t) = 2∂2

x lnϑ(x, t) gives rise to a Weierstrass ex-pansion of the form (3.8), with an arbitrary prefactor exp(c(x, t)), for an arbitraryfunction c(x, t), entire inx. Substitution of this ansatz in the KdV equation thendetermines thatcxx is doubly periodic and meromorphic inx. The only suchc isa constant. Hence, all finite-gap elliptic solutions of the KdV equation are of theform (1.2).? ‘Real KdV solution’ refers to a solution of the KdV equation which is real whenx is restricted

to the realx-axis.


4.2. IF L1/L2� 1, ANY NUMBER OF POLESN 6= 2 IN THE FUNDAMENTAL

DOMAIN IS ALLOWED

This immediately follows from the soliton case, which is a special case of theelliptic case. Thickstun’s results [17] already implied that any value ofN 6= 2occurs. We do not provide a direct proof for finite values ofL1. On the other hand,the numerical evidence presented in Section 5 supports this statement.

We have already argued that the equations (2.9) and (2.11a–b) are obtainedfrom (1.2) and (3.12a–b) in the limitL1 → ∞. On the other hand, (3.13) allowsus to rewrite each term in (1.2) as an infinite sum of solitons, each of which islocalized in a different real-period interval of the solution. In the limitL1 → ∞,only two terms in each one of these series remain: the constant term and a one-soliton term. This gives rise to the soliton limit of the solution (2.9). The terms thatvanish asL1→∞ are then regarded as a deformation of the soliton limit. Similardeformations are valid at the level of the dynamical system and the constraints. Thesuccess of the numerical method prompts us to formulate the following

CONJECTURE. Every nonsingular soliton solution that is periodic inix withperiodiL2 is obtained as the limit in whichL1 → ∞ of an elliptic solution withperiodsiL2 andL1.

At this point, it is appropriate to remark that if one is interested in ellipticsolutions of the Kadomtsev–Petviashvili (KP) equation,

∂x (−ut + 6uux + uxxx)+ 3σ 2uyy = 0 (4.1)

(with parameterσ ), then the Equations (1.2), (3.12a–b) are replaced by [4]

u(x, y, t) = −2N∑n=1

℘(x − xn(y, t)), (4.2a)

∂xn

∂t= 3σ 2

(∂xn

∂y

)2

+ 12N∑

j=1,j 6=n℘ (xj − xn), (4.2b)

σ 2∂2xn

∂y2= −16

N∑j=1,j 6=n

℘′(xj − xn). (4.2c)

This clarifies the appearance of the constraint (3.12b) on the motion of the polesof elliptic solutions of the KdV equation, where the poles are independent ofy.For y-independent solutions, the KP equation reduces to the KdV equation, andEquations (4.2a–c) reduce to (1.2), (3.12a–b), forcing the poles to remain on theinvariant manifold defined by (3.12b). For the KP equation, no such constraintexists and the number of polesN in the fundamental domain can be any integer,not equal to two.


4.3. FOR ANY N > 4, NONEQUIVALENT CONFIGURATIONS EXIST, FORL1/L2

SUFFICIENTLY LARGE

Consider the asymptotic behavior fort →−∞ of limL1→∞ u(x, t). In this solitonlimit, as t → −∞, the poles are collected in groups corresponding to one-solitonsolutions.

In this section, two configurations are callednonequivalentif the above asymp-totic behavior results in two different groupings of the poles.

ForN = 3, all configurations are equivalent to one configuration. In the limitL1→∞, this configuration corresponds to the two-soliton case withk1 : k2 = 2 :1. This configuration is discussed in Section 5.1.

ForN = 4, all configurations are again equivalent to one configuration. Thisconfiguration corresponds to the two-soliton case withk1 : k2 = 3 : 1. Recallthatk1 andk2 are not allowed to be equal, hence a configuration with two poles tothe left and two poles to the right does not exist. Another way of expressing thatonly one configuration exists is thatN = 4 can only be decomposed in one wayas the sum of distinct positive integers without common factor (> 1), namely asN = 3+ 1. Again, allN = 4 configurations are equivalent. This configuration isdiscussed in Section 5.2. That section also discusses another example of anN = 4potential which does not have a nonsingular soliton limit. This potential is a specialcase of one of the Treibich–Verdier potentials (1.8).

Any integerN > 4 can be written as a sum of distinct positive integers withoutoverall common factor in more than one way.? Let the number of terms in them-thdecomposition ofN be denoted asNm, thenN = ∑Nm

i=1 ni, with theni distinctand having no overall common factor. This configuration corresponds to theNm-soliton case with wavenumber ratios(k1 : k2 : . . . : kNm) = (n1 : n2 : . . . : nNm).A solution with these wave numbers hasNm phases and is anNm-soliton solution.Hence for anyN > 4 there exist at least as many different configurations as thereare decompositions ofN into distinct positive integers, without overall commonfactor. These configurations need not have the same number of phases.

Two nonequivalent configurations corresponding toN = 5 are discussed inSection 5.3.

4.4. THE POLES ONLY COLLIDE IN TRIANGULAR NUMBERS

A collision of poles is a local process in which only the colliding poles play asignificant role. The analysis of the collisions is identical to that of the rationaland the soliton cases because close to the collision point, the Weierstrass functionreduces to 1/x2. Kruskal [15] already noticed that the poles do not collide in pairs,but triple collisions do occur. In fact, any triangular number of poles can participatein a collision, in which case the solution of the KdV equation at the collision timetc, nearby the collision pointxc is given byu(x, tc) = −g(g + 1)/(x − xc)2.? 5= 1+ 4= 2+ 3,N = 1+ (N − 1) = 1+ 2+ (N − 3), forN > 5.


Asymtotically near the collision pointxc, before the collision, i.e.,t < tc, the poleslie on the vertices of a regular polygon [2] withg(g + 1)/2 vertices. Fort > tc,the poles emanate from the collision points, again forming a regular polygon withg(g + 1)/2 vertices [2]. Ifg(g + 1)/2 is even, this polygon is identical to thepolygon before the collision. Ifg(g + 1)/2 is odd, the polygon is rotated aroundthe collision point by 2π/g(g + 1) radians.

Of all these collision types, the one where three poles collide (corresponding tog = 2) is generic. It is the one observed in the examples illustrated in Section 5.

Since the poles only collide in triangular numbers, it is possible that at any giventime tc the solution of the KdV equation has the form (1.9), with not allgi = 1.At almost every other timet , such a solution hasN = ∑M

i=1 gi(gi + 1)/2 distinctpoles.

4.5. THE SOLUTIONS(1.2) ARE FINITE-GAP POTENTIALS OF THE STATIONARY

SCHRÖDINGER EQUATION(1.6)

By construction the solutions (1.2) are periodic inx because they are obtained asa Weierstrass factorization of the theta function appearing in (3.1), upon whichwe have imposed the double periodicity. Hence the solutions (1.2) are finite-gappotentials of the Schrödinger operator. In [11], another proof of this can be found.

For solutions that are elliptic deformations of the nonsingular solitons of Sec-tion 2, more can be said: an elliptic deformation of ag-soliton solution is ag-gap potential of the Schrödinger equation. The reasoning is as follows: we al-ready know that any elliptic deformation results in a finite-gap potential of theSchrödinger equation. On the other hand, any finite gap potential of the Schrödingerequation is of the form (3.1). The soliton limit of such a finite-gap solution withg-phases is ag-soliton solution [3]. Hence the number of phases of an ellipticdeformation of ag-soliton solution is equal tog.

This limit is the soliton limit of the periodic solutions, in which the fundamentaldomain reduces to the fundamental strip. In order to have ag-soliton solution ofthe KdV equation, we remarked in Section 2 that at leastN = g(g + 1)/2 =1+2+· · ·+g poles are required in the fundamental strip. Hence, this many poles arerequired in the fundamental domain to obtain ag-gap potential of the Schrödingerequation that is an elliptic deformation of a nonsingular soliton solution.

5. Examples

In this section, some explicit examples of elliptic solutions of the KdV equationare discussed. These are illustrated with figures displaying the motion of the polesin the fundamental domain. Other figures display the solution of the KdV equationu(x, t) as a function ofx and t . All these figures were obtained from numericalsolutions of the corresponding constrained dynamical system. In all cases, the con-strained dynamical system was solved using a projection method: the dynamical


system (3.12a) is used to evolve the system for some time. Subsequently, the newsolution is projected onto the constraints (3.12b) to correct numerical errors, afterwhich the process repeats.

In all examples given,L1 = 4 andL2 = π . This seems to indicate that one canwander far away fromL1/L2 � 1 and still obtain soliton-like elliptic solutionsof the KdV equation. This is not surprising as (3.13) indicates that as perturbationparameter on the soliton case one should useε = exp(−2πL1/L2). For the valuesgiven above, this givesε = 0.00034.

5.1. THE SOLUTION OF DUBROVIN AND NOVIKOV [7]: N = 3

Dubrovin and Novikov [7] integrated the KdV equation with the Lamé–Ince poten-tial u(x,0) = −6℘(x − xc) as initial condition. They found the solution to be el-liptic for all time, withN = 3. They gave explicit formulae for the solution, whichthey remarked was probably the simplest two-gap solution of the KdV equation.The dynamics of the poles in the fundamental domain is displayed in Figure 2a.Figure 2b displays the corresponding two-phase solution of the KdV equation. An-imations of the behavior of the poles and ofu(x, t) ast changes are also available athttp://amath-www.colorado.edu/appm/other/kp/papers. Notice the soliton-like interactions of the two phases in the solution. In terms of the classification ofLax [16], these are interactions of type (c) (i.e.,u(x, t) has only one maximumwhile the larger wave overtakes the smaller wave).

From Figure 2a, it appears that the Dubrovin–Novikov solution is periodic intime. This was indeed proven by Ènol’skii [8].

For this specific solution only one of the three constraint equations is inde-pendent: since the derivative of the Weierstrass function is odd, the sum of theconstraints is zero. Furthermore, labelling the three poles byx1, x2 and x3, forreality x2 = x∗1 + iL2 andx3 is on the centerline. Hence, the second constraint isthe complex conjugate of the first constraint. The constraints (3.12b) reduce to thesingle equation

℘′(x1 − x∗1)+ ℘′(x1− x3) = 0. (5.1)

This equation was solved numerically to provide the initial condition shown in Fig-ure 2a. The initial guess required for the application of Newton’s method is basedon the knowledge of the soliton limit. In that case two poles on the right representa faster soliton, one pole on the left represents the slower soliton. The periodic caseis not that different: the vertical line of poles with the smallest vertical distancebetween poles has poles closer to the realx-axis than the others and correspondto the wave crest with the highest amplitude, as seen in Figure 2b. We refer to theDubrovin–Novikov solution as a(2 : 1)-solution because of the natural separationof the poles in a group of 2 poles(x1, x2) and a single pole(x3).


(a)

(b)Figure 2. The solution of Dubrovin and Novikov, withL1 = 4 andL2 = π . (a) The motionof the poles in the fundamental domain. The initial position of the poles is indicated by theblack dots. The arrows denote the motion of the poles. (b) The KdV solutionu(x, t).

Equatingx1 = x3 + ε and only considering the singular terms of (5.1), it ispossible to examine the location of the poles close to a collision pointsxc. With℘′(x) = −2/x3 in this limit andε = εr + iεi, one finds

ε3r − 3ε2

i εr = 0, 9ε3i − 3ε2

r εi = 0. (5.2)

This set of equations has three solutions, corresponding to the three distancesbetween the poles:εr/εi ∈ {0,

√3,−√3}. This allows for two triangular con-

figurations of the poles: an equilateral triangle pointing left of the collision pointand one pointing right.


Using the dynamical system (3.12a) in the same way and only retaining singularterms results in

εr = − 3

4ε2i

+ 3ε2r − ε2

i(ε2r + ε2

i

)2 , εi = 6εrεi(ε2r + ε2

i

)2 . (5.3)

Since the constraints (3.12b) are invariant under the flow (3.12a), the solutions to(5.2) give invariant directions of the system (5.3). Along these invariant directions,one obtains ordinary differential equations for the motion of the poles as they ap-proach the collision point. It follows from these equations that the poles approachthe collision pointxc with infinite velocity. Integrating the equations with initialconditionε(tc) = 0 gives

ε = 35/6

2(√

3+ i)(tc − t)1/3. (5.4)

Using the three branches of(tc − t)1/3 results in the dynamics of each edge of theequilateral triangle. Ift < tc this triangle is pointing left, fort > tc it is pointingright.

5.2. N = 4: AN ELLIPTIC DEFORMATION AND A TREIBICH–VERDIER

SOLUTION

The next solution we discuss has 4 poles in the fundamental domain and is anelliptic deformation of a soliton solution. In the limitL1 → ∞, this solutioncorresponds to a two-soliton solution with wavenumber ratiok1/k2 = 3/1, so thissolution is refered to as a(3 : 1)-solution.

The motion of the poles in the fundamental domain is displayed in Figure 3.Corresponding to the given wavenumber ratio, the amplitude ratio of the two phasespresent in the solution is roughlyk2

1/k22 = 9/1. As a consequence, the form of

u(x, t) is not very illuminating and it has been omitted. Animations with the timedependence of both the positions of the poles and ofu(x, t) are again available athttp://amath-www.colorado.edu/appm/other/kp/papers.

Note that the poles of the(3 : 1)-solution do not collide. This is in agree-ment with the results of Thickstun [17] who outlined which configurations leadto collisions and which do not, in the hyperbolic case. As mentioned before, theexamination of collision behavior is essentially local and no differences appearamong the rational, hyperbolic and elliptic cases.

Another configuration withN = 4 exists. Consider the potential

u(x, t = tc) = −2℘(x − x0)− 6℘(x − x0− ω1), (5.5)

with x0 on the centerline. This is a Treibich–Verdier potential, obtained from (1.9)with M = 2, g1 = 1, g2 = 2, α1 = x0 andα2 = x0 + ω1. It is referred to as aTreibich–Verdier potential because the position of the poles is given in terms of the


Figure 3. N = 4: the pole dynamics of a(3 : 1)-solution. The initial positions of the fourpoles are indicated. The arrows on the centerline indicate that the poles there move in bothdirections.

periods of the Weierstrass function, as in (1.8). Also, it can be obtained from (1.8)as a degenerate case. As beforeN =∑2

i=1 gi(gi+1)/2= 4, hence for all times thatare not collision times, this solution has 4 distinct poles in the fundamental domain.The timet = tc is a collision time. Immediately after the collision timet = tc, the3 poles located atx0+ω1 separate, as in the Dubrovin–Novikov solution, along anequilateral triangle. The result appears to be a three-phase solution. However, it isknown that the potential (5.5) is a two-gap potential of the Schrödinger equationand its hyperelliptic Riemann surface is given explicitly in [3]. This solution isnot an elliptic deformation of a nonsingular soliton solution and the separationinto different phases does not make sense. This is also seen from the followingargument: if, for a fixed time which is not a collision time, we attempt to take thelimit asL1 = 2ω1→∞, the poles seem to separate in three distinct solitons withrespective wave numbers(k1 : k2 : k3) = (2 : 1 : 1). Such a nonsingular solitonsolution does not exist for the KdV equation and the separation into different phasesdoes not make sense.

The dynamics of the poles is illustrated in Figure 4a. The corresponding KdVsolution is shown in Figure 4b.

The dynamics of the poles illustrated in Figure 4a exhibits behavior that ap-pears qualitatively different from any other solution discussed here. The trajectoriestraced out by the motion of the poles in the fundamental domain appear to havesingular points (cusps), away from the collision points. Upon closer investigation,these ‘cusps’ are only a figment of the resolution of the plot and the poles traceout a regular curve as a function of time, away from the collision times. Exactlywhy the global pole dynamics of the Treibich–Verdier potential (5.5) under theKdV flow appears so different from the pole dynamics of elliptic deformations ofsoliton solutions of the KdV equation is an open problem. Another question onemay ask is whether similar behavior is observed for other solutions originatingfrom Treibich–Verdier potentials.


(a)

(b)Figure 4. An N = 4,M = 2 Treibich–Verdier solution, withL1 = 4 andL2 = π . (a) Themotion of the poles in the fundamental domain. The initial position of the poles is indicatedby the black dots. The initial timet = 0 was chosen different from the collision timestc. Thearrows denote the motion of the poles. (b) The KdV solutionu(x, t).

5.3. N = 5: TWO DIFFERENT POSSIBILITIES

ForN = 5, two soliton configurations are possible, and corresponding to each ofthese is an elliptic solution. The first solution is a(4 : 1)-solution. The secondsolution is a(3 : 2)-solution.

The(4 : 1)-solution offers no new pole-dynamics: initially 1 pole is located onthe centerline, at the left in the fundamental domain. The other poles are located atthe right of the fundamental domain, symmetric with respect to the centerline. Thethree poles closest to the centerline interact as the(2 : 1)-solution. The two outer


(a)

(b)

(c)Figure 5. N = 5: (a) The pole dynamics of a(4 : 1)-solution in the fundamental domain.(b) The pole dynamics of a(3 : 2)-solution in the fundamental domain. (c) The KdV solutionu(x, t) corresponding to the pole dynamics in (b). In (a) and (b), the initial positions of thepoles are indicated. Both solutions are quasiperiodic in time.


Figure 6. The pole dynamics of a(3 : 2 : 1)-solution. The black dots mark the initial positionof the poles; the grey dots mark the position of the poles att = 0.4.

poles behave as the two outer poles of the(3 : 1)-solution. The pole dynamics ofthe(4 : 1)-solution is displayed in Figure 5a.

The (3 : 2)-solution is more interesting. It is displayed in Figure 5c, togetherwith the motion of the poles in the fundamental domain 5b. Again, the two crestsof u(x, t) interact in a soliton-like manner. In Lax’s classification [16], this is aninteraction of type (a), where at every time two maxima are observed. Figure 5bonly displays the motion of the poles for a short time, in order not to clutter thepicture. The motion of the poles is presumably quasiperiodic in time, as is the casefor the (4 : 1)-solution. It appears that the two poles above (or below) the middleline of the fundamental domain share a common trajectory. It is an open problemto establish whether or not this is the case.

5.4. N = 6: TWO DIFFERENT POSSIBILITIES. A THREE-PHASE SOLUTION

ForN = 6, two distinct pole configurations are possible. The first one correspondsto a(5 : 1)-solution and results in a two-gap potential of the Schrödinger equation.It essentially behaves as the(3 : 1)-solution with two more poles added, which alsobehave as the outer poles of the(3 : 1)-solution.

The second configuration is a(3 : 2 : 1)-solution, which limits to a three-solitonsolution with wavenumber ratio(k1 : k2 : k3) = (3 : 2 : 1). This elliptic solution isa three-phase solution of the KdV equation.

The amplitude ratio of the(3 : 2 : 1)-solution is(9 : 4 : 1), which explainswhy the third phase is hard to notice in Figure 7. Animations of the pole dy-namics and of the time dependence of the(3 : 2 : 1)-solution are available athttp://amath-www.colorado.edu/appm/other/kp/papers.


Figure 7. A (3 : 2 : 1)-solution of the KdV equation, corresponding to the pole dynamics inFigure 6.

Acknowledgements

The authors acknowledge useful discussions with B. A. Dubrovin, S. P. Novikov,C. Schober, A. Treibich and A. P. Veselov. This work was carried out at the Uni-versity of Colorado and the Mathematical Sciences Research Institute. It was sup-ported in part by NSF grants DMS 9731097 and DMS-9701755.

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8. Ènols’kii, V. Z.: On solutions in elliptic functions of integrable nonlinear equations associatedwith two-zone Lamé potentials,Soviet Math. Dokl.30 (1984), 394–397.

9. Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M.: Method for solving theKorteweg–de Vries equation,Phys. Rev. Lett.19 (1967), 1095–1097.

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75

On Bianchi and Bäcklund Transformations ofTwo-Dimensional Surfaces inE4

YU. AMINOV ? and A. SYMInstitute of Theoretical Physics of Warsaw University, Hoza street 69, 00-681 Warsaw, Poland

(Received: 22 July 1999)

Abstract. In this paper we discuss the existence and properties of the Bianchi transformations forpseudospherical surfaces inE4. The results of the paper show that the theory of Bianchi transfor-mations in the discussed case is essentially different from the well-known case of pseudosphericalsurfaces inE3 (in generaln-manifolds of constant and negative curvature inE2n−1).

Mathematics Subject Classifications (2000):53A05, 53A25, 37K35.

Key words: Bianchi transformations, Bäcklund transformations, differential geometry, Gauss curva-ture, pseudospherical surface.

1. Historical Remarks

These remarks serve as an introduction to the fascinating history of both Bianchitransformations and their generalization known as Bäcklund transformations.Simultaneously, we fix a terminology of the problem following the tradition ofclassical differential geometry.

To begin, we recall the classical notion of an evolute of a surfaceS in E3 [1].Consider a family of all normals toS. In general, there exist two uniquely definedsurfaces, sayF andF , such that each normal toS is tangent to both surfacesF andF . ThenF together withF is an evolute ofS.

Luigi Bianchi in his 1879 habilitation thesis [2] introduced a notion of ‘com-plementary surfaces’. There are exactly two piecesF andF of an evolute of somesurface inE3. In this way, for Bianchi, the primary objects areF andF , while Sis a secondary one. Indeed, at the very beginning of his thesis he formulated themain problem as follows: “It is particularly interesting to consider complementarysurfaces to a surface of constant negative curvatures”.

In order to solve this problem, Bianchi made use of the following observation[1]: The tangents to curves of geodesic foliation of surfaceF are normal to aninfinity of parallel surfaces.

? On leave of absence from B. I. Verkin Institute for Low Temperature of NAN of the Ukraine,47 Lenin Ave, 310164, Kharkov, Ukraine.

76 YU. AMINOV AND A. SYM

Let F be a pseudospherical surface of Gaussian curvatureK = −b2 = const.F can be always equipped with coordinates(x, y) such that the metric form ofFis equal to

ds2 = e2bydx2 + dy2. (1)

The system(x, y) is called an ‘horocyclic system’ since the curvesy = constare horocycles. Simultaneously, the linesx = const are geodesics. Consider now aset of all tangents to the curves of the geodesic foliation ofF defined by geodesicsx = const. Relying on the earlier work of Weingarten [3], Bianchi pointed out(without proof) that the complementary surfaceF is also pseudospherical of thesame Gaussian curvatureK = −b2. The corresponding analytical expression isvery simple. Letr = r(x, y) be a position vector toF . A position vectorr =r(x, y) to F is given by

r = r − 1

bry. (2)

In fact, (2) is the Example 4 in [1], p. 290. To summarize, one can speak of thewell-defined transformation of the pseudospherical(K = −b2) surfaceF intothe pseudospherical surfaceF of the same curvature. After Darboux [4], we callF → F the ‘Bianchi transformation’.

In 1883, A. V. Bäcklund published a paper, [5], which contains a general-ization of the Bianchi transformation. This is still a transformation between twopseudospherical surfaces of the same Gaussian curvature. This paper is an essen-tially analytical one. Bäcklund representedF(F ) in the Monge form z=f (x, y)(z′ = f ′(x′, y′)) and replaced the original Bianchi transformation by asystem of four equations relating(x, y, z, p, q) and(x′, y′, z′, p′, q ′), wherep =zx, q = zy, etc. Next, he formulated a generalization of this system containinga real parameter which defines an angle between normals to the correspondingsurfaces at the corresponding points. Within the theory of pseudospherical surfacesin E3, this generalization is called the ‘Bäcklund transformation’.

Bäcklund paper [5] is based on his earlier paper [6].In any case, the Bäcklund transformation of [5] gave rise to further devel-

opments which were finally completed by the French school of J. Clairin andE. Goursat as a theory of Bäcklund transformations [7].

In a remarkable paper, Bianchi [8] reformulated the Bäcklund transformation ina purely geometric setting. In the same paper, he pointed out, for the first time thepossibility, of interpreting Bäcklund’s results within Kummer’s theory of congru-ences. We recall that according to the classical terminology, a congruence inE3 isa 2-parameter family of straight lines. Indeed, the Bianchi transformation, defines,for instance, a congruence: this is a 2-parametric family of tangent lines definedby ry . The important point is thatF andF can be now interpreted as the so-calledfocal surfaces of the congruence. This idea is fully developed in [9] published in1887. For some other historical and modern aspects of this topic, see [10].

ON BIANCHI AND BÄCKLUND TRANSFORMATIONS 77

2. Modern Developments

Surprisingly enough, the topic of Bäcklund transformations has been out fashionfor many years. A strong resurgence in interest in them has been evident sincethe creation of the theory of solitons (the theory of completely integrable sys-tems of partial differential equations) [11]. In certain respects, a series of papersby G. L. Lamb [12, 13] and H. Wahlquist and F. Estabrook [14, 15] is a directcontinuation of the work produced by the school of Clairin and Goursat.

We have already pointed out that the Bianchi transformation (‘complementarytransformation’ in Bianchi’s terminology) is a particular case of the Bäcklundtransformation for pseudospherical surfaces corresponding to the special choiceof Bäcklund parameters. This is why – despite its geometrical beauty – not muchof attention was paid to this transformation in the past. There is, for instance,a discussion of Bianchi transformations by G. Darboux [4]. A generalization ofthe Bianchi transformation (with replacements: ‘pseudospherical surface’ by ‘anisometric immersion of the domain of then-dimensional Lobachevski space’ andE3 by E2n−1) was formulated and proved in 1978 [16], almost one hundred yearsafter Bianchi [2].

On the other hand, a similar generalization of the Bäcklund transformationfor pseudospherical surfaces was presented by Tenenblat and Terng in 1979 [18],see also [19]. Their approach is based on Bianchi’s formulation of the Bäcklundtransformations in terms of congruences.

Paper [16] initiated further investigations concerning the generalized Bianchitransformations [17]. Similarly, papers [18, 19] are a point of departure for furtherstudies [20].

3. The Questions

To the best of our knowledge, investigation of the Bäcklund (Bianchi) transfor-mations for isometric immersions ofn-dimensional Lobachevski space intoEN

with N > 2n − 1 has never been undertaken. In this paper we attempt to answersome questions related to the Bianchi transformations for isometric immersions ofdomains of Lobachevski planeL2 intoE4.

We start withL2 equipped with the horocyclic system of coordinates(x, y).The metric ofL2 in these coordinates is given by

ds2 = e2ydx2 + dy2. (3)

Note that (3) is in fact (1) withb = 1, (K = −1). Coordinate curvesx = const aregeodesics while curvesy = const are of the geodesic curvature equal to 1.F 2 denotes some isometric immersion of the domain ofL2 intoE4. r = r(x, y)

stands for a position vector toF 2. Guided by formula (2), we putr = r − ry . Asurface inE4 defined by its position vectorr is called a Bianchi transform ofF 2

and is denoted byF 2 while a mapF 2→ F 2 is called a Bianchi transformation inE4. The Gauss curvature ofF 2 is denoted byK.


A few questions can be asked here.

QUESTION 1. When, in the definition above,E4 is replaced byE3, we areassured thatF 2 is a pseudospherical surface as well. Does the same hold in thediscussed case(K = −1?)?

QUESTION 2. Is the vectorry tangent toF 2 as well?

Soon we shall see that the answers to the above-mentioned questions are nega-tive. Hence, one can weakes these two questions.

QUESTION 3. Does a pseudospherical surfaceF 2 in E4 exist (the caseF 2 ⊂ E3

is excluded) for whichF 2 is pseudospherical as well?

QUESTION 4. Does a pseudospherical surfaceF 2 inE4 exist such that the vectorry is tangent toF 2 as well? In this case, we say that the vector fieldry onF 2 enjoysa bi-tangency property. For simplicity, we also say thatF 2 enjoys a bi-tangencyproperty.

QUESTION 5. How many pseudospherical surfaces with bi-tangency propertyexist inE4?

This question in turn inspires the following general question.

QUESTION 6. Consider now an arbitrary surfaceF 2 in E4. Let a be a field ofunit tangent vectors toF 2. Let F 2 be given by its position vectorr = r+ha, whereh = h(u1, u2) is a function onF 2. The question is: under which conditions doesthe vector fieldha exist onF 2 with the bi-tangency property?

It is evident that this transformationr → r is a generalization of the Bäcklundtransformation.

4. The Answers

The answer to Questions 1 and 2 is given by the following theorem:

THEOREM 1. There exist pseudospherical surfacesF 2 inE4 such that its Bianchitransformation hasK 6= −1.

For surfacesF 2 ⊂ E4 in general position, the vectorry is not tangent toF 2.If the bi-tangency property is fulfilled, then the Bianchi transformation has itsremarkable property as inE3:

THEOREM 2. If F 2 enjoys the bi-tangency property, thenK = −1 (F 2 is apseudospherical surface as well).


Theorem 2 answers Question 3. The answer to Questions 4 and 5 is given bythe following theorem:

THEOREM 3. The set of all pseudospherical surfaces inE4 with a bi-tangencyproperty is arbitrarily of four functions, each with a single variable.

Consider now Question 6. A partial answer (a necessary condition) to Question6 follows below (Theorem 4). First of all, we recall that the set of allm-dimensionalsubspaces ofn-dimensional Euclidean space through fixed pointO (called a Grass-mann manifold and denoted byGm,n) is a Riemannian space [25, 26]. The corre-sponding sectional curvature for the element of the area tangent to the Grassmannimage ofF 2 is denoted byKG. We recall that the Grassmann image ofF 2 is elliptic(parabolic, hyperbolic) if and only ifKG > 1, (KG = 0,KG < 1).

THEOREM 4. (1)If the Grassmann image ofF 2 is elliptic, then the vector fieldatangent to bothF 2 andF 2 never exists.

(2) If the Grassmann image is hyperbolic, then at every pointx ∈ F 2 there existat most two directions of the vector field tangent to bothF 2 andF 2.

One can complete Theorem 4 by the following theorem:

THEOREM 5. SupposeF 2 ⊂ E4 is of a flat normal connection and its GaussiancurvatureK 6= 0. If, additionally, there exists a vector fieldha with constant lenghth and tangent to regular surfaceF 2, then necessarilyF 2 ⊂ E3.

If F 2 6⊂ E3, then for every field of principal directionsτi , it is possible to definethe functionh – by putting it equal to the minus radius of the geodesic curvature ofthe orthogonal curvature line – such that the vector fieldhτi will be tangent toF 2

andF 2.

5. The Proofs

PROOF OF THEOREM1

We putx = y1, y = y2. Select any field of the normalized normal vectorsnσ , σ =1,2.Correspondingly,Lσij are the coefficients of the second fundamental forms. Wedenote the first and second derivatives ofr by ri andrij . The Gauss decompositionsfor F 2 imply

ri = ri − r2i = ri − 0j2irj − Lσ2inσ .Since01

21= 1 and0222 = 0, we have

r1 = −Lσ21nσ , r2 = r2 − Lσ22nσ .


Then we obtain the formula for the metric ofF 2.

ds2 = (r1dy1 + r2dy2)2 = (−Lσ21nσdy1 + (r2− Lσ22nσ )dy2)

2

=2∑

σ=1

(Lσ12dy1 + Lσ22dy2)2+ dy2

2. (4)

Now we make an additional assumption: the normal connection ofF 2 is flat.Hence, one can choose the fieldn1, n2 with the torsion coefficientsµασ |i = 0.Fortunately, this implies an existence of two functionsφσ = φσ (y1, y2) such that

Lσ12dy1 + Lσ22dy2 = e−y2dφσ . (5)

In other words, (5) is equivalent to

∂Lσ12ey2

∂y2− ∂L

σ22e

y2

∂y1= 0. (6)

But the Codazzi equations forF 2 have the formLσ12,2− Lσ22,1 = 0 or, in detail,

∂Lσ12

∂y2− ∂L

σ22

∂y1+ Lσ12= 0.

Therefore, Equations (5), (6) are true. The metric ds2 of F 2 can be written as

ds2 = e−2y2(dφ21 + dφ2

2)+ dy22. (7)

If φ1, φ2, y2 were independent coordinates, (6) could serve as a metric of the three-dimensional Lobachevski space of the curvature equal to−1. This means that,apart fromF 2 ⊂ E4, we can consider some immersion of its metric intoL3. Thisresulting surface is denoted byM2 ⊂ L3.

The Gauss equation forF 2 reads

L111L

122− (L1

12)2+ L2

11L222− (L2

12)2 = −e2y2. (8)

The assumption of the flat normal connection is expressed by

e2y2(L111L

112− L1

12L211)+ (L1

12L222− L1

22L221) = 0. (9)

We can solve (8) and (9) with respect toL111 andL2

11 and the resulting expres-sions can be substituted into the Codazzi equations:Lσ11,2 − Lσ12,1 = 0, σ = 1,2.Let

P = (e2y − e−2y(u2x + v2

x))ux + vy(uxvy − uyvx),Q = (e2y − e−2y(u2

x + v2x))vx + uy(uyvx − uxvy),

where we have putφ1 = u, φ2 = v. Finally, we obtain the following system:

∂

∂y

(P

uxuy + vxvy)+ uxxe−2y = −uy, (10)

∂

∂y

(Q

uxuy + vxvy)+ vxxe−2y = −vy. (11)


The important point is that the whole system of Gauss–Codazzi–Ricci equationsin the discussed case reduces to system (10)–(11). This system is of the form

uxx = A(y, ux, uy, vx, vy, uxy, uyy, vxy, vyy), (12)

vxx = B(y, ux, uy, vx, vy, uxy, uyy, vxy, vyy), (13)

whereA andB are analytical functions of their variables.As initial data, we can select the following functions:

u(0, y), ux(0, y), v(0, y), vx (0, y).

We can certainly, use these functions as parameters labelling isometric immer-sions ofL2 intoE4. The arbitrariness is four functions of a single argument.

After these preliminary remarks, we go over the proof of Theorem 1. In fact,we present two proofs (one indirect and one direct).

The indirect proof.Let us consider the surfaceM2 ⊂ L3. For surfaces inL3,we haveK = KL

e − 1, whereK(KLe ) is the Gauss (extrinsic) curvature ofM2.

ObviouslyK = −1 iff KLe = 0. As a convenient model ofL3 we take an open ball

D ⊂ E3 of the unit radius equipped with the metric

ds2l =

(ldl)2+ (1− l2)(dl)2(1− l2)2 , (14)

wherel is a position vector starting at the center of the ball and ending at the pointofD. In this way, any surfaceM2 ⊂ D can be interpreted either as a Lobachevskiansurface or as a Euclidean one. Sidorov [21] in particular, derived the followingformula relatingKL

e (extrinsic curvature ofM2 ⊂ L3) toKE (Gauss curvature ofM2 ⊂ E3)

KLe = KE

(1− l2

1− (ln)2)2

,

wheren is a unit normal toM2 ⊂ E3. This formula implies thatKLe = 0 iff

KE = 0. Now the set of all developable(KE = 0) surfaces is parametrized by twofunctions of a single argument. In other words, of the set of all surfacesF 2 of theGaussian curvature= −1 the arbitrariness is smaller than the arbitrariness of allisometric immersions ofL2 intoE4 with a flat normal connection.

The direct proof.Consider again metric (4) rewritten as follows:

ds2 = e−2y(du2+ dv2)+ dy2.

The two-dimensional metric

ds20 = e−2ydu2 + dy2


is of the Gaussian curvature equal to−1 as well. The equations of the immersionread

u = u(x, y), v = v(x, y).We want to compute the value ofK at a given pointP0, sayx = 0, y = 0. Let

us assume thatvx(0,0) = vy(0,0) = 0 atP0. According to the Frobenius formulafor the Gaussian curvature [22], one can write

K = −Eyy − 2Fxy +Gxx

2(EG− F 2)+9(E,F, . . . , Ex, . . . ,Gy),

where we have

E = e−2y(u2x + v2

x), F = e−2y(uxuy + vxvy),G = e−2y(u2

y + v2y)+ 1.

We point out that all expressions for the first derivatives ofE,F andG at pointP0 do not contain the derivatives of the functionv. At pointP0 we have

Eyy − Fxy +Gxx = −2(vxxvyy − v2xy)+3(ux, uy, . . . , uyy).

EG− F 2 = u2x.

Moreover, the expression of9 atP0 does not involve the derivatives ofv. TheGaussian curvature atP0 is given by

K = −1+ vxxvyy − v2xy

u2x

.

As initial data we select

v(0, y) = 0, vx(0, y) = y, ux(0,0) 6= 0, uy(0,0) 6= 0.

Thus, at pointP0

vyy = 0, vxy = 1, vx = vy = 0.

So, finally, we obtain atP0

K = −1− 1

u2x

6= −1.

Theorem 1 is proved in a direct way.


PROOF OF THEOREM2

Let ξ1 andξ2 denote two fields of normalized normals toF 2. We selectξ1 andξ2 asfollows. It is not difficult to show that e−y2ry1 is orthonormal toF 2. Hence, we put

ξ1 = e−y2ry1,

ξ2 = λ[ry1ry2ry1](skew product inE4), andλ = |[· · ·]|−1 is a normalized factor.

Trilinearity of the skew product implies

ξ2 = λ(L112[n1ry2ry1] + L2

12[n2ry2ry1] + (L122L

212− L1

12L222)[n1n2ry1]). (15)

This in turn implies

(ry2ξ2) = (L122L

212− L1

12L222)(ry2n1n2ry1) = 0,

where the second factor of the the RHS is a four-dimensional ‘mixed product’. Forregular surfaces the ‘mixed product’ is nonzero everywhere.

The vectorry2 is tangent toF 2 iff (ry2, ξ2) = 0, or iff

L122L

212− L1

12L222= 0. (16)

If so, then (15) reduces to

ξ2 = −L212n1+ L1

12n2√(L1

12)2+ (L2

12)2. (17)

(17) implies that normal planes toF 2 andF 2 at the corresponding point inter-sect along a line generated byξ2. It is natural to chooseξ2 = n1 (L

112 = 0), while

(16) implies

L122L

212= 0. (18)

The caseL212= 0 is excluded(F 2 is not regular). So we are left with

L112= L1

22 = 0.

The Codazzi equation reduces to

∂L212

∂y2− ∂L

222

∂y1+ L2

12= 0.

Thus

L212dy1 + L2

22dy2 = e−y2dφ,

and the metric ofF 2 assumes the form

ds2 = e−2y2dφ2 + dy22. (19)


Certainly, the Gauss curvature of (19) is equal to−1. Theorem 2 has been proved.

PROOF OF THEOREM3

We recall (see the previous proof) that in the discussed case

L112= L1

22 = 0, (20)

L212= e−y2φy1, L2

22= e−y2φy2. (21)

Thanks to (20) and (21) one of the Codazzi equations is satisfied. Certainly, theother Codazzi equation is to be satisfied

L112,2− L1

22,1 = µ21|2L212− µ21|1L2

22, (22)

where theµ coefficients are called ‘torsion coefficients’ [23].From (20) it follows that the right-hand side of (22) vanishes. We can put

µ21|2 = λL222, µ21|1 = λL2

12, (23)

whereλ = λ(y1, y2) is an unknown factor.The Gauss equation

L211L

222− (L2

12)2 = −e2y2 (24)

implies

L211=

e−2y2φ2y1− e2y2

e−y2φy2

.

Combining the Codazzi equation

L211,2− L2

12,1 = µ12|2L111− µ12|1L2

12= µ12|2L111

and (23) we obtain

∂

∂y2

(e−2y2φ2

y1− e2y2

φy2

)− e−2y2φy1y1 = −λφy2L

111e−2y2. (25)

By using (23) and (24) we can rewrite the last Codazzi equation

L111,2− L1

12,1 = µ21|2L211− µ21|1L2

12

as

∂L111

∂y2− L1

11= λ[L211L

222− (L2

12)2] = −λe2y2. (26)

(25) is equivalent to

∂

∂y2(L1

11e−y2) = −λey2. (27)


Finally we consider the (single) Ricci equation

µ21|2,1− µ21|1,2+ gll(L2l2L

1l1− L2

l1L1l2) = 0,

wheregll stands for contravariant components of the metric tensor. By using (23)we obtain

∂

∂y1(λe−y2φy2)−

∂

∂y2(λe−y2φy1)+ e−3y2L1

11φy1 = 0. (28)

(27) givesλ which we insert into (25) and (28). Letθ stand forL111. We conclude

the calculation of the current proof with the following statement: the system ofGauss–Codazzi–Ricci equations in the discussed case reduces to the system of twoequations for two functionsφ(y1, y2) andθ(y1, y2) of the form

∂

∂y2

(e−2y2φ2

y1− e2y2

φy2

)− e−2y2φy1y1 = e−3y2φy2θ

∂θ

∂y2,

− ∂

∂y1

(e−2y2φy2

∂e−y2θ

∂y2

)+ ∂

∂y2

(e−2y2φy1

∂e−y2θ

∂y2

)+ e−3y2φy1θ = 0.

Rewrite these equations in the brief form

φy2y2 = A(y2, θ, φy1, φy2, θy2, φy1y1, φy1y2),

θy2y2 = B(y2, θ, φy1, φy2θy1, θy2, φy1y2, θy1y2).

As usualφ(y1,0), φy2(y1,0), θ(y1,0) and θy2(y1,0) denote the initial condi-tions.

Thus, the arbitrariness of isometric immersions ofL2 into E4 (no flat normalcondition is assumed) with the bi-tangency property is of four functions of a singleargument.

Moreover, we can select the initial functions in such a way that the Gausstorsion κ0 := gll(L1

l1L2l2 − L1

l2L2l1) 6= 0 and hence the discussed immersion of

L2 essentially lies inE4 not inE3.

PROOF OF THEOREM4

Let F 2 ⊂ E4 be any regular surface and(u1, u2) any orthogonal coordinates on it.The metric ofF 2 reads

ds2 = Edu21 +Gdu2

2. (29)

We write

r = r + ha, (30)

wherer = r(u1, u2) is a point vector toF 2, while a is a field tangent toF 2 of unitlenght:|a| = 1, r is a position vector to a surfaceF 2. Instead of (29) we can write

r = r + h(cosγ τ1+ sinγ τ2),


whereτi (i = 1,2) are normalized coordinate vectors. Letb = − sinγ τ1+cosγ τ2.

We recall that our aim is to state necessary conditions forha to be tangent toF 2 as well. To this end we compute

rui = rui + huia+ hbAi + h(

cosγLσi1√E+ sinγLσi2√

G

)nσ , (31)

where

A1 = ∂γ

∂u1− 1

2√EG

∂E

∂u2, A2 = ∂γ

∂u2+ 1

2√EG

∂G

∂u1. (32)

We simplify the right-hand side of (31) as

ru1 = a1τ1+ a2τ2 + c1n1+ c2n2, (33)

ru2 = b1τ1+ b2τ2 + d1n1+ d2n2, (34)

whereai, . . . , di are some coefficients. The vectora is a linear combination of thevectorsru1, ru2 iff the following matrix

D = a1 a2 c1 c2

b1 b2 d1 d2

cosγ sinγ 0 0

(35)

has the rankD 6 2. Hence

c1d2− c2d1 = 0. (36)

In the left-hand side of (36) we replaceci anddi by their explicit forms givenin terms of the coefficients ofII 1 andII 2. We arrive at

cos2 γ

E(L1

11L212− L1

12L211)+

sinγ cosγ√EG

(L111L

222− L1

22L211)+

+ sin2 γ

G(L1

12L222− L1

22L212) = 0. (37)

So far we have not specified orthogonal coordinatesu1, u2. Now we select themas those defined by the indicatrix of the normal curvature ofF 2 ⊂ E4 [24]. Werecall it is an ellipse in our case. Ifn1 andn2 are parallel to its axes then

L111= E(α + a), L2

11= Eβ, (38)

L112= 0, L2

12=√EGb, (39)

L122= G(α − a), L2

22= Gβ, (40)

wherea andb are lengths of semi-axes whileα andβ are coordinates of the ellipsecenter.

Substituting the expressions (38)–(40) into (37) gives

cos2 γ b(α + a)+ 2 cosγ sinγ aβ − sin2 γ b(α − a) = 0. (41)


This quadratic equation has two roots iff

a2β2 + α2b2− a2b2 > 0. (42)

To conclude our proof, let us recall the formula for the Gaussian curvatureKGof the Grassmann image ofF 2 inG2,4 expressed in terms of the Gaussian curvatureK of F 2 [25, 26]

KG = K2+ 4a2b2

K2+ 4(a2β2+ b2α2). (43)

It is evident that (41) impliesKG < 1. According to the definition given in [25,26] the Grassmann image ofF 2 is hyperbolic. This ends the proof of Theorem 4.

PROOF OF THEOREM5

In this proof we employ all the notations of the previous proof. The assumptionof a flat normal connection impliesb = 0. Thus the ellipse of normal curvature isdegenerated and Equation (40) reduces to

sinγ cosγ aβ = 0. (44)

One should consider three cases: (1) sinγ = 0 (cosγ = 0 is treated similarly),(2) a = 0, and (3)β = 0.

In case (1)a= τ1 and we have

r = r + hτ1,

ru1 = a1τ1+ a2τ2 + c1n1+ c2n2,

ru2 = hu2τ1+ (√G+ hA2)τ2. (45)

Let ξi (i = 1,2) be normalized normal vectors toF 2. Consider at first the caseh = const and denoteb2 =

√G + hA2. But (ru2ξi) = 0 implies b2 = 0 or

(τ2ξi) = 0. If b2 = 0, hu2 = 0, thenF 2 is necessarily degenerated. So, letb2 6= 0.Now (τ2ξi) = 0 when combined with(τ1ξi) = (aξi) = 0 gives an identity ofnormal spaces toF 2 and F 2 at the corresponding points. Thusc1 = c2 = 0 andthereforec2 = L2

11 = β = 0. By the result of [27] we know that the identitiesβ = 0, b = 0 and inequalityK 6= 0 imply F 2 ⊂ E3.

In case (2) ora = 0 andb = 0 impliesF 2 is a standard sphereS2 ⊂ E3.Case (3) has been already discussed above.The consideration above and (44) imply:if F 2 ⊂ E4 is of a flat normal connection,F 2 6⊂ E3, K 6= 0 and if the field

ha enjoys the bi-tangency property, thena must be necessarily a field of principaldirections.As we have proved above,hmust be some nonconstant function onF 2.

Now we shall prove that in general it is possible to select the functionh in sucha way that the fieldhτi will enjoy the bi-tangency property.


Consider, for example, the transformationr = r + hτ1. We recall formula (44).To determine the functionh, we put

√G+ hA2 =

√G+ h 1

2√EG

∂G

∂u1= 0.

This equation can be rewritten in the following simple form

1+ h 1

ρgv= 0,

where 1/ρgv denotes the geodesic curvature of the curvature lineu = const. So,if the geodesic curvature of the orthogonal curvature line is not equal to zero, thenwe can take−h equal to the radius of the geodesic curvature of this line. If thisgeodesic curvature is not constant, thenhu2 6= 0, ru2 6= 0 and F 2 is a regularsurface. In this case, matrixD has rank= 2. Therefore fieldha is tangent to bothF 2 andF 2.

Acknowledgement

Work partially supported by the Polish Committee of Scientific Researches (KBNgrant 2 PO3B 185 09).

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11. Ablowitz, M. J. and Clarkson, P. A.:Solitons, Nonlinear Evolution Equations and InverseScattering, Cambridge University Press, Cambridge, 1991.

12. Lamb, G., Jr.: Bäcklund transformations at the turn of the century, In: R. M. Miura (ed.),Bäcklund Transformations, Lecture Notes in Math. 515, Springer, New York, 1976.


13. Lamb, G. L., Jr.: Bäcklund transformations for certain nonlinear evolution equations,J. Math.Phys.15 (1974), 2157–2165.

14. Wahlquist, H. D. and Estabrook, F. B.: Bäcklund transformations for solutions of the Korteweg–de Vries equation,Phys. Rev. Lett.31 (1973), 1386–1390.

15. Wahlquist, H. D. and Estabrook, F. B.: Prolongation structures of nonlinear evolution equations,J. Math. Phys.16 (1975), 1–7.

16. Aminov, Yu.: Bianchi transform for the domain of multidimensional Lobachevsky space,Ukrain. Geom. Sb.21 (1978), 3–5 (in Russian).

17. Masal’tzev, L.: Pseudospherical Bianchi congruencies inE2n−1, Math. Phys. Anal. Geom.1(3/4) (1994), 505–512.

18. Tenenblat, K. and Terng, C.-L.: A higher dimension generalization of the sine-Gordon equationand its Bäcklund transformation,Bull. Amer. Math. Soc. (N. S.)1 (1979), 589–599.

19. Tenenblat, K. and Terng, C.-L.: Bäcklund theorem forn-dimensional submanifolds ofR2n−1,Ann. of Math.111(1980), 477–490.

20. Tenenblat, K.: Bäcklund’s theorem for submanifolds of space form and a generalized waveequation,Bol. Soc. Brasil Mat.16 (1985), 67–92.

21. Sidorov, L.: Some properties of surfaces of negative extrinsic curvature in the Lobachevskyspace,Mat. Zametki4 (1968), 165–169.

22. Blaschke, W.:Einfuhrung in die Differentialgeometrie, Springer-Verlag, Berlin, 1950.23. Eisenhart, L. P.:Riemannian Geometry, Princeton University Press, Princeton.24. Cartan, E.:Léçons sur la géométrie des espaces de Riemann, 2nd edn, Gauthier-Villars, Paris,

1946.25. Aminov, Yu.: Grassmann transform of a two-dimensional surface in four-dimensional Euclid-

ean space,Ukrain. Geom. Sb.23 (1980), 3–16.26. Aminov, Yu.: Determination of a surface in 4-dimensional Euclidean space by means of its

Grassmann image,Mat. USSR Sb.45 (1983), 155–167.27. Aminov, Yu.: Torsion of two-dimensional surfaces in Euclidean spaces,Ukrain. Geom. Sb.17

(1975), 3–14.


91

The Ground State of Certain Coulomb Systemsand Feynman–Kac Exponentials

AMÉDÉE DEBIARD and BERNARD GAVEAUUniversité Pierre et Marie Curie (Paris 6), Mathématiques, T-46, 5ème étage, Boîte 172,4 place Jussieu, 75252 Paris Cedex 05, France

(Received: 10 January 2000)

Abstract. We give a lower bound for the ground state energy of certain Coulomb Hamiltoniansusing the Feynman–Kac formula. We show that this bound is very precise for two electron atoms.

Mathematics Subject Classifications (2000):58J70, 60J65, 81S40.

Key words: Coulomb systems, Brownian motion, ground state.

1. Introduction

In this short note, we show that it is possible to use the Feynman–Kac formulato obtain a very sharp (indeed, almost exact) lower bound for the ground stateof a two-electron atom, using only the basic facts of Brownian motion theoryand elementary calculus. The starting point is the Feynman–Kac formula whichprovides a way of writing the solution of the heat equation with a potential, using aWiener expectation on the Brownian motion. A consequence is the formula for theupper bound of the spectrum of1

21−V in terms of the asymptotic estimate of suchWiener expectations (see [1] for the original reference, [2, 3] for a recent review ofthe subject and, more recently, [4] for another method of derivation which can begeneralized to other situations). The estimation of such Wiener expectations havebeen the subject of many works (see [5 – 7], and their references). We thus obtain anupper estimate of the spectrum of1

21−V , or a lower bound for the spectrum of theHamiltonianH . The usual variational method (Rayleigh–Ritz) would provide anupper bound of the ground state ofH . We obtain, in the special case of a Coulombpotential, an upper bound of the Wiener expectation, which uses the fact that theCoulomb potential is homogeneous of degree−1. That fact allows us to reduce theproblem to an ergodic theorem for the spherical Brownian motion. For the two-electron system, the absolute ground state is also the physical ground state. Theestimation that we obtain in this case can be compared to the experimental valueand is surprisingly sharp (1.2% of error), indicating the power of the Feynman–Kacformula and path integrals.

92 AMEDEE DEBIARD AND BERNARD GAVEAU

In Section 2, we recall the basic facts about the Feynman–Kac formula andin Section 3, we specialize to the case of the Coulomb system. Section 4 givesthe main estimate, Section 5 gives the final result. The Appendix contains thecalculations of various integrals on a sphere.

2. The Feynman–Kac Formula for the Ground State

We consider the operatorH = −121+V where1 is the standard Laplace operator

in anN-dimensional spaceRN andV is a real function, such thatH is essentiallya self-adjoint operator. We consider the Cauchy problem

∂ψ

∂t= −Hψ, ψ|t=0 = ψ0. (2.1)

This can be solved, using the Feynman–Kac formula as follows. We consider theN-dimensional standard Brownian motionb(t) = (b1(t), . . . , bN(t)), wherebj (t)are independent one-dimensional standard Brownian motions. Then, ifx ∈ RN ,the solution of Equation (2.1) is

ψ(x, t) = E

{exp

(−∫ t

0V (b(s))ds

)ψ0(b(t))

∣∣∣∣ b(0) = x}, (2.2)

whereE{. . . | b(0) = x} is the conditional expectation on Brownian paths, startingfrom x at timet = 0. We refer to [1 – 3] and also to [4] for a different proof.

BecauseH is essentially a self-adjoint operator, it has a spectral decompositionand its spectrum is bounded from below. Call3 an upper bound of the spectrumof −H . If ψ0 is not orthogonal to the generalized eigenfunction of eigenvalue3,then

ψ(x, t) ∼ C exp(3t) (2.3)

and comparing Equations (2.2) and (2.3), we deduce Kac’s formula for the upperbound3:

3 = limt→∞

1

tlogE

{exp

(−∫ t

0V (b(s))ds

)ψ0(b(t))

∣∣∣∣ b(0) = x}. (2.4)

We shall apply this formula to the situation where3 is an eigenvalue of−H ,the corresponding eigenfunction is integrable (and, as usual, square integrable),and it does not change sign, so that we can use anyψ0 > 0, bounded. This lasthypothesis is normally fulfilled for standard Schrödinger Hamiltonians becausethe ground state does not change sign.

3. N Electrons Atom Hamiltonian

We consider the situation ofN electrons (positions−→r1 , . . . ,−→rN ) in the Coulombfield of a nucleus of chargeZe. In atomic units, the Hamiltonian is

H = −1

2

N∑i=1

1i + V (−→r1 , . . . ,−→rN ), (3.1)

THE GROUND STATE OF CERTAIN COULOMB SYSTEMS 93

where1i is the three-dimensional Laplace operator for the variables−→ri and

V (−→r1 , . . . ,−→rN ) = −N∑i=1

Z

ri+

∑16i<j6N

1

|−→ri −−→rj | . (3.2)

The HamiltonianH satisfies all the hypotheses of the previous section. We shallapply Equation (2.4) withV given as the Coulomb potential Equation (3.2) andwith ψ0 = exp(−ϕ), whereϕ is a real function such thatψ0 is bounded. Ourproblem is, then, to find the large time behavior of the quantity

Q(N,Z, t)

= E

{exp

(−∫ t

0V (−→r1 (s), . . . ,−→rN (s))ds − ϕ(−→r1 (t), . . .

. . . ,−→rN (t))) ∣∣∣∣ −→ri (0) = −→ri (0)}, (3.3)

where−→r1 (s), . . . ,−→rN (s) areN independent three-dimensional Brownian motions.

4. Transformation of Q(N,Z, t)

To obtain a bound ofQ, we shall use a method related to the one used in [7] forcompact manifolds. Here, the fact that the Coulomb potential is homogeneous ofdegree−1 replaces the compactness. We define

ϕ(−→r1 , . . . ,−→rN ) =N∑j=1

Z−→rj − 1

2

∑16i<j6N

|−→ri −−→rj |. (4.1)

It is easy to see that, using121r = 1r,(

1

2

N∑i=1

1i

)ϕ = −V. (4.2)

Now, we use Itô’s formula for the Brownian motion (see, e.g., [8]), under the form

ϕ(−→r1 (t), . . . ,−→rN (t))= ϕ(−→r1 (0), . . . ,−→rN (0))+

+∫ t

0

N∑j=1

(−→∇jϕ)(−→r1 (s), . . . ,−→rN (s)) · d−→rj (s)+

+∫ t

0

1

2

(N∑j=1

1j

)ϕ(−→r1 (s), . . . ,−→rN (s))ds. (4.3)


Equations (4.2) and (4.3) allow us to rewrite the Feynman–Kac exponential insideEquation (3.3) as

exp(−∫ t

0V(−→r1 (s), . . . ,−→rN (s)) ds − ϕ(−→r1 (t), . . . ,−→rN (t)))

= exp(−ϕ(−→r1 (0), . . . ,−→rN (0))) exp

(−∫ t

0

N∑j=1

−→∇jϕ · d−→rj (s)). (4.4)

We use the exponential martingale (see [8]). Given aN-dimensional Brownian mo-tion−→b (s) and a vector-valued random function

−→f (s, ω) (ω being the simple path

of the Brownian motion,s the time) such that−→f is nonanticipating, the quantity

Mt ≡ exp

(∫ t

0

−→f (s, ω)d

−→b (s)− 1

2

∫ t

0|−→f |2 ds

)(4.5)

is a martingale and, in particular, its expectation is equal to 1. We shall rewrite theexponential of the stochastic integral in Equation (4.4) as

exp

(−∫ t

0

N∑j=1

−→∇jϕ · d−→rj (s))= F ·G

with F andG given as

F = exp

(−∫ t

0

N∑j=1

−→∇jϕ(−→r (s)) · d−→rj (s)− p2N∑j=1

|−→∇jϕ|2 ds

),

G = exp

(p

2

∫ t

0

N∑j=1

|−→∇jϕ|2 ds

)andp is a positive number.

ThenQ can be rewritten

Q(N,Z, t) = exp(−ϕ( Erj (0)))E(FG) (4.6)

and we apply the Hölder–Young inequality with(1/p)+ (1/q) = 1

E(FG) 6(E(Fp)

)1/p(E(Gq)

)1/qto the expectation in Equation (4.6).

Thus,E(Fp) is of the typeE(Mt) for a certainMt , as in Equation (4.5), with

−→f = (−→fj ), −→

fj = −p((−→∇jϕ)(−→r (s))

)


so thatE(Mt) = 1 and, thus,

Q(N,Z, t)

6 exp(−ϕ(r(0)))E{exp

(pq

2

∫ t

0

N∑j=1

∣∣−→∇jϕ(−→r (s))∣∣2 ds)∣∣∣∣−→rj (0) = −→r(0)j }1/q

. (4.7)

Now,∑N

j=1 |−→∇jϕ|2 is a function homogeneous of degree 0 so that it is a functionon the(3N−1)-dimensional unit sphere inR3N and we can use the ergodic theoremon the sphere [9] to get

pq

2

∫ t

0

N∑j=1

∣∣−→∇jϕ(−→r (s))∣∣2 ds ∼ pq

2

⟨N∑j=1

|−→∇jϕ|2⟩t, (4.8)

where〈 · 〉 is the average on the(3N − 1)-dimensional sphere. Then, from Equa-tions (4.7)–(4.8),

limt→∞

1

tlogQ(N,Z, t) 6 p

2

⟨N∑j=1

|−→∇jϕ|2⟩.

This upper bound is valid for anyp > 1, noticing thatq has disappeared and so,finally, we have the upper bound:

3 = limt→∞

1

tlogQ(N,Z, t) 6 1

2

⟨N∑j=1

|−→∇jϕ|2⟩≡ 8. (4.9)

This estimate gives an upper bound of the spectrum of−H = 121 − V , and so a

lower bound of the ground state of the Coulomb Hamiltonian.

5. Estimation of3

We need to calculate the second member of Equation (4.9), namely the quantity8

defined there. We have

−→∇jϕ = Z−→rjrj− 1

2

∑i 6=j

−→rj −−→ri|−→rj −−→ri |

and soN∑j=1

|−→∇jϕ|2 =N∑j=1

∣∣∣∣Z−→rjrj − 1

2

∑i 6=j

−→rj −−→ri|−→rj −−→ri |

∣∣∣∣2= NZ2+ N(N − 1)

4+ N

2

∑i,j 6=1i 6=j

(−→riri

∣∣∣∣ −→rjrj)+

+NZ∑i<j

( −→ri −−→rj|−→ri −−→rj |

∣∣∣∣ −→rj|−→rj | −−→ri|−→ri |

),


where, in the last summation, each couple(i, j) is counted once. Moreover, callingθij the angle between−→ri and−→rj , one has( −→ri −−→rj|−→ri −−→rj |

∣∣∣∣ −→rj|−→rj | −−→ri|−→ri |

)= ri + rj|−→ri −−→rj |(1− cosθij )

so that, finally, using〈cosθij 〉 = 0,

8 = 1

2

⟨N∑j=1

|−→∇jϕ|2⟩

= N

2

(Z2+ N − 1

4

)− N(N − 1)

4Z

⟨r1 + r2|−→r1 −−→r2 |(1− cosθ12)

⟩. (5.1)

We obtain a trivial upper bound, saying that

r1+ r2|−→r1 −−→r2 | > 1, 〈cosθ12〉 = 0,

namely

8 6 N

2

(Z2+ N − 1

4− (N − 1)Z

2

). (5.2)

In the Appendix, it is proved that⟨r1+ r2|−→r1 −−→r2 |(1− cosθ12)

⟩= 4

π, (5.3)

so that

8 = N

2

(Z2+ N − 1

4− 2(N − 1)Z

π

). (5.4)

ForN = 2, the Pauli principle states that the physical ground state is indeed theabsolute ground state, while this is no more correct forN > 3.

ForN = 2, we can restore the units (see Appendix), and obtain

− from Equation (5.2)8 6 88 eV;− from the more exact Equation (5.4)8 ' 80 eV

to be compared with the actual value of the physical ground state 79 eV. See,e.g., [10] or [11].


Appendix: Calculation of Spherical Integrals

In this Appendix we calculate the spherical integral

I =⟨r1+ r2|−→r1 −−→r2 |(1− cosθ12)

⟩(A.1)

and prove Equation (5.3). In particular, we prove that this is independent ofN .

(I) COORDINATES ON THE SPHERE

Call

r ′ =√−→r1 2+−→r2 2

,

r ′′ =√−→r3 2+ . . .+−→rN 2

,

R =√r ′2+ r ′′2.

In the space−→r1 , we use the traditional polar angles(θ1, ϕ1) and write

d−→r1 = r21 dr1 sinθ1 dθ1 dϕ1.

In the space−→r2 , we use polar angles around the axis defined by−→r1 , namely angles(θ12, ϕ12)

d−→r2 = r22 dr2 sinθ12 dθ12 dϕ12.

Call

r1 = r ′ cosγ,

r2 = r ′ sinγ (06 γ 6 π/2),dr1 dr2 = r ′ dr ′ dγ.Moreover, in the space(−→r3 , . . . ,−→rN ), we use any polar coordinate system and write

d−→r3 . . . d−→rN = r ′′3N−7 dr ′′ dσ(−→r3 , . . . ,−→rN ),where dσ is the spherical volume element of the unit sphere inR3N−6. Finally, wewrite

r ′ = R cosρ, r ′′ = R sinρ (06 ρ 6 π/2).

In this way, we obtain

d−→r1 d−→r2 . . . d−→rN= R3N−1 dR cos5ρ(sinρ)3N−7 dρ cos2 γ sin2 γ dγ×× sinθ1 dθ1 dϕ1 sinθ12 dθ12dϕ12 dσ(−→r3 , . . . ,−→rN )


and thus the spherical volume of the unit sphere inR3N

dσ(−→r1 , . . . ,−→rN )= cos2 γ sin2 γ dγ sinθ1 dθ1 dϕ1 sinθ12 dθ12 dϕ12×× cos5 ρ(sinρ)3N−7 dρ dσ(−→r3 , . . . ,−→rN ). (A.2)

The function to be integrated is

r1+ r2|−→r1 −−→r2 |(1− cosθ12) = cosγ + sinγ√

1− sin 2γ cosθ12(1− cosθ12). (A.3)

It is independent of−→r3 , . . . ,−→rN by definition, but also onρ, θ1, ϕ1, ϕ12, so that theaverage value is the average value with respect to the measure

dσ = cos2 γ sin2 γ dγ · sinθ12 dθ12, (A.4)

06 γ 6 π

2, 06 θ12 6 π.

(II ) CALCULATION OF THE INTEGRAL

First of all we calculate∫dσ =

∫ π/2

0

(sin 2γ )2

4dγ∫ π

0sinθ12 dθ12 = π

8. (A.5)

Then we need to calculate the integral of Equation (A.3), namely abbreviatingθ =θ12,

J ≡ 1

4

∫ π/2

0(sin 2γ )2 dγ

∫ π

0sinθ dθ

cosγ + sinγ√1− sin 2γ cosθ

(1− cosθ) (A.6)

so that

I = 8J

π. (A.7)

For convenience, we give details of the calculation:

J = 1

2

∫ π/4

0(sin 2γ )2 dγ

∫ 1

−1du

cosγ + sinγ√1− (sin 2γ )u

(1− u). (A.8)

Thus∫ 1

−1du

1− u√1− (sin 2γ )u

= − 2

sin 2γ

(√1− sin 2γ −√1+ sin 2γ

)+


+ 1

sin 2γ

∫ 1

−1

−(sin 2γ )u√1− (sin 2γ )u

du

= − 2

sin 2γ

((1− sin 2γ )1/2− (1+ sin 2γ )1/2

)++ 2

sin2 2γ

((1− sin 2γ )1/2− (1+ sin 2γ )1/2

)−− 2

3 sin2 2γ

((1− sin 2γ )3/2− (1+ sin 2γ )3/2

).

But

(1+ sin 2γ )1/2− (1− sin 2γ )1/2 = 2 sinγ(06 γ 6 π

4

),

(1+ sin 2γ )3/2− (1− sin 2γ )3/2 = (2 sinγ )(1+ 2 cos2 γ ),∫ 1

−1du

1− u√1− (sin 2γ )u

= 2

cosγ− 2 sinγ

3 cos2 γ. (A.9)

From Equations (A.8)–(A.9), we deduce

J = 4

3

∫ π/4

0sin2 γ (4 cos2 γ + 2 sinγ cosγ − 1)dγ = 1

2

because∫ π/4

04 sin2 γ cos2 γ dγ = π

8,∫ π/4

02 sin3 γ cosγ dγ = 1

8,∫ π/4

0sin2 γ dγ = π

8− 1

4,

so thatI = 4/π .

(III ) RESTORING THE UNITS

In the unit of this article, the exact ground state of an hydrogenoïd atom of chargeZ would beZ2/2.

ForZ2 = 1 (hydrogen), we obtain 13.5 eV, so 1 unit corresponds to 27 eV, fromwhich we deduce the numerical values given in this article.

Acknowledgement

We thank the referee for comments.


References

1. Kac, M.: Probability and Related Topics in the Physical Sciences, Interscience, New York,1989.

2. Kac, M.:Integration in Function Spaces and Some of its Applications, Lezioni Fermiane, Pisa,1980.

3. Schulman, L. S.:Techniques and Applications of Path Integration, Wiley, New York, 1981.4. Gaveau, B. and Schulman, L. S.: Grassmann-valued processes for the Weyl and the Dirac

equations,Phys. Rev. D36 (1987), 1135–1140.5. Berthier, A.-M. and Gaveau, B.: Convergence des exponentielles de Kac et applications en

physique et en géométrie,J. Funct. Anal.29 (1978), 416–424.6. Gaveau, B. and Mazet, E.: Divergence des fonctionnelles de Kac et diffusion quantique,Publ.

RIMS, Kyoto18 (1982), 365–377.7. Gaveau, B.: Estimation des fonctionnelles de Kac sur une variété compacte et première valeur

propre de1+ f , Proc. Japan Acad. Sci.60 (1985), 361–364.8. McKean, H. Jr.:Stochastic Integrals, Academic Press, New York, 1969.9. Itô, K. and McKean, H. Jr.:Diffusion Processes and their Sample Paths, Academic Press, New

York, 1964.10. Levine, I. N.:Quantum Chemistry, Prentice-Hall, Englewood, 1991.11. Karplus, M. and Porter, R. N.:Atoms and Molecules, Benjamin Cummings, Reading, 1970.


101

Periodic Ground State Configurationsin a One-Dimensional HubbardModel of Statistical Mechanics

M. M. KIPNISChelyabinsk Pedagogical University, 69 Lenin Ave, Chelyabinsk, 454080, Russiae-mail: [email protected]

(Received: 1 July 1999; in final form: 17 April 2000)

Abstract. This paper considers an averaging procedure for the description of a particles arrangementin a Hubbard model with antiferromagnetic interactions. The arrangements are described by thedevil’s staircase. Completeness of the staircase is proved.

Mathematics Subject Classification (2000):82B20.

Key words: statistical mechanics, Hubbard model, periodic ground state configurations, symbolicdynamics, phenomenon of even two-colouring, averaging procedure, devil’s staircase, completeness.

1. Introduction

In the Hubbard model [1, 3, 8, 13, 14] of statistical mechanics, the ground states aredescribed by bilateral sequences(un) ((un): Z→ {−1;1}). The values of variableun may be interpreted either as an electron(un = 1) and a hole(un = −1) or as aparticle with anup (un = 1) or down (un = −1) spin. The ground states providethe minimum of the formally defined HamiltonianH :

H = −ψ∑i∈Z

ui +∑

i>j; i,j∈Zγi−juiuj , (1)

whereγi ((γi): N → R) may be interpreted as the interaction energy of twoparticles at the distance ofi units;ψ stands for a chemical potential. We shall givea euristical procedure for calculating the minimal arrangement for the Hamiltonian(1) in the spirit of dynamic programming. Let us suppose that the valuesum (m <

n) are constructed and we have no information aboutum for m > n. Excludingumfor m > n from (1) and defining the value ofun, we get the energy increment

1Hn = un(−ψ +

∞∑i=1

γiun−i). (2)

102 M. M. KIPNIS

To decrease energy according to (2), it is quite natural to provide1Hn 6 0 byputting

un = sgn

(ψ −

∞∑i=0

γiun−i). (3)

Equation (3) will be called a Boolean averaging system. It will be the mainobject of our investigations. Another method of minimization of Hamiltonian (1)is given in [3].

We may consider that in (3) one selects eitherun = 1 or un = −1 in anattempt to provide the equality ofψ and a weighted average of the sequenceun, un−1, . . . with weightsγ1, γ2, . . . That is why we refer to (3) as an averagingsystem. Equation (3) was introduced by the author [13, 14].

The averaging system (3) describes a variety of systems: relay periodic processesin a sampled-data control system [6, 11]; periodic output in analog-to-digital con-verter with sigma-delta modulation and leaky integration [4, 5, 14, 18]; the pointitineraries in the iterates of some maps on an interval [10, 14, 15]; the rotation ofthe circle through a rational angle [17]. The common property of these systems isthe even mixture of two kinds of objects on a circle in a given proportion, i.e. theeven 2-colouring [16, 17, 21]. We treat of two equivalent formal descriptions of theperiodic configurations in the above-mentioned systems (all as words in the−+alphabet): Hubbard configurations [3] and the setJ [11]. The frequency of plusesin the periodic configurations is called its rotation number. The rotation numbersare described in a parameter space by the devil’s staircase. The completeness of thedevil’s staircase is the main result in the article. It is stated in Section 3 (Theorem 2)and proved in Section 8. Some results of this paper were given earlier withoutproofs by the author [13, 14].

2. Periodic Ground State Configurations

It is convenient to consider the periodic ground state configurations in Booleanaveraging system (3) as words in the−+ alphabet.

DEFINITION 1. Let for eachi (i ∈ N, 1 6 i 6 p, p ∈ N) εi = − or εi = +.The wordε1 . . . εp is called a periodic ground state configuration in Boolean aver-aging system (3) for the givenψ , if there exists a bilateral sequence(un) (n ∈ Z),satisfying Equation (3), such that

(1) for eachi (16 i 6 p) if εi = − thenui = −1; if εi = + thenui = 1;(2) for each integern un+p = un.

Let A be a word in the−+ alphabet. The set of pointsψ , such thatA is aperiodic ground state configuration in (3) for a givenψ , is calledthe domainof theperiodic configurationA.

PERIODIC GROUND STATE CONFIGURATIONS 103

EXAMPLE 1. Let γ1 = 10, γ2 = 9, γ3 = 6, γ4 = 4, andγn = 0 whenn > 4.The complete list of periodic ground state configurations (up to shifts and degrees)is−+ for ψ ∈ (0,3); −−++ for ψ ∈ (0,1); −+−++ for ψ ∈ (3,11); −++for ψ ∈ (11,17); −+++ for ψ ∈ (17,21); −++++ for ψ ∈ (21,29); + forψ > 29, and symmetrically forψ < 0 (interchanging plusses and minuses). Wehave here non-uniqueness of periodic configurations: there exist simultaneouslytwo configurations−+ and−−++ for ψ ∈ (−1,1).

DEFINITION 2. For the wordA the ratio of the number of plusses to the totalnumber of letters is called the rotation number ofA.

The rotation number ofA is denoted asω(A). For example,ω(−+−++) =3/5. In the following definition [a] stands for the integer part ofa. For each wordA in the−+ alphabet the symbolA0 denotes the empty word. For the nonnegativeintegern An+1 = AnA. The wordB is called the shift ofA if there exist wordsC,D such thatA = CD andB = DC.

DEFINITION 3 [3]. The wordA in −+ alphabet is called a Hubbard configu-ration, if in each of its shiftsB there are either [i/ω(A)] or [i/ω(A)] − 1 lettersbetween every plus andith plus on the right-hand side of it. Besides, the words−n(n > 0),−n+ (n > 0) and their shifts are also called Hubbard configurations.

EXAMPLE 2. The words−+,−+−++,−−+−+ are Hubbard configurations,while the word−−++ is not.

3. Statement of the Main Result

Now we are ready to connect the periodic ground state configurations in averagingprocedure (3) and the Hubbard configurations.

THEOREM 1 [13]. Let for eachi (1 6 i < ∞) γi > γi+1 > 0, let the series∑∞i=1 γi be convergent and(γi) be convex(i.e. γi+1 < (γi + γi+2)/2 for i > 1).

Then the wordA is a periodic configuration in Boolean averaging system(3) if andonly if it is a Hubbard configuration.

We avoid giving the proof of Theorem 1 because in Theorem 2 which comeslater, we’ll deal with the wider class of interaction functions, free from the con-dition of convexity. In addition, results, similar to Theorem 1, were stated earlier[3, 8] for systems, which are not described by Equation (3), but also originate froma Hamiltonian minimization problem. The dependence of rotation numbers of theperiodic ground state configurations on the values ofψ is described by the devil’sstaircase [3, 14]: it is an increasing function, whose derivative equals to zero almosteverywhere. It is a discontinuous function, but it may be extended to a continuousfunction. The case of nonconvex(γi) had not been investigated until the author’spaper [13]. It is considered in the following theorem:

104 M. M. KIPNIS

THEOREM 2. If for eachi (1 6 i < ∞) γi > γi+1 > 0, and the series∑∞

i=1 γiis convergent, then:

(1) Every Hubbard configuration is the periodic ground state configuration inBoolean averaging system(3) with some value ofψ .

(2) Let periodic ground state configurationA in system(3) not be a Hubbard con-figuration. LetB be a Hubbard configuration with the same rotation numberasA. Then the domain ofA is a proper subset of theB domain.

(3) LetP be the set of values ofψ such that there exist no periodic ground stateconfigurations in Boolean averaging system(3) with a givenψ . Then the setP is a Cantor perfect set of the Lebesgue zero measure.

Theorem 2 is our main result. It is proved in Section 8.When (γi) is nonconvex, someψ values may correspond to more than one

periodic ground state configuration (see Example 1). However, Theorem 2 statesthe preservation of the devil’s staircase in this case. Part (3) of Theorem 2 meansthe completeness of the staircase.

4. The Point Itineraries in the Maps of Interval in Itself

Consider the variant of Boolean averaging procedure (3) with the exponentialweight sequence(γn):

γn = e−nα(eα − 1) (α > 0). (4)

If we replace (3) by two equationsun = sgnσn; σn = ψ −∑∞i=1 γiun−i , we getfrom (4):

σn+1 ={

e−ασn + (ψ − 1)(1− e−α), if σn > 0;e−ασn + (ψ + 1)(1− e−α), if σn < 0.

(5)

Map (5) is a piecewise linear discontinuous transformation of the line in itself.The orbits of transformation (5) were investigated earlier [15]. Write+ (plus) whenσn > 0 and− (minus) whenσn < 0. Then the periodic trajectories in map (5) aredescribed by words in the−+ alphabet. By Theorem 1, the periodic ground stateconfigurations in map (5) are Hubbard configurations andvice versa. Besides, byTheorem 2 the devil’s staircase associated with map (5) is complete. Hence, Theo-rem 2 gives a new proof of the result obtained earlier [10, 18]. Transformation (5)is associated with selfsimilar structures [14] and deterministic chaos [11].

Changing the variables in Equation (5) by means of equationsσn = 2(δn + ω− 1); ψ = 2ω − 1, and lettingα go to zero (5) gives the map

δn+1 ={δn + ω − 1, if δn + ω > 1;δn + ω, if δn + ω < 1,

(6)

whereω is a rotation number, 06 ω 6 1. Map (6) is the classical model ofcircle rotation through the angleω. Write + (plus) if δn ∈ [0, ω) and− (minus)


otherwise. Ifω is rational, then the sequence of letters−,+ is periodic and itforms the so-called Sturmian chain [17]. The latter are equivalent to the Hubbardconfigurations.

5. Even 2-Colouring and the Set J

The even 2-colouring problem has been widely discussed [2, 3, 16, 19, 21]. Theproblem is how to distributea objects of the first kind (say, pluses) andb objectsof the second kind (say, minuses) on a circle as evenly as possible. In the definitionof the Hubbard configuration (Section 2), we found the requirement of the evendistribution of plusses in words.

The second description of the words in the−+ alphabet with the even distribu-tion of plusses is given by the linear ordered setJ [11]. To defineJ , we define thesequence of finite linear ordered setsJn, each having words in the−+ alphabet ascomponents.

DEFINITION 4. (1)J0 = (−,+). The order inJ0: − < +.(2) AssumeJn be defined. Let the wordsA andB be arbitrary components ofJn,

letA < B and let there be noC in Jn, such thatA < C < B. Then the wordsA, BandAB are the components ofJn+1 andA < AB 0) in the order induced by the order in the setsJn.

The setsJn are constructed by the operations of concatenation and insertion.J0 consists of two words:− (minus) and+ (plus). We getJ1 by inserting theconcatenation of the− (minus) and+ (plus) between the aforesaid words:J1 =(−,−+,+). Further, we insert the concatenation of the neighboring words be-tween the neighbors:

J2 = (−,−−+,−+,−++,+),J3 = (−,−−−+,−−+,−−+−+,−+,−+−++,−++,−+++,+),and so on. The setsJn are the redefinitions of the Farey tree [7, 19]. The twodefinitions mentioned above are equivalent in a certain sense.

THEOREM 3 [12]. For each wordA in−+ alphabet the following two assertionsare equivalent:

(1) A is a Hubbard configuration(2) A is a shift of a nonzero degree of some component ofJ .

EXAMPLE 3. The word− + − + + − + − + + − + + is simultaneously aHubbard configuration and a component ofJ (namely,J5).

106 M. M. KIPNIS

6. Properties of the Words in J

The number of the minuses and plusses in the wordA we denote byq−(A) andq+(A) accordingly. We call the wordL the left neighborof the wordR in Jn, ifL andR are components ofJn, L < R and there exist noM in Jn, such thatL < M < R. For example, the word− + − + + is the left neighbour of− + +in J3. If L is the left neighbor ofR in Jn, thenL andR are calledthe leftandtheright predecessorsof the wordLR accordingly. IfL is the left neighbour ofR inJn (n > 0), then eitherR is the right predecessor ofL, orL is the left predecessorof R. (For example, the word−++ is the right predecessor of−+−++.) Thisdichotomy will be used in the forthcoming lemmas.

LEMMA 1. If L is the left neighbor ofR in Jn, then

q−(L)q+(R)− q+(L)q−(R) = 1. (7)

Proof (by induction onn). The casen = 0 is evident.Induction Step. Let the assertion be valid forn. LetL be the left neighbor ofR

in Jn+1.Case 1: L is the left predecessor ofR. ThenL is a component ofJn and there

exists a wordR1, such thatL is the left neighbor ofR1 in Jn andLR1 = R. Byinduction hypothesisq−(L)q+(R1) − q+(L)q−(R1) = 1. From the latter equalityand equalitiesq−(R) = q−(R1) + q−(L) and q+(R) = q+(R1) + q+(L), weget (7).

Case 2: R is the right predecessor ofL. The proof is similar. Lemma 1 isproved. 2LEMMA 2. For each two pairs of nonnegative integers(a, b) and (c, d) if ad −bc = 1, then there exist the wordsL,R and the integern, such thatL is the leftneighbor ofR in Jn andq−(L) = a, q+(L) = b, q−(R) = c, q+(R) = d.

Proof (by induction onb + d). The caseb + d = 1: sincead − bc = 1, we getb = 0, d = 1. PutL = −, R = +, n = 0.

Induction Step. Supposeb + d > 1 and for each quadruple(a1, b1, c1, d1) ifb1+ d1 d. Sincead−bc = 1, we havea > c. Thena−c > 0 andb−d > 0.Take two pairs(a − c, b − d) and(c, d). We have(a − c)d − (b − d)c = 1. Thesum of the second components of the pairs in question is less thanb + d, henceby the induction hypothesis, there exist two wordsL1, R1 and a natural numbern,such thatL1 is the left neighbor ofR1 in Jn, andq−(L1) = a− c, q+(L1) = b− d,q−(R1) = c, q+(R1) = d. Then the wordL1R1 is the left neighbor ofR1 in Jn+1.PutL = L1R1, R = R1 and we are done.

Case 3: b < d. The proof is similar to that of case 2. Lemma 2 is proved.2


THEOREM 4. (1)If A is a word inJ , thenq−(A) andq+(A) are coprime num-bers.

(2) If A < B in J , thenω(A) < ω(B) (recall thatω(A) is the rotation number,ω(A) = q+(A)/(q−(A)+ q+(A))).

(3) For each pair of coprime natural numbers(a, b) there exists the only wordA in J , such thatq−(A) = a, q+(A) = b.

Proof. (1) Part (1) is an evident consequence of Lemma 1.(2) LetA < B in J , then there existsn, such thatA andB are inJn. If A is the

left neighbor ofB, then the proof is attained by Lemma 1; otherwise there exist thewordsAi (1 6 i 6 m) in J , such thatA is the left neighbor ofA1, Ai is the leftneighbor ofAi+1 (16 i 6 m− 1), Am is the left neighbor ofB. By Lemma 1 foreveryi (16 i 6 m− 1) ω(A) < ω(Ai) < ω(Ai+1) < ω(B), as required.

(3) Leta, b be coprime numbers. Construct numbersc, d, such thatad−bc = 1,and by Lemma 2 we get the wordA, such thatA is inJ andq−(A) = a, q+(A) = b.The uniqueness of the wordA is followed from part (2) of Theorem 4 by the linearordering ofJ . Theorem 4 is proved. 2EXAMPLE 4 (to Theorem 4). (1) The wordA = −+−+−++ is a componentof J . We haveq−(A) = 3, q+(A) = 4, and 3, 4 are coprime numbers.

(2)A = −+−+−++ < −+−++ = B in J , andω(A) = 4/7< ω(B) =3/5.

(3) There exists a unique wordA in J , such thatq−(A) = 11,q+(A) = 71. ThewordA is (−+6)2−+7(−+6−+7)4.

7. Domain of the Periodic Ground State Configurations in BooleanAveraging System

Let us introduce certain notations. Define the function sg:{−,+} → {−1,1} bythe equations sg(−) = −1, sg(+) = 1. Given a sequenceγ = (γn) and integersj, p (16 j 6 p) we defineKj

p(γ ) by the equationKjp(γ ) =∑∞m=0 γj+mp.

For each integerj we putKj+pp (γ ) = Kj

p(γ ).LetA = ε1 . . . εp, whereεi = − or εi = + (16 i 6 p). Define

F(γ,A) =p∑i=1

Kp−ip (γ ) sgεi =

p∑i=1

Kip(γ ) sgεp−i .

Let Shiftp(A) (respectively,Shiftn(A)) stand for the set of the shifts ofA, whichend with the letter+ (plus) (respectively, with− (minus)).

THEOREM 5 (Domain Theorem). Let Shiftp(A) 6= φ and Shiftn(A) 6= φ. Thenthe wordA is a periodic ground state configuration in Boolean averaging sys-tem(3) with the givenψ , if and only if

minB∈Shiftn(A)

F (γ, B) > ψ > maxB∈Shiftp(A)

F (γ, B). (8)

108 M. M. KIPNIS

Proof. (1) Let A = ε1 . . . εp, whereεi = − or εi = + (1 6 i 6 p), andA be a periodic ground state configuration in Boolean averaging system (3) withthe givenψ . Let (un) be a bilateral sequence(n ∈ Z) satisfying Equation (3) andthe conditions (1), (2) of Definition 1. Thenun+mp = sgεn for all integersm and1 6 n 6 p. Denote the various shifts ofA: A1 = ε2 . . . εpε1, Ap = A, An =εn+1 . . . εpε1 . . . εn (26 n 6 p−1). Introduce the notationσn = ψ−∑∞i=1 γiun−i(16 n 6 p). With 16 n 6 p we get

σn = ψ −p∑i=1

sgεiKn−ip (γ ) = ψ − F(γ,An).

If εn = + (1 6 n 6 p) then by part (1) of Definition 1σn = ψ − F(γ,An) > 0;if εn = − (16 n 6 p) thenσn = ψ − F(γ,An) < 0. Hence, we get (8).

(2) Let the condition (8) be hold. Then put for each integern σn = ψ −∑p

i=1 sgεiKn−ip (γ ) andun = sgnσn. If 1 6 n 6 p, thenσn = ψ − F(γ,An).

By (8) with εn = + (1 6 n 6 p), we haveσn = ψ − F(γ,An) > 0 andun = 1,with εn = − (1 6 n 6 p) we haveσn = ψ − F(γ,An) < 0 andun = −1. Thesequence(σn) is p-periodic by thep-periodicity ofKi

p(γ ) as to the superscript,henceun+mp = un (1 6 n 6 p,m ∈ Z). The sequence(un) satisfies Equation (3)because

ψ −∞∑i=1

sgnσn−iγi = ψ −p∑i=1

sgεiKn−ip (γ ) = σn.

Theorem 5 is proved. 2COROLLARY 1. The wordA in the−+ alphabet is a periodic configuration inBoolean averaging system(3) if and only if

minB∈Shiftn(A)

F (γ, B) > maxB∈Shiftp(A)

F (γ, B).

8. Proof of Theorem 2

8.1. SUBSIDIARY ASSERTIONS TO THE PROOF OF THEOREM2

We need some definitions and lemmas. LetA be a component ofJ ,A 6= −,A 6= +.By +A (resp. byA−) let us denote the wordA where the first (resp., last) letter isreplaced by the letter+ (plus) (resp.,− (minus)). By+A− denote the result of thetwo operations mentioned above.

Define the functionsλ andρ overJn by recursion onn.

DEFINITION 6. (1) For words inJ0: λ(−) = −, λ(+) = 3 (3 is an emptyword),ρ(−) = 3, ρ(+) = +.


(2) Let λ andρ be defined inJn. Let A be a component ofJn+1 andL, R becomponents ofJn,L andR be the left and the right predecessors ofA respectively.

Case 1: L is the left predecessor ofR. Takeλ(A) = λ(LR) = Lλ(R), ρ(A) =ρ(LR) = ρ(R).

Case 2: R is the right predecessor ofL. Takeλ(A) = λ(LR) = λ(L), ρ(A) =ρ(LR) = ρ(L)R.

EXAMPLE 5. LetA = −++−++−+++−++−+++. Thenλ(A) =−++−++−, ρ(A) = +++−++−+++.

LEMMA 3. If the wordA is a component ofJn (n > 1) andL, R are the leftand the right predecessors ofA respectively, thenρ(A) = +L, λ(A) = R−,ρ(A)λ(A) = +A−.

Proof (by induction onn). The casen = 1 is obvious.Induction Step. Suppose the assertion be valid forn. LetA be a component of

Jn+1 (n > 1), letL andR be the left and the right predecessors ofA, respectively.Case 1: R is the right predecessor ofL. ThenL = L1R, whereL1, R are

the components ofJn−1. In this caseλ(LR) = λ(L), ρ(LR) = ρ(L)R. Hence,ρ(A) = ρ(L)R = +L1R = +L; λ(A) = λ(LR) = λ(L) = R−. We’ve usedthe equalitiesρ(L) = +L1 and λ(L) = R− which are valid by the inductionhypothesis. Thus,ρ(A)λ(A) = +LR− = +A−.

Case 2:L is the left predecessor ofR. The proof is similar. Lemma 3 is proved.2LEMMA 4. If the wordA is a component ofJ andA 6= −, A 6= +, then+A− ∈Shiftn(A).

Proof follows immediately from Lemma 3 and the evident equalityλ(A)ρ(A)= A. Lemma 4 is proved. 2DEFINITION 7. Define the functiong(i,A) for the wordsA in the−+ alphabetand the integersi (16 i 6 q+(A)). Let us number the positions of the letters in thewordA. Let the position of the extreme right letter ofA be 0; the other positionsof the letters from right to left are the integers from 1 toq−(A) + q+(A) − 1. IfA = B− for someB and 16 i 6 q+(A), theng(i,A) is the number of the positionof ith plus from the right edge ofA. If A = B+ for someB and 16 i < q+(A),theng(i,A) is the number of the position of(i + 1)th plus from the right edge ofA. Besides, ifA = B+ andi = q+(A), theng(i,A) = q−(A)+ q+(A).EXAMPLE 6. If A = + + − + − + −, then g(1, A) = 1, g(2, A) = 3,g(3, A) = 5,g(4, A) = 6. If A = −+−+−++, theng(1, A) = 1,g(2, A) = 3,g(3, A) = 5, g(4, A) = 7.

LEMMA 5. For each wordA in Jn (n > 0, A 6= −) and for each naturalnumberm

g(i,Am) = [i + iq−(A)/q+(A)] (16 i 6 q+(Am)).

110 M. M. KIPNIS

Proof (by induction onn) is being demonstrated form = 1. The expansion onarbitrarym is obvious. The casen = 1 is evident.

Induction Step. Suppose the lemma is valid forn. Let L be the left neighborof R in Jn. Denoteq−(L) = a, q+(L) = b, q−(R) = c, q+(R) = d. We haveg(i, L) = [i + ia/b] (1 6 i 6 b), g(i, R) = [i + ic/d] (1 6 i 6 d). Weaim to show thatg(i, LR) = [i + i(a + c)/(b + d)] (1 6 i 6 b + d). Inview of ad − bc = 1 (Lemma 1), if 16 i 6 d, then 0< i(a + c)/(b + d) −ia/b = i/(d(b + d)) 6 1/(b + d). Hence,[ic/d] = [i(a + c)/(b + d)] andg(i, LR) = g(i, R) = [i + i(a + c)/(b + d)], as required. Ifd < i 6 b + d,theng(i, LR) = c + d + g(i − d,L) = c + d + [i − d + (i − d)a/b]. Hence, by0 < i(a + c)/(b + d) − c − (i − d)a/b = (b + d − i)/(b(b + d)) 6 1/(b + d)(the latter equality is the concequence ofad−bc = 1 as well) we haveg(i, LR) =[i + i(a + c)/(b + d)], as required. Lemma 5 is proved. 2LEMMA 6. For any wordA (A 6= −n,A 6= +n, n > 0) in the−+ alphabet thereexist the wordsB ∈ Shiftp(A) andC ∈ Shiftn(A) such that for eachi (1 6 i 6q+(A))

g(i, B) 6 i + [iq−(A)/q+(A)]; g(i, C) > i − 1+ iq−(A)/q+(A).Proof.Given the wordA we define the numberj by

max16i6q+(A)

(g(i, A)− i − iq−(A)/q+(A)) = g(j,A)− j − jq−(A)/q+(A). (9)

Let the wordB be a shift ofA, such that thej th plus ofA is the rightmost letterof B, i.e.

g(k, B) ={g(j + k,A)− g(j,A), if 1 6 k 6 q+(A)− j;g(k − q+(A)+ j,A)− g(j,A)+ q−(A)+ q+(A),

if q+(A)− j < k 6 q+(A).(10)

ThenB ∈ Shiftp(A). When 16 k 6 q+(A) − j , we get by (9)g(j,A) − j −jq−(A)/q+(A) > g(j + k,A)− (j + k)− (j + k)q−(A)/q+(A). Hence, by (10)g(k, B) = g(j+k,A)−g(j,A) 6 k+kq−(A)/q+(A), as required. Whenq+(A)−j < k 6 q+(A), we get by (9) and (10) similarlyg(k, B) 6 k + kq−(A)/q+(A).The wordC we construct analogously. Lemma 6 is proved. 2LEMMA 7. If A is a component ofJ , A 6= −, A 6= +, andB ∈ Shiftp(A),C ∈ Shiftn(A), then for eachi (16 i 6 q+(A))

g(i, B) > g(i,A); g(i, C) 6 g(i, +A−).

Proof.By Lemma 5 for eachi (16 i 6 2q+(A))

g(i, A2) = i + [iq−(A)/q+(A)]. (11)


Let us assume thatB ∈ Shiftp(A). ThenB is the subword ofA2. Let the last letterof B be in the positiong(j,A2) in the wordA2. Then for anyi (1 6 i 6 q+(A))by (11) we get

g(i, B) = g(i + j,A2)− g(j,A2)

= i + [(i + j)q−(A)/q+(A)] − [jq−(A)/q+(A)]> i + [iq−(A)/q+(A)] = g(i,A).

Let us assume thatC ∈ Shiftn(A). If q+(A) = 1, then the conclusion of the Lemmafollows from the inequality

g(q+(A),C) 6 q+(A)+ q−(A)− 1= g(q+(A),+A−).Now, consider the variantq+(A) > 1. The wordC is a subword ofA2. We mayassume, without loss of generality, that the first letter to the right of the subwordC

in the wordA2 is the letter+ (plus). Let the position of this letter inA2 beg(j,A2).Then for anyi (16 i 6 q+(A)− 1) by (11) and Definition 7

g(i, C) = g(i + j,A2)− g(j,A2)− 1

= i − 1+ [(i + j)q−(A)/q+(A)] − [jq−(A)/q+(A)]6 i + [iq−(A)/q+(A)] = g(i,+A−).

Besides, ifi = q+(A), we have

g(i, C) 6 q+(A)+ q−(A)− 1= g(i,+A−).Lemma 7 is proved. 2LEMMA 8. If A, B are components ofJ and A 0). Hence, byLemma 1

q−(A)q+(B)− q−(B)q+(A) = 1. (12)

Then there exist integersm,n (m > 0, n > 0), such thatq−(Am) + q+(Am) =q−(Bn) + q+(Bn) = p. By (12) q+(Am) < q+(Bn). By Lemma 5 if 16 i 6q+(Bn) theng(i, Bn) = i + [iq−(B)/q+(B)]. Besides, if 16 i 6 q+(Am) andi/q+(A) 6∈ N then by Lemma 5g(i, (+A−)m) > g(i, Bn). Consider the casei/q+(A) = r ∈ N and 16 i 6 q+(Bn). By Lemma 5 and by (12)g(i, Bn) =i+[iq−(B)/q+(B)] = rq+(A)+[rq−(A)− r/q+(B)] 6 rq+(A)−1+ rq−(A) =g(i, (+A−)m). So, for eachi (1 6 i 6 q+(Am) < q+(Bn)) g(i, (+A−)m) >g(i, Bn). Then by the equalitiesF(γ, (+A−)m) = F(γ,+A−) andF(γ,Bn) =

112 M. M. KIPNIS

F(γ,B) and by the monotony ofKjp(γ ) as to superscribe with 16 j 6 p (j ∈ Z)

we get

F(γ,+A−) = F(γ, (+A−)m)

= 2q+(Am)∑i=1

Kg(i,(+A−)m)p (γ )−K(γ )

< 2q+(Bn)∑i=1

Kg(i,(+A−)m)p (γ )−K(γ )

6 2q+(Bn)∑i=1

Kg(i,Bn)p (γ )−K(γ ) = F(γ,Bn) = F(γ,B).

Lemma 8 is proved. 2

8.2. PROOF OF PART(1) OF THEOREM2

Evidently, the word+ (plus) is a periodic ground state configuration in Booleanaveraging system (3) with the givenψ if and only ifψ >

∑∞i=1 γi = K(γ ). For−

(minus) we have analogouslyψ < −K(γ ). Now, consider the components ofJ oflength more than 1. LetA = ε1 . . . εp, whereεi = − or εi = + (1 6 i 6 p). Wehave

F(γ,A) =p∑i=1

Kp−ip (γ ) sgεi = 2

q+(A)∑i=1

Kg(i,A)p (γ )−

p∑i=1

Kip(γ ). (13)

Because of the monotonicity of(γi) we getKip(γ ) > K

jp(γ )when 16 i < j 6 p.

Hence, for the wordA of length greater than 1 inJ by Lemma 7

F(γ,A) = maxB∈Shiftp(A)

F (γ, B); F(γ,+A−) = minB∈Shiftn(A)

F (γ, B); (14)

(Recall that+A− ∈ Shiftn(A)). By (13) we get

F(γ,+A−) = F(γ,A)+ 2(Kp−1p (γ )−Kp

p (γ ) > F(γ,A). (15)

Hence by (14), (15) and the corollary of Theorem 5 the domain ofA is nonempty.Part (1) of Theorem 2 is proved.


Let the periodic ground state configurationA in Boolean averaging system (3) notbe a Hubbard configuration. By part (3) of Theorem 4 there exists a wordB, suchthatB is a component ofJ andq+(A)q−(B) = q+(B)q−(A). Thenq+(Bn) =


q+(A), q−(Bn) = q−(A), wheren = q+(A)/q+(B). By Lemma 6 we’ll findC ∈Shiftp(A), such that for eachi (16 i 6 q+(A)) g(i, C) 6 i + [iq+(A)/q−(A)] =g(i, Bn) (the latter equality is valid by Lemma 5). Besides, there existsi (16 i 6q+(A)), such that we have a strict inequality in the latter inequality; otherwise thewordsC andA would be Hubbard configurations. Hence,

F(γ,C) = 2q+(A)∑i=1

Kg(i,C)p (γ )−K(γ ) > F(γ,Bn)

= 2q+(A)∑i=1

Kg(i,Bn)p (γ )−K(γ ).

Similarly we’ll find D ∈ Shiftn(A), such thatF(γ,D) < F(γ, (+B−)n). ByTheorem 5 the domain of A is defined by the inequalities

F(γ, (+B−)n) > F(γ,D) > minE∈Shiftn(A)

F (γ,E) > ψ > maxE∈Shiftp(A)

F (γ,E)

> F(γ,C) > F(γ,Bn). (16)

The domain ofBn coincides with that ofB and is defined by the inequalities (seeTheorem 5 and Equation (14))

F(γ, (+B−)n) > ψ > F(γ,Bn). (17)

In (16) and (17), we proceed from the fact that for any wordE in the−+ alphabetand for any integern (n > 0) F (γ,En) = F(γ,E). Comparing the domains ofψ ,defined by inequalities (16) and (17), we get what is required. Part (2) of Theorem2 is proved.


By Theorem 5, Lemma 8 and part (1) of Theorem 2 the domain of the periodicground state configurations inJ is a disjoint system of intervals on theψ axis.By part (2) of Theorem 2 the setP is a complement to the system mentionedabove on theψ axis. The intervals(−∞;−K(γ )) and(K(γ );∞) in theψ axisare the domains of the periodic ground state configurations− (minus) and+ (plus)respectively, which are the components ofJ0. The interval[−K(γ ),K(γ )] intheψ axis is being divided in three parts:[−K(γ );K2

2(γ ) − K12(γ )], (K2

2(γ ) −K1

2(γ );K12(γ )−K2

2(γ )), [K12(γ )−K2

2(γ );K(γ )]. The middle part is the domainof the periodic ground state configuration−+ in J1. Furthermore, the domainsof − − + and− + + are being deleted respectively from the left-handed andright-handed intervals mentioned above. Thus we have deleted the domains of thecomponents ofJ2. This procedure goes on according to the definition ofJn. Sowe have constructed the Cantor perfect setP . Let us prove that the total lengthof the deleted intervals is equal to 2K(γ ). True, any periodic configuration in

114 M. M. KIPNIS

J , consisting ofp letters, is located in theψ axis in the interval of the length2(Kp−1

p (γ )−Kpp (γ )) (Theorem 5 and Equations (14), (15)). By Theorem 4 there

areϕ(p) components ofJ of the lengthp (ϕ is a number-theoretic Euler function,ϕ(p) is the number of integersk with 16 k 6 p andk relatively prime top). So,we must prove that

∑∞p=2 ϕ(p)(K

p−1p (γ )−Kp

p (γ )) = K(γ ).Indeed,∞∑p=2

ϕ(p)(Kp−1p (γ )−Kp

p (γ ))

=∞∑p=2

ϕ(p)

∞∑m=0

(γmp+p−1− γmp+p)

= γ1+∞∑n=2

γn

( ∑p|(n+1),p>1

ϕ(p)−∑

p|n,p>1

ϕ(p)

)=∞∑n=1

γn = K(γ ).

In this chain of equalities, we have used the Gauss theorem about the Euler func-tion [7]:

∑p|n,p>1 ϕ(p) = n− 1. Part (3) of Theorem 2 is proved.

Acknowledgement

The author would like to thank B. Slepchenko and M. Zelikin for useful discus-sions.

References

1. Bak, P. and Bruinsma, R.: One-dimensional Ising model and the complete devil’s staircase,Phys. Rev. Lett.49 (1982), 249–251.

2. Bernoulli, J., III.: Sur une nouvelle espèce de calcul,Recueil pour les astronomes (Berlin),Vol. 1, 1772, pp. 255–284.

3. Burkov, S. and Sinay, Ya.: Phase diagrams of one-dimensional lattice models with long-rangeantiferromagnetic interactions,Russian Math. Surveys38(4) (1983), 235–257.

4. Delchamps, D.: Nonlinear dynamics of oversampling A-to-D converters,Proc. 32nd IEEECDC, San-Antonio, 1993.

5. Feely, O. and Chua, L.: The effect of Integrator leak in Sigma-delta modulation,IEEE Trans.Circuits Systems38 (1991), 1293–1305.

6. Gelig, A. and Churilov, A.:Stability and Oscillations in Nonlinear Pulse-modulated Systems,Birkhäuser, Basel, 1998.

7. Hardy, G. and Wright, E.:Introduction to the Theory of Numbers, Clarendon Press, Oxford,1976.

8. Hubbard, J.: Generalized Wigner lattices in one dimension and some applications to tetra-cianoquinodimethane (TCNQ) salts,Phys. Rev. B.17 (1978), 494–505.

9. Jury, E.:Sampled-data Control Systems, Wiley, New York, 1958, 2nd edn, Krieger, 1977.10. Kieffer, J. C.: Analysis of dc input response for a class of one-bit feedback encoders,IEEE

Trans. Comm.38(3) (1990), 337–340.11. Kipnis, M. M.: Symbolic and chaotic dynamics of a pulse-width control system,Soviet Phys.

Dokl. 324(2) (1992), 273–276.


12. Kipnis, M. M.: On the formalizations of the even 2-colouring,Proc. Chelyabinsk PedagogicalUniv. Series 4. Natural Sciences1 (1996), 96–104.

13. Kipnis, M. M.: One-dimensional model of statistical mechanics with the Hubbard Hamiltonianand the interaction function, free from the convexity condition,Phys. Dokl.336(3) (1994),316–319.

14. Kipnis, M. M.: Boolean Averaging in a statistical mechanics model and in an analog-to-digitalconverter,Russian J. Math. Phys.14(3) (1996), 397–402.

15. Leonov, N. N.: On the pointwise transformation of the line in itself,Izv. Vyssh. Uchebn. Zaved.Radiofiz.2(6) (1959), 942–956 (in Russian).

16. Markoff, A.: Sur une question de Jean Bernoulli,Mat. Ann.19 (1882), 27–36.17. Morse, M. and Hedlund, G.: Symbolic dynamics II: Sturmian trajectories,Amer. J. Math.62

(1940), 1–42.18. Park, S. and Gray, R.: Sigma-delta modulation with leaky integration and constant input,IEEE

Trans. Inf. Theory38 (1992), 1512–1533.19. Rockmore, D., Siegel, R., Tongring, N. and Tresser, C.: An approach to renormalization on

n-torus,Chaos1(1) (1991), 25–30.20. Siegel, R., Tresser, C. and Zettler, G.: A decoding problem in dynamics and in number theory,

Chaos2(4) (1992), 473–493.21. Smith, H. J. S.: Note on continued fraction,Messenger Math.VI (1877), 1–14.


117

Polynomial Asymptotic Representation ofSubharmonic Functions in a Half-Plane

P. AGRANOVICHInstitute for Low Temperature Physics, Mathematical Division, Lenin Ave. 47,61164 Kharkov, Ukraine

(Received: 5 August 1999; in final form: 1 May 2000)

Abstract. Let u(z) be a subharmonic function in a half-plane such that its Riesz measure is concen-trated on the finite system of rays.

In the paper the connection between the behavior ofu(z) and the distribution of its measure(including boundary measure) is investigated in terms of polynomial asymptotic representations.

Mathematics Subject Classifications (2000):30E15, 31A05.

Key words: half-plane, subharmonic function, measure, asymptotic representation.

1. Introduction

The relation between the asymptotic distribution of zeros of a holomorphic func-tion and the growth of this function at infinity is one of the most important ques-tions of function theory.

We will say that a functionf (t), t > 0, has polynomial asymptotics if it can berepresented in the following way:

f (t) = 11tρ1 +12t

ρ2 + · · · +1ntρn + κ(t), t →∞, (1)

where1j , j = 1, . . . , n, are real constants, 0< [ρ1]? < ρn < ρn−1 < · · · < ρ1,and the last term on the right, i.e., the functionκ(t), is small in a certain sense incomparison with the previous term. Similarly, we will understand the expression‘polynomial asymptotics of a functionf (z), z→∞’. In this case, the coefficients1j are functions of 0= argz andt = |z|.

The theory of functions of completely regular growth establishes a close con-nection between the growth of an entire function and the distribution of its zerosby means of one-term asymptotic representations. This theory was constructedby B. Levin and A. Pflüger simultaneously and independently in the thirties andimmediately found intensive use in different parts of mathematics and physics.Later, this theory?? was extended to other classes of functions, including holomor-phic functions in a half-plane (N. Govorov [5]). Note that, in the latter case, new? [a] is the integral part ofa.?? An extensive bibliography is in [9].

118 P. AGRANOVICH

effects appeared. They are connected with possible singularity of the holomorphicfunction on the boundary of the half-plane.

In addition to such generalizations of this theory, a number of authors have con-sidered problems relating to the connection between the behavior of a subharmonicfunction and the growth of its Riesz measure in terms of polynomial asymptoticrepresentation. They have shown that it is not possible to use the methods of thefunction theory of completely regular growth to answer these questions ( see, forexample, [4]). The relation between the existence of polynomial asymptotic rep-resentations of a subharmonic function in the plane and its Riesz measure wasinvestigated in [1–3,6,7].

In this paper, we consider this problem for a subharmonic function in a half-plane, where the Riesz masses are concentrated on a finite system of rays.

For the formulation of the main results, we need the following notations anddefinitions.

LetUρ be the class of subharmonic functions in the upper half-planeC+ = {z :Imz > 0}, which are bounded from above on every semi-circleCR = {z : z ∈C+, |z| < R}, 0 < R < ∞, and have noninteger orderρ in C+, and where theRiesz measures are concentrated on the positive ray of the imaginary axis.

Recall that the orderρu of a subharmonic functionu(z) in C+ is the followingvalue (see, for example [9]):

ρu = lim supr→∞

ln+max{81(r),82(r)}ln r

,

where

81(r) = sup0<θ<π

u(reiθ )

and

82(r) =∫ π

0|u(reiθ )| sinθ dθ.

If u(z) ∈ Up, then the distribution of its Riesz measureµu can be characterizedby the functionτu(r) = µu(Ir), whereIr = {iy : 0< y 6 r}.

In addition, we will use the so-called “boundary measure”µ∂,u defined as

µ∂,u(φ) = limh→+0

∫u(x + ih)φ(x)dx,

whereφ(x) is a test function onR and the limit is considered in the distributionspaceD′(R). It is known [9] that this limit exists and that it is a real measure(charge).

As was shown in [5], the functionu(z) ∈ Uρ1 can be represented in the form(p = [ρ1])

u(z) ={

ImPp(z)− 1

π

∫ 1

−1Im

1

t − z dµ∂,u +

POLYNOMIAL ASYMPTOTIC REPRESENTATION OF SUBHARMONIC FUNCTIONS 119

+∫ 1

−1Re

{ln

(1− z

it

)− ln

(1+ z

it

)}dτu(t)

}+

+{∫ ∞

1

(Re{

ln(

1− z

it

)− ln

(1+ z

it

)}+

+p∑j=1

1

j

[(z

it

)j−(z

−it)j])

dτu(t)+

+∫

1<|t |<∞Im[

1

t − z −1

t− · · · − zp

tp+1

]dµ∂,u(t)

}= U1+ U1

for p > 1 or

u(z) ={C − 1

π

∫ 1

−1Im

1

t − z dµ∂,u +

+∫ 1

1Re

{ln

(1− z

it

)− ln

(1+ z

it

)}dτu(t)+

+∫ ∞

1Re{

ln(

1− z

it

)− ln

(1+ z

it

)}dτu(t)+

+∫

1<|t |<∞Im

z

t (t − z) dµ∂,u(t)

}= U2+ U2

for p = 0, wherePp(z) is a polynomial of degreep. Sinceρ1 is noninteger and thefunctionsU1, U2 are harmonic for|z| > 1,

limr→∞ r

−ρ1|Uj(reiθ )| = 0, j = 1,2,

uniformly for θ ∈ [0, π ].Without loss of generality, we further consider only the casep > 1 and assume

thatτu(t) = 0, t ∈ (0,1], so

u(z) =∫ ∞

1

(Re

{ln

(1− z

it

)− ln

(1+ z

it

)}+

+p∑j=1

1

j

[(z

it

)j−(z

−it)j])

dτu(t)+

+∫

1<|t |<∞Im

[1

t − z −1

t− · · · − zp

tp+1

]t dτu(t), (2)

where

τu(t) := τ+u (t) = −1

2π

∫ t

1

1

xdµ∂,u(x)

120 P. AGRANOVICH

for t ∈ (1,∞) and

τu(t) := τ−u (t) =1

2π

∫ −1

−r1

xdµ∂,u(x)

for t ∈ (−∞,−1). All results below will be formulated, without these additionalrestrictions.

The smallness of the functionκ(t) in (1) will be expressed in terms of thefollowing definition:

DEFINITION. We will say that a functionf (t) satisfies the condition(q, ρ) if∫ 2T

T

|f (t)|qdt = o(T ρq+1), T →∞,whereρ > 0 is any fixed number.

Theorem 1, which will be formulated below, shows the existence of polynomialasymptotics of a subharmonic functionu(z) if there are polynomial asymptoticrepresentations of the functionsτu, τ+u andτ−u .

THEOREM 1. Let u(z), z ∈ C+, be a subharmonic function of the classUρ1,

ρ1 /∈ Z, and let the support of its Riesz measure be contained in the ray{z : Rez =0, Im z > 0}. Suppose that

τu(r) =n∑j=1

1jrρj +8(r),

τ+u (r) =n∑j=1

δ+j rρj + γ +(r), (3)

τ−u (r) =n∑j=1

δ−j rρj + γ −(r),

wherep = [ρ1] < ρn < ρn−1 < · · · < ρ1 and the functions8(r), γ ±(r) satisfythe condition(q, ρn) for some fixedq > 1.

Then there exists aC0,1-setE? such that, in any sectorη 6 θ 6 π − η,0 <η < π/2, u(z) has the representation

u(reiθ ) =n∑j=1

{πrρj1j

sinπρj

[cos%j

(∣∣∣∣θ − π2∣∣∣∣− π)− cos%j

(θ − π

2

)]−

− δ+j πρjrρj

sinπρjsinρj(θ − π)+

δ−j πρjrρj

sinπρjsinρjθ

}+

+ κ(reiθ ), (4)? The setE is called aC0,1 set if it can be covered by a system of circlesKj = {|z− zj | < rj }

such that limj→∞ 1R

∑j :|zj |<R rj = 0.


where

limz→∞,z/∈E

|z|ρnκ(z) = 0.

Moreover, if in the conditions(q, ρn) for the functions8(r), γ ±(r), the numberq is strictly greater than1, then the functions

supη6θ6π−η

|κ(reiθ )| and∫ π

0κ(reiθ ) sinθ dθ

satisfy the condition(q, ρn).

Remark 1. The assumptionρn > [ρ1] is a natural condition. Indeed, letρn <[ρ1]. It is not difficult to show that one can slightly modify the Riesz measure of thesubharmonic functionu(z)without changing the asymptotics (3) and the conditions(q, ρn) for the functions8(r), γ ±(r) and change the asymptotics ofu(z).

In a certain sense, the following theorem is the converse to Theorem 1.

THEOREM 2. Letu(z), z ∈ C+, be a function of classUρ1, ρ1 /∈ Z, such that itsRiesz measure is concentrated on the positive ray of the imaginary axis. Assumeequality (4) holds on the union0 of the positive ray of the imaginary axis andsectors{0< argz < θ1}, {θ2 < argz < π}, 0< θ1 < π/2< θ2 < π , i.e.

u(reiθ ) =n∑j=1

{πrρj1j

sinπρj

[cos%j

(∣∣∣∣θ − π2∣∣∣∣− π)− cos%j

(θ − π

2

)]−

− δ+j πρjr

ρj


δ−j πρjrρj

sinπρjsinρjθ

}+ κ(reiθ ),

where[ρ1] < ρn < ρn−1 < · · · < ρ1; 11 > 0 and for some fixedq ∈ (1,∞) andη > 0 the functions

κ1(r) :={

supη6θ6θ1

|κ(reiθ )|, supθ26θ6π−η

|κ(reiθ )|, |κ(ir)|},

and

κ2(r) :=∫ θ1

0κ(reiθ ) sinθ dθ +

∫ π

θ2

κ(reiθ ) sinθ dθ

satisfy the condition(q, ρn).Then the functionsτu(r), τ+u andτ−u in (2) have the asymptotic representations

τu(r) =n∑j=1

1jrρj +8(r),

122 P. AGRANOVICH

τ+u (r) =n∑j=1

δ+j rρj + γ +(r),

τ−u (r) =−∑j=1

δ−j rρj + γ −(r),

where the remainder terms8(r) andγ ±(r) satisfy the condition(q, ρn).

Remark 2. The main difficulties in the proof of this theorem are connected withthe presence of singularities of the subharmonic functionu(z) on the boundary ofthe half-plane (see Theorem 3 in Section 2).

2. Proof of Theorem 1

Proof. ? As was noted above we may suppose the functionu(z) to have repre-sentation (2). Integrating by parts, we obtain

u(z) = limR→∞Re

[(ln

1− zit

1+ zit

+n∑j=1

1

j

(z

it

)j−

p∑j=1

1

j

(z

−it)j)

τu(t)|R1 −

−∫ R

1

2izp+1(z sinpπ2 − t cospπ2 )

tp+1(t2 + z2)τu(t)dt

]+

+ Im[

zp+1

tp(t − z) τu(t)|R1 + pzp+1

∫1<|t |<R

τu(t)

tp+1(t − z) dt +

+ zp+1∫

1<|t |<Rτu(t)

tp(t − z)2 dt

].

As follows from (3) and condition(q, ρ2) for the functionsτu(t) andτu(t),

limR→∞

Re

[(ln

1− zit

1+ zit

+p∑j=1

1

j

(z

it

)j−

−p∑j=1

1

j

(z

−it)j)

τu(t)

∣∣∣∣R+Imzp+1

t (p+1)τu(t)

∣∣∣∣|t |=R] = 0. (5)

Since

τu(1) = 0, τ+u (1) = 0 and τ−u (−1) = 0,

? Only for simplicity in this paper we will consider the case of two-term asymptotic representa-tion.


then the expression in (5) is zero when|t| = 1. Thus

u(z) = −Re

[2izp+1

∫ ∞1

z sinpπ2 − t cospπ2 )

tp+1(t2+ z2)τu(t)dt

]+

+ Im

[zp+1

(p

∫1<|t |<∞

τu(t)

tp+1(t − z) dt +∫

1<|t |<∞τu(t)

tp(t − z)2 dt

)]= I1+ I2. (6)

Let us consider the integralI1. Substituting inI1 the expressions ofτu(t) from(3) and integrating the main members with the help of residues, we obtain

I1(reiθ ) =

2∑j=1

πrρj1j

sinπρj

[cosρj

(∣∣∣∣θ − π2∣∣∣∣− π)− cosρj

(θ − π

2

)]+

+ψ1(reiθ ), (7)

where

ψ1(reiθ ) = Re

[2i(reiθ )p+1

∫ ∞1

t cospπ2 − reiθ sinpπ2tp+1(t2 + r2e2iθ )

8(t)dt]. (8)

To estimate the functionψ1(reiθ ), let us divide the ray[1,∞) into intervals[2k,2k+1), k = 0,1, . . . . Let |z| ∈ [2k,2k+1). We can represent the functionψ1(z)

as the sum of three terms:

ψ1(z) = Re∫ 2k−1

1+Re

∫ 2k+2

2k−1+Re

∫ ∞2k+2= I (k)1,1 + I (k)1,2 + I (k)1,3,

whereI (k)1,2 ls taken as the principal value when Rez = 0.

Estimates ofI (k)1,1 andI (k)1,3 can be obtained with the help of the following lemma:

LEMMA A [7] . Let the functionγ (t) be from the spaceLq, q > 1, on any finiteinterval ofR and

∫0 |γ (t)|t−p−1 dt <∞, wherep > 0 is an integer. Suppose also

that, for anyT > 0 the functionγ (t) satisfies the condition(q, ρ),0 1 the asymptoticestimates

supA∈[0,k1r]

{sup

06θ62π

∣∣∣∣∫ A

0

γ (t)

tp+1(t − reiθ ) dt

∣∣∣∣} = o(rρ−p−1), r →∞,

and

supB∈[k2r,∞]

{sup

06θ62π

∣∣∣∣∫ ∞B

γ (t)

tp+1(t − reiθ ) dt

∣∣∣∣} = o(rρ−p−1), r →∞,

are valid.

124 P. AGRANOVICH

From our hypothesis on8, Lemma A and the fact thatρ2 ∈ ([ρ1], ρ1), weobtain

|z|−ρ2(I(k)

1,1 + I (k)1,3)→ 0, (9)

when|z| → ∞, uniformly for θ = argz ∈ [0, π ].Let us divide the expressionI (k)1,2 into two terms:

I(k)

1,2 = Re{zp+1

∫ 2k+2

2k−1

t cospπ2 − z sinpπ2tp+2(z− it) 8(t)dt

}−

−Re

{zp+1

∫ 2k+2

2k−1

t cospπ2 − z sinpπ2tp+2(z+ it) 8(t)dt

}= I

(k)

1,2,1+ I (k)1,2,2.

Sincez ∈ C+, it is easy to see that

|I (k)1,2,2| = o(|z|ρ2), |z| → ∞. (10)

For the estimate of the integralI (k)1,2,1, we will use the following theorem about theCauchy-type integrals:?

THEOREM A. (a)For eachq ∈ (1,∞) there exists a constantCq < ∞ suchthat, for each functiong(t) ∈ Lq(R) ands ∈ R+ the function

g(s) = sup{z:|Im(z−is)|>|Re(z−is)|}

∣∣∣∣∫ ∞−∞ g(t)

z− it dt

∣∣∣∣satisfies the estimate‖g(t)‖Lq 6 Cq‖g‖Lq .

(b) There also exists a constantC <∞ such that, for any functiong(t) ∈ L1(R)and anyh > 0

mes{s : g(s) > h} 6 C‖g‖L1h−1.

The integralI (k)1,2,1 consists of two terms:

i1 = zp+1∫ 2k+2

2k−1

cospπ2tp+1(z− it)8(t)dt

and

i2 = zp+2∫ 2k+2

2k−1

sinpπ2tp+2(z− it)8(t)dt.

To estimate the integrali1, set

gk(t) ={

cospπ/28(t)t−(p+1), t ∈ [2k−1,2k+2),

0, t /∈ [2k−1,2k+2).

? In [2] this theorem was formulated for the half-plane{z : Rez > 0}.


Since the function8(t) satisfies the condition(q, ρ2), whereq > 1 is fixed, wehave that

‖gk‖Lq = o(2k(ρ2−p− 1q′ )), k→∞, q ′ = q

q − 1.

From this and part (a) of Theorem A, forq > 1 we obtain∫ 2k+2

2k−1

{sup

{z:|Im(z−is)|>|Re(z−is)|}|i1|q

}ds

6∫ ∞−∞

{sup

{z:|Im(z−is)|>|Re(z−is)|}

(2(k+1)(p+1)

∣∣∣∣∫ ∞−∞ gk(t)

z− it dt

∣∣∣∣)q}ds

= 2(k+1)(p+1)q‖gk‖qLq 6 Cq2k(p+1)q‖gk‖qLq = o(2k(ρ2q+1)), k→∞.Now let {εk} be a sequence of positive numbers. Then the measure of the set

Ek ={s ∈ [2k,2k+1) : sup

{z:|Im(z−is)|>|Re(z−is)|}|i1| > εk2kρ2

}satisfies

mesEk = ε−qk o(2k), k→∞.If q = 1, then using part (b) of Theorem A withh = εk2kρ2, we obtain that

mesEk = ε−1k o(2k), k→∞.

Thus, for any fixedq > 1

mesEk = ε−qk o(2k), k→∞.Since the integrali2 has similar estimates, we conclude that ifq > 1, then∫ 2k+1

2k−1

{sup

{z:|Im(z−is)|>|Re(z−is)|}|I (k)1,2,1|

}qds = o(2k(ρ2q+1)), k→∞. (11)

and forq > 1 the measure of the set

Ek ={s : [2k,2k+1) : sup

{z:|Im(z−is)|>|Re(z−is)|}|I (k)1,2,1| > εk2kρ2

}is ε−qk o(2k), k→∞.

If the sequence{εk} tends to zero sufficiently slowly, then it is easy to see that therelative measure? of the sete = ⋃k Ek is zero. By comparing this with estimates(9) and (10) we conclude that if

z /∈ E :=⋃s∈e{z : |Im(z− is)| > |Re(z− is)|}

? The relative measure of a setG ⊂ (0,∞) is defined to be the limit limt→∞ t−1mes(G∩(0, t)).

126 P. AGRANOVICH

then

r−ρ2|ψ1(reiθ )| → 0, r →∞

uniformly for θ ∈ [0, π ], whereψ1(z) is defined by (8). The relative measure ofE

is zero. Evidently we can assume that the setE is open, soE =⋃ Ij whereIj is aninterval. Let us consider squares with diagonalsIj , j = 1, . . . , and circumscribecircles around each such square. The union of these circles coversE and it is easyto see that their radii satisfy the condition from the definition ofC0,1-set. So the setE isC0,1-set.

If q > 1, then direct calculation shows that∫ 2k+1

2k−1

{sup

θ∈[0, π4 ]∪[ 3π4 ,π]|I (k)1,2,1(re

iθ )|q}

dr = o(2k(ρ2q+1)), k→∞.

From this and (9), (10), (11) we obtain that∫ 2k+1

2k−1sup

06θ6π|ψ1(re

iθ )|q dr = o(2k(ρ2q+1)), k→∞,

and, hence, the function sup06θ6π |ψ1(reiθ )| satisfies the condition(q, ρ2).Now let us consider the integralI2 (see (6)). From (3), integrating the main

terms with the help of residues, we conclude that

I2 =2∑j=1

−πδ+ρjrρjsinπρj

sinρj(θ − π)+

+2∑j=1

πδ−j ρj rρj

sinπρjsinρjθ + ψ2(re

iθ ), (12)

where

ψ2(reiθ ) = Im

{zp+1

(p

∫ ∞1

γ +(t)tp+1(t − z) dt + (−1)pp

∫ ∞1

γ −(t)tp+1(t + z) dt +

+∫ ∞

1

γ +(t)tp(t − z)2 dt + (−1)p

∫ ∞1

γ −(t)tp(t + z)2 dt

)}= Im

{zp+1

4∑j=1

I2,j (z)

}. (13)

Let η,0 < η < π/2, be fixed. Ifz = reiθ ,0 < η 6 θ 6 π − η, then|t − z| > 2

√tr sinη/2 and, hence, it is easy to see that

|ψ2(reiθ )|r−ρ2 → 0, r →∞.


From here taking into account (7), (12) and thatψ1 satisfies condition(q, ρ2),we obtain (4) with the functionκ(reiθ ) = ψ1(reiθ ) + ψ2(reiθ ) such thatsupη6θ6π−η |κ(reiθ )| satisfies condition(q, ρ2).

For the completion of the proof of Theorem 1 we have to establish that thefunction

∫ π0 κ(re

iθ ) sinθ dθ also satisfies the condition(q, ρ2).In fact,∫ 2T

T

∣∣∣∣∣∫ π


∣∣∣∣∣q

dr

6 Bq(∫ 2T

T

∣∣∣∣∣∫ π

0ψ1(re

iθ ) sinθ dθ

∣∣∣∣∣q

dr +∫ 2T

T

∣∣∣∣∣∫ π

0ψ2(re

iθ ) sinθ dθ

∣∣∣∣∣q

dr

)with some constantBq > 0.

By virtue of the condition(q, ρ2) for the function sup06θ6π |ψ1(reiθ )| it followsthat ∫ 2T

T

∣∣∣∣∣∫ π

0ψ1(re

iθ ) sinθ dθ

∣∣∣∣∣q

dr = o(T ρ2q+1), T →∞. (14)

Also, from Theorem A (the case of the half-plane{z : Rez > 0}), we have (see(13))∫ 2T

T

dr

∣∣∣∣∣∫ π

0rp+1ei(p+1)θ )I2,j (re

iθ )dθ

∣∣∣∣∣q

= o(T ρ2q+1), T →∞, j = 1,2.

The integralsI2,3 andI2,4 can be estimated in the same way, so we will consideronly one of them, for exampleI2,3.

Let us introduce the function

γ +(reiθ ) ={γ +(r) sinθei(p+1)θ , r ∈ [1,∞), θ ∈ [0, π ],0, otherwise.

It is easy to see thatγ +(reiθ ) ∈ Lq(C) and

‖γ +(reiθ )‖qLq(C) = o(rρ2q+1), r →∞.Then∫ π

0sinθrp+1ei(p+1)θI2,3(re

iθ )dθ = rp+1∫C

γ +(teiθ )tp+1(t − reiθ )2 dλ,

where dλ is the Lebesgue measure onC. It is clear that this integral is the Beurlingtransformation [10] of the functionγ +(teiθ )t−(p+1) and, hence, forq > 1

‖rp+1I2,3(reiθ )‖qLq{T6r62T } = o(T ρ2q+2), T →∞.

128 P. AGRANOVICH

So in view of (13), we have∫ 2T

T

∣∣∣∣∣∫ π

0ψ2(re

iθ ) sinθ dθ

∣∣∣∣∣q

dr = o(T ρ2q+1), T →∞, q > 1.

Hence, by virtue of (14), we have finished the proof of Theorem 1. 23. Proof of Theorem 2

For the proof of this statement, we need the following theorem.

THEOREM 3. Letu(z), z ∈ C+, be a subharmonic function of the classUρ1, ρ1 /∈Z, and let its Riesz measure be concentrated on the ray{z : Rez = 0, Im z > 0}.In some sector letY (θ1, θ2) = {z : θi 6 argz 6 θ2,0 < θ1 < π/2< θ2 < π} andthe functionu(z) have the asymptotic representation(4) with the remainder termκ(reiθ ) such thatsupθ16θ6θ2 |κ(reiθ )| satisfies the condition(q, ρn). Thenτu maybe represented as

τu(t) =n∑j=1

1jtρj +8(t),

where the function8(t) satisfies the condition(q, ρn).

The proof of this theorem is analogous to the proof of Theorem 6 of [2].

Proof of Theorem 2.First we consider the sectorY1 = {z : θ1 6 argz 6π/2}. Under the conformal mappingz → (zeiθ1)π/(π/2−θ1) (so thatθ → ψ ≡π(θ − θ1)/((π/2)− θ1)) the sectorY1 is transformed into the upper half-plane andthe functionu(z) turns into a functionv(z) of orderρ1/π((π/2) − θ1) which isharmonic in the upper half-plane. Without loss of generality we can assume thatρ2/π((π/2)− θ1) is noninteger. Otherwise we will take

θ ′1,0< θ′1 < θ1, such that

ρ2

π

(π

2− θ ′1

)/∈ Z

and will do all further reasonings for thisθ ′1.It is easy to see that the hypotheses of Theorem 2 yield that corresponding to

θ = θ1, θ = π/2

v(t) =2∑j=1

{π1j t

ρjπ (

π2−θ1)

sinπρj

(cosρj

(θ1+ π

2

)− cosρj

(θ1− π

2

))+

+ δ+j πρj t

ρjπ( π2−θ1)

sinπρjsinρj(π − θ1)+

δ−j πρj tρjπ( π2−θ1)

sinπρjsinρjθ1

}+

+ κ(t 1π (

π2−θ1)eiθ1), t > 0; (151)


and

v(−t) =2∑j=1

{π1j t

ρjπ (

π2−θ1)

sinπρj(cosρjπ − 1)+

+ πρj tρjπ( π2−θ1)

sinπρjsinρj

π

2(δ+j + δ−j )

}+

+ κ(t(π2−θ1) 1

π eiπ2 ), t > 0; (152)

Nevanlinna’s formula ([8], p. 34) implies that

v(z) =s∑k=1

akrk sinkθ + 1

π

∫|t |>1

v(t)Im{

zs+1

t s+1(t − z)}

dt +O(1),

where

s =[ρ1

π

(π

2− θ1

)], z = reiθ1.

Substituting expressions(151) and(152) into this, we obtain

v(reiθ1) =s∑k=1

a′krk sinkθ +

+2∑j=1

{π1jr

ρjπ (

π2−θ1)

sinπρj

(cosρj

(π

2+ θ

(π

2− θ1

)1

π+ θ1

)−

− cosρj

(θ

(π

2− θ1

)1

π+ θ1− π

2

))−

− δ+j πρjr

ρjπ (

π2−θ1)

sinπρjsinρj

(θ

(π

2− θ1

)1

π+ θ1− π

)+

+ δ−j πρjr

ρjπ( π2−θ1)

sinπρjsinρj

(θ

(π

2− θ1

)1

π+ θ1

)}+ I +O(1),

where the remainderI is

1

π

∫ ∞1κ(t(

π2−θ1) 1

π eiθ )Im{

zs+1

t s+1(t − z)}

dt+

+ 1

π

∫ −1

−∞κ(|t|( π2−θ1) 1

π eiπ2 )Im

{zs+1

t s+1(t − z)}

dt.

130 P. AGRANOVICH

Define the function κ(t) as κ(t(π2−θ1) 1

π eiθ1)t−(s+1) when t > 1 andκ(|t|( π2−θ1) 1

π eiπ2 )t−(s+1) when t 6 −1. Then, by virtue of the condition(q, ρ2)

for κ we have∫T6|t |62T

|κ(t)|qdt = o(T ρ2(π2−θ1) 1

π −s−1)q+1), T →∞, (16)

Let us estimate the remainder termI using (16). For this we representI as thesum of three integrals:

I = 1

π

{∫16|t |6 |z|2

+∫|z|2 6|t |62|z|

+∫

2|z|6|t |6∞

}κ(t)Im

{zs+1

t − z}

dt

= I1+ I2+ I3.From Theorem A (for the case of the half-plane{z : Rez > 0}) it follows that∫ 2T

T

sup06θ6π

|I2(reiθ )|q dr = o(Tρ2π (

π2−θ1)q+1), T →∞, (17)

Sinces < ρ2/π(π2 − θ1), then by virtue of Lemma A

|z|− ρ2π( π2−θ1)(|I1| + |I3|)→ 0,

when|z| → ∞. From this and (17) we conclude that∫ 2T

T

sup06θ6π

|I (reiθ )|q dr = o(Tρ2π (

π2−θ1)q+1), T →∞, (18)

Moreover, it is clear that we can adjoin the sum∑m

k=1 a′krk sinkθ to the remainder

termI without changing its estimate.Returning to the original functionu(z), we deduce that forθ1 6 θ 6 π/2

u(reiθ ) =2∑j=1

{πrρj1j

sinπρj

[cosρj

(θ + π

2

)− cosρj

(θ − π

2

)]−

−δ+j πρjr

ρj


δ−j πρjrρj

sinπρjsinρjθ

}+ κ1(re

iθ ), (191)

and by virtue of (18) the function supθ16θ6 π2|κ1(reiθ )| satisfies the condition

(q, ρ2).Now let us consider the sectorY2 = {z : π/2 6 argz 6 θ2} and repeat the

reasonings which we have made for the sectorY1. Then we obtain that, inY2,

u(reiθ ) =2∑j=1

{πrρj1j

sinπρj

[cosρj

(θ − 3π

2

)− cosρj

(θ − π

2

)]−


−δ+j πρjr

ρj


δ−j πρjrρj

sinπρjsinρjθ

}+ κ2(re

iθ ), (192)

and the function supπ26θ6θ2 |κ2(reiθ )| satisfies the condition(q, ρ2).

So in the whole half-planeC+, in view of (191), (192) and the conditions ofTheorem 2, the functionu(z) can be represented in the following form:

u(reiθ ) =2∑j=1

{πrρj1j

sinπρj

[cosρj

(∣∣∣∣θ − π2∣∣∣∣− π)− cosρj

(θ − π

2

)]−

−δ+j πρjr

ρj


δ−j πρjrρj

sinπρjsinρjθ

}+ κ(reiθ ), (19′)

where forq ∈ (1,∞) the functions

supη6θ6π−η

|κ(reiθ )| and∫ π


satisfy the condition(q, ρ2).Using Theorem 3, we now conclude that

τu(t) = 11tρ1 +12t

ρ2 +8(t) (20)

and the function8(t) satisfies the condition(q, ρ2).Let us now consider in the half-planeC+ the subharmonic functionu1(z) of the

classUρ1, which has the Riesz masses the same asu(z) and with the distribution ofthe boundary measureτ±u1

(t) ≡ 0. Then in view of Theorem 1 and (20), we have

u1(reiθ ) =

2∑j=1

πrρj1j

sinπρj

[cosρj

(∣∣∣∣θ − π2∣∣∣∣− π)− cosρj

(θ − π

2

)]+

+w1(reiθ ),

where the function sup06θ6π |w1(reiθ )| satisfies the condition(q, ρ2).From (19′), it is easy to see, inC+

u2(reiθ ) := u(reiθ )− u1(re

iθ )

=2∑j=1

{−δ+j πρjr

ρj


δ−j πρjrρj

sinπρjsinρjθ

}+

+w2(reiθ ),

where the function supη6θ6π−η |w2(reiθ )|, η > 0, satisfies the condition(q, ρ2).From this we obtain that

limr→∞

u2(reiθ )

rρ1= πρ1

sinπρ1[δ+j sinρ1(π − θ1)+ δ−1 sinρ1θ1]

132 P. AGRANOVICH

and

limr→∞

u2(reiπ2 )

rρ1= πρ1 sinρ1

π2

sinπρ1[δ+1 + δ−1 ].

So we can calculateδ+1 andδ−1 .Now let us consider the function(reiθ ∈ C+)

u3(reiθ ) := u2(re

iθ )+ δ+1 πρ1r

ρ1

sinπρ1sinρ1(θ − π)− δ

−1 πρ1r

ρ1

sinπρ1sinρ1θ.

Evidently, the condition(q, ρ2) for the function supη6θ6π−η |w2(reiθ )| guaran-tees the existence of a sequence{rj } such that limj→∞ rj = ∞ and

limj→∞

w2(rjeiπ2 )

rρ2j

= limj→∞

w2(rjeiθ1)

rρ2j

= 0.

Going over to the limit with respect to the sequence{rj }, we have

limj→∞

u3(rjeiπ2 )

rρ2j

= πρ2

sinπρ2[δ+2 + δ−2 ] sinρ2

π

2

and

limj→∞

u3(rjeiθ1)

rρ2j

= πρ2

sinπρ2[δ+2 sinρ2(π − θ1)+ δ−2 sinρ2θ1].

So we can calculateδ+2 andδ−2 , too.Thus, taking into account (20), we obtain that

τu(r) = 11tρ1 +12t

ρ2 +8(t),τ+u (r) = δ+1 rρ1 + δ+2 rρ2 + γ +(r),τ−u (r) = δ−1 rρ1 + δ−1 rρ2 + γ −(r),

where the function8(t) satisfies the condition(q, ρ2).It remains to estimate the termsγ +(r) andγ −(r).First we assume that

supIm z>0

u(z) < 0. (21)

By virtue of the generalized Carleman formula [5, 9] we have the equality∫ r

1

(1

x2− 1

r2

)dσ(x) = 1

πr

∫ π

0u(reiθ ) sinθ dθ +A(u, r), (22)

where

σ(r) =∫ r

1t dτu(t)−

∫16|t |6r

t dτu(t);


A(u, r) = C11r2 + C2 with the constantsC1 andC2 independent ofr.

Integrating (22) by parts and taking into account thatσ(1) = 0, we concludethat ∫ r

1

σ(x)

x3dx = 1

2πr

∫ π

0u(reiθ ) sinθ dθ + 1

2A(u, r).

If we replacer by kr, 0 < k < 1, and subtract the second equation from thefirst, we obtain∫ r

kr

σ (x)

x3dx = 1

2πr

∫ π

0u(reiθ ) sinθ dθ − 1

2πkr

∫ π

0u(kreiθ ) sinθ dθ +

+ d(1− k2)

4πr2k2. (23)

By virtue of the condition (21), the measuret dτu(t) is nonpositive onR, andthat implies monotonic growth of the functionσ(x). Hence,

σ(kr)(kr)−2 1− k2

26∫ r

kr

σ (r)

x3dx 6 σ(r)(rk)−2 1− k2

2.

From this and (23) we have

σ(kr) 6 kr

(1− k2)π

[k

∫ π

0u(reiθ ) sinθ dθ −

∫ π

0u(kreiθ ) sinθ dθ +

+ d(1− k2)

2kr

]6 σ(r).

Let us calculate∫ π

0 u(reiθ ) sin dθ . According to (19′),∫ π

0u(reiθ ) sinθ dθ =

2∑j=1

πρjrρj

ρ2j − 1

(1j − δ+j − δ−j )+∫ π

0κ(reiθ ) sinθ dθ.

Therefore

σ(kr) 6 kr

(1− k2)π

[k

2∑j=1

πρjrρj

ρ2j − 1

(1j − δ+j − δ−j )+

+k∫ π

0κ(reiθ ) sinθ dθ −

2∑j=1

πρjrρj

ρ2j − 1

(1− δ+j − δ−j )−

−∫ π

0κ(kreiθ ) sinθ dθ + d(1− k

2)

2kr

]6 σ(r). (24)

It is easy to show that

τu(r)− τ+u (r)− τ−u (r) =σ(r)

r+∫ r

1

σ(x)

x2dx.

134 P. AGRANOVICH

From this and (24), we obtain

τu(r)− τ+u (r)− τ−u (r)

> k

(1− k2)π

[2∑j=1

πρjrρj

ρ2j − 1

(1j − δ+j − δ−j )(k − kρj )+

+k∫ π


∫ π

0κ(kreiθ ) sinθ dθ + d(1− k

2)

2kr+

+2∑j=1

π(rρj − 1)

ρ2j − 1

(1j − δ+j − δ−j )(k − kρj )+

+k∫ r

1

dt

t

∫ π

0κ(teiθ ) sinθ dθ −

∫ r

1

dt

t

∫ π

0κ(kteiθ ) sinθ dθ+

+d(1− k2)

2k

(1− 1

r

)]. (25)

On the other hand, from (24) it follows that(t = kr)τu(t)− τ+u (t)− τ−u (t)

6 1

(1− k2)π

[ 2∑j=1

πρj tρj

ρ2j − 1

(1j − δ+j − δ−j )(k1−ρj − 1)+

+∫ π

0κ

(t

keiθ)

sinθ dθ −∫ π

0κ(teiθ ) sinθ dθ + d(1− k

2)

2t+

+2∑j=1

π

ρ2j − 1

(1j − δ+j − δ−j )(k1−ρj − 1)(tρj − 1)+

+ k∫ t

1

dx

x

∫ π

0κ

(x

keiθ)

sinθ dθ−

−∫ t

1

dx

x

∫ π

0κ(xeiθ ) sinθ dθ + d(1− k

2)

2

(1− 1

t

)]. (26)

Since

limk→1

k1−ρj − 1

1− k2= lim

k→1

k(k − kρj )1− k2

= −1− ρj2

,

from (25) and (26) we have that

τu(r)− τ+u (r)− τ−u (r)−2∑j=1

(1j − δ+j − δ−j )rρ

> lim supk→1

[k

∫ π


∫ π

0κ(kreiθ ) sinθ dθ+


+k∫ r

1

dt

t

∫ π

0κ(teiθ ) sinθ dθ −

∫ r

1

dt

t

∫ π

0κ(kteiθ ) sinθ dθ

]k

1− k2,

and

τu(r)− τ+u (r)− τ−u (r)−2∑j=1

(1j − δ+j − δ−j )rρj

6 lim infk→1

[∫ π

0κ

(r

keiθ)

sinθ dθ−

−∫ π

0κ(reiθ ) sinθ dθ + k

∫ r

1

dt

t

∫ π

0κ

(t

keiθ)

sinθ dθ−

−∫ r

1

dt

t

∫ π

0κ(teiθ ) sinθ dθ

]1

1− k2.

By virtue of the condition(q, ρ2) for the function∫ π

0 κ(teiθ ) sinθ dθ , it follows

that ∫ 2T

T

∣∣∣∣∫ r

1

dt

t

∫ π

0κ(teiθ ) sinθ dθ

∣∣∣∣qdr 6 o(T ρ2q+1), T →∞,

hence

τu(r)− τ+u (r)− τ−u (r)−2∑j=1

(1j − δ+j − δ−j )rρj = γ (r),

where the functionγ (r) satisfies the condition(q, ρ2).So Theorem 2 is proved for this special case.For the complete proof of Theorem 2, we have to remove restriction (21). To

this end, it suffices to consider the function

U(z) = u(z)−KRezρ1e−iρ1π2 − lnM,

whereK andM are sufficiently large constants. Theorem 2 is proved. 2

4. General Case

The statements of Theorems 1 and 2 can be extended to the case when the Rieszmeasure of a subharmonic function is concentrated on a finite system of rays.

Let us formulate the corresponding results.

THEOREM 4. Letu(z), z ∈ C+, be a subharmonic function of the classUρ1, ρ1 /∈Z, and let the support of its Riesz measureµu be contained on a finite system ofrays

{argz = θj }, j = 1, . . . ,m,0= θ0 < θ1 < · · · < θm < θm+1 = π.

136 P. AGRANOVICH

Suppose that

τu,j (r) =n∑k=1

1(j)

k rρk +8j(r), j = 1, . . . ,m,

where

τu,j (r) = µu({teiθj : 0< t 6 r}),and

τ+u (r) =n∑k=1

δkrρk + γ +(r),

τ−u (r) =n∑k=1

δ−k rρk + γ −(r).

Here[ρ1] < ρn < ρn−1 < · · · < ρ1 and the functions8j(r), j = 1, . . . ,m, γ ±(r)satisfy the condition(q, ρn), q > 1.

Then

u(z) =n∑k=1

{πrρk

sinπρk

m∑j=1

1(j)

k [cosρk(|θ − θj | − π)− cosρk(θ − θj )] −

−δ+k πρkr

ρk

sinπρksinρk(θ − π)+ δ

−k πρkr

ρk

sinπρksinρkθ

}+ κ(reiθ ),

where the functionκ(reiθ ) = o(rρn ) when r → ∞ uniformly for θ ∈ [η,π − η], 0< η < π/2, if the pointz = reiθ does not belong to anyC0,1-set.

If q > 1, then the functions

supη6θ6π−η

|κ(reiθ )|, 0< η <π

2,

and ∫ π


satisfy the condition(q, ρn).

THEOREM 5. Letu(z), z ∈ C+, be a subharmonic function of the classUρ1, ρ1 /∈Z, let its Riesz measure be concentrated on a finite system of rays{argz = θj },j = 1, . . . ,m,0= θ0 < · · · < θm < θm+1 = π . Suppose, further, that

u(reiθ ) =n∑k=1

{πrρk

sinπρk

m∑j=1

1(j)

k [cosρk(|θ − θj | − π)− cosρk(θ − θj )] −

− δ+k πρkr

ρn

sinπρkρk(θ − π)+ δ

−k πρkr

ρn

sinπρksinρkθ

}+ κ(reiθ ),


θ ∈ 0 = (0, α1) ∪ (α2, π) ∪( m⋃j=1

{θj }),

where

[ρ1] < ρn < · · · < ρ1,ρn

π(θj+1− θj ) /∈ Z

for any

j, j = 1,2, . . . ,m− 1,0< α1 < θ1, θm < α2 < π

and the functions

sup{|κ(reiθ )| : {η 6 θ 6 α1} ∪ {α2 6 θ 6 π − η} ∪

(⋃θj

), η > 0

}and ∫ α1

0κ(reiθ ) sinθ dθ +

∫ π

α2

κ(reiθ ) sinθ dθ

satisfy the condition(q, ρn). Then

τu,j (r) =n∑k=1

1(j)

k rρk +8j(r),

τ+u (r) =n∑k=1

δkrρk + γ +(r),

τ−u (r) =n∑k=1

δ−k rρk + γ −(r),

where the functions8(r), j = 1, . . . ,m, andγ ± satisfy the condition(q, ρn).

Acknowledgements

With deep sorrow I say my last thankful words to Professor L. Ronkin for somevery useful conversations related to this problem.

I am grateful to the referee for constructive criticism directed at improving thequality of my exposition.

References

1. Agranovich, P. Z.: Polynomial asymptotic representations of subharmonic functions withmasses on the finite system of rays,MAG 3(3/4) (1996), 219–230 (Russian).

138 P. AGRANOVICH

2. Agranovich, P. Z. and Logvinenko, V. N.: The analogue of the Valiron–Titchmarsh theorem fortwo-term asymptotics of the subharmonic function with masses on a finite set of rays,Sibirsk.Mat. Z.24(5) (1985), 3–19 (Russian).

3. Agranovich, P. Z. and Logvinenko, V. N.: Polynomial asymptotic representation of subhar-monic function in the plane,Sibirsk. Math. Z.32(1) (1991), 3–21 (Russian).

4. Anderson, J. M.: Integral functions and Tauberian theorems,Duke Math. J.32(4) (1965), 145–163.

5. Govorov, N. V.:Riemann Boundary Value Problems with Infinite Index, Nauka, Moscow, 1986(Russian).

6. Logvinenko, V. N.: About entire functions with zeros on a half-line. I, Theory of functions,Funct. Anal. Appl.16 (1972), 154–158.

7. Logvinenko, V. N.: About entire functions with zeros on a half-line. II, Theory of functions,Funct. Anal. Appl.17 (1973), 84–99.

8. Nevanlinna, R.: Uber die Eigenschaften Meromorpher Funktionen in einem Winkebraum,ActaSoc. Sci. Fenn.50(12) (1925), 1–45.

9. Ronkin, L. I: Functions of Completely Regular Growth of Several Variables, Kluwer Acad.Publ., Dordrecht, 1992.

10. Zygmund, A.:Integrales Singulieres, Lecture Notes in Math. 204, Springer-Verlag, New York,1971.


139

Differential Equations Compatiblewith KZ Equations

G. FELDER1, Y. MARKOV2, V. TARASOV3 and A. VARCHENKO2?

1Departement Mathematik, ETH-Zentrum, 8092 Zürich, Switzerland. e-mail: [email protected] of Mathematics, University of North Carolina, Chapel Hill, NC 27599 – 3250, U.S.A.e-mail: {markov, av}@math.unc.edu3St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011,Russia. e-mail: [email protected]

(Received: 3 April 2000)

Abstract. We define a system of ‘dynamical’ differential equations compatible with the KZ differen-tial equations. The KZ differential equations are associated to a complex simple Lie algebrag. Theseare equations on a function ofn complex variableszi taking values in the tensor product ofn finitedimensionalg-modules. The KZ equations depend on the ‘dual’ variable in the Cartan subalgebra ofg. The dynamical differential equations are differential equations with respect to the dual variable.We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamicalequations. As an application we give a new determinant formula for the coordinates of a basis ofhypergeometric solutions.

Mathematics Subject Classifications (2000):Primary 35Q40; secondary 17B10.

Key words: hypergeometric solutions, Kac–Moody Lie algebras, KZ equations.

1. Introduction

In the theory of the bispectral problem [5, 12], one considers a commutative algebraA of differential operatorsL(z, ∂/∂z) acting on functions of one complex variablez. Such an algebra is called bispectral if there exists a non-trivial familyu(z, µ) ofcommon eigenfunctions depending on a spectral parameterµ

Lu(z,µ) = fL(µ)u(z, µ), L ∈ A, (1)

which is also a family of common eigenfunctions of a commutative algebraB ofdifferential operators3(µ, ∂/∂µ) with respect toµ:

3u(z,µ) = θ3(z)u(z, µ), 3 ∈ B. (2)

J. Duistermaat and A. Grünbaum [5] studied the case whereA is the algebra ofdifferential operators that commute with a Schrödinger operator(d2/dz2) − V (z)with meromorphic potentialV (z). They give a complete classification of bispectral? The last author is supported in part by NSF grant DMS-9801582.

140 G. FELDER ET AL.

algebras arising in this way. In particular they show thatA is bispectral ifV (z) isa rational KdV potential (a rational function which stays rational under the flow ofthe Korteweg–de Vries equation). G. Wilson [12] classified bispectral algebras ofrank one, i.e., such that the greatest common divisor of the orders of the differentialoperators inA is one. He showed that the maximal bispectral algebras of rankone are in one to one correspondence with conjugacy classes of pairs(Z,M) ofsquare matrices so thatZM−MZ+I has rank one. The bispectrality then followsfrom the existence of the involution(Z,M) 7→ (MT ,ZT ), which corresponds toexchangingz andµ.

The higher-dimensional version of the bispectral problem, in whichA consistsof partial differential operators inz ∈ Cn is open. However, O. Chalykh, M. Feiginand A. Veselov [3, 2] constructed examples of algebras in higher dimensions whichhave the bispectral property (see Veselov’s contribution to [7]). In these examples,A consists of differential operators commuting with ann-particle Schrödinger op-erator with certain special rational potentials, including those of Calogero–Moser.These potentials are in many respects the natural generalization of rational KdVpotential associated to rank one algebras. In these examples, the Baker–Akhiezerfunctionu(z, µ) is symmetric in the two arguments, thusB = A.

A good source of material on the bispectral problem is the volume [7].In this paper we study a class of examples of commutative algebras of partial

differential operators acting onvector-valuedfunctions with the bispectral prop-erty. This means that in (1), (2),u takes values in a vector space andfL(µ), θ3(z)are endomorphisms of the vector space. In our class of examples, the algebraA isgenerated by Knizhnik–Zamolodchikov differential operators. They are commut-ing first-order differential operators associated to a complex simple Lie algebrag with a fixed non-degenerate invariant bilinear form and a non-zero complexparameterκ. They act on functions ofn complex variableszi taking values inthe tensor product ofn finite-dimensionalg-modules. The ‘dual’ variableµ is in aCartan subalgebra ofg. The first set of Equations (1) is then the set of generalizedKnizhnik–Zamolodchikov equations(

κ∂

∂zi−∑j :j 6=i

�(ij)

zi − zj)u(z, µ) = µ(i)u(z, µ), i = 1, . . . , n.

Here� ∈ g⊗ g is dual to the invariant bilinear form and�(ij) acts as� on theithandj th factors of the tensor product and as the identity on the other factors. Sim-ilarly µ(i) is µ acting on theith factor. It is well-known that these equations forma compatible system, i.e., they are the equations defining horizontal sections for aflat connection. Forµ = 0 they reduce to the classical Knizhnik–Zamolodchikovequations. The algebraB is generated by rank(g) first order partial differentialoperators inµ with rational coefficients. We call the corresponding Equations (2)dynamical differential equations, and show that they form, together with the gen-eralized Knizhnik–Zamolodchikov equations, a compatible system. We also givesimultaneous solutions of both systems of equations in terms of hypergeometric

DIFFERENTIAL EQUATIONS COMPATIBLE WITH KZ EQUATIONS 141

integrals for a more general class of Lie algebras, which includes in particular allKac–Moody Lie algebras.

In the case ofg = sl2, the algebraB is generated by one ordinary differentialoperator. In this case, the corresponding equations where first written and solvedby H. Babujian and A. Kitaev [1], who also related the equations to the Maxwell–Bloch system.

Our paper is organized as follows. In Section 2 we introduce the systems ofKnizhnik–Zamolodchikov and dynamical differential equations for arbitrary sim-ple Lie algebra and prove their compatibility. We then give formulae for hyperge-ometric solutions in Section 3 and give as an application a determinant formula.The fact that the hypergeometric integrals provide solutions is a consequence ofa general theorem valid for a class of Lie algebras with generic Cartan matrix,introduced in [10]. We introduce in Section 4 the Knizhnik–Zamolodchikov anddynamical differential equations in this more general context and explain in thenext Section the results on complexes of hypergeometric differential forms from[10]. In Section 6 we prove that the hypergeometric integrals for generic Lie alge-bras satisfies the dynamical differential equations. Finally in Section 7 we provethat our hypergeometric integrals are solutions of both systems of equations forany Kac–Moody Lie algebra. We also find a determinant formula, which implies acompleteness result for solutions in the case of generic parameters.

2. Dynamical Differential Equations

2.1. Let g be a simple complex Lie algebra with an invariant bilinear form( , )and a root space decompositiong = h ⊕ (⊕α∈1Ceα). The root vectorseα arenormalized so that(eα, e−α) = 1. Then the quadratic Casimir element ofg⊗ g hasthe form� =∑s hs ⊗ hs +

∑α∈1 eα ⊗ e−α, for any orthonormal basis(hs) of the

Cartan subalgebrah. We also fix a system of simple rootsα1, . . . , αr .Consider the Knizhnik–Zamolodchikov (KZ) equations with an additional pa-

rameterµ ∈ h, for a functionu on n variables taking values in a tensor productV = V1 ⊗ · · · ⊗ Vn of highest weight modules ofg with corresponding highestweights31, . . . ,3n,

κ∂u

∂zi= µ(i)u+

∑i 6=j

�(ij)

zi − zj u, i = 1, . . . , n, (3)

whereκ is a complex parameter. We are interested in a differential equation foru

with respect toµ which are compatible with KZ equations. Ifµ′ ∈ h, denote by∂µ′ the partial derivative with respect toµ in the direction ofµ′.

THEOREM 2.1. The equations

κ∂µ′u =n∑i=1

zi(µ′)(i)u+

∑α>0

〈α,µ′〉〈α,µ〉 e−αeαu, µ′ ∈ h, (4)


form together with the KZ equations(3), a compatible system of equations for afunctionu(z, µ) taking values inV = V1⊗ · · · ⊗ Vn.

Equations (4) will be calleddynamical differential equations.

EXAMPLE. Let g = slN = glN/C. View slN -modules as glN -modules by lettingthe center of glN act trivially. Denote byEa,b ∈ glN the matrix whose entries arezero except for a one at the intersection of theath row with thebth column. Thefundamental coweights$a = (1 − a/N)∑b6a Eb,b − (a/N)

∑b>a Eb,b, a =

1, . . . , N − 1 form a basis of the standard Cartan subalgebra of slN . Write µ =∑N−1a=1 µa$a. Then our equations may be written as

κ∂u

∂zi=

N−1∑a=1

µa$(i)a u+

∑j :j 6=i

∑a,b

E(i)a,bE

(j)

b,a

zi − zj u,

κ∂u

∂µa=

n∑i=1

zi$(i)a u+

∑b,c:b6a<c

∑i,j

E(i)b,cE

(j)

c,b

µb + µb+1+ · · · + µc−1u.

2.2. PROOF OF THEOREM2.1

It is rather easy to verify that most terms of the compatibility equations vanish.The only non-trivial thing to check is that the operators

∑α>0(〈α, λ〉/〈α,µ〉)e−αeα

commute for different values ofλ. The operators obtained by extending the sumto all roots differ from the sum over positive roots by an element of the Cartansubalgebra. Since the operators commute with the Cartan subalgebra, it is sufficientto prove the following proposition.

PROPOSITION 2.2. Let for λ,µ ∈ h, T (λ,µ) = ∑α∈1(〈α, λ〉/〈α,µ〉)e−αeα.

Then for allλ,µ ∈ h,

T (λ,µ)T (ν, µ) = T (ν,µ)T (λ,µ).The proof is based on the following fact.

LEMMA 2.3. Let α, β ∈ 1 with α 6= ±β, and let S = S(α, β) be the setof integersj such thatβ + jα ∈ 1. Then

∑j∈S[eα, eβ+jαe−β−jα] = 0 and∑

j∈S[e−α, eβ+jαe−β−jα] = 0.Proof.For rootsγ, δ such thatγ+δ is a root, letNγ,δ = ([eγ , eδ], e−γ−δ), so that

[eγ , eδ] = Nγ,δeγ+δ . By considering the adjoint action ong of the sl2 sub-algebragenerated bye±α, we see that forβ 6= ±α, S is a finite sequence of subsequentintegers. We may thus assume thatS = {0, . . . , k} by replacingβ by β − jα forsomej if necessary.


We then havek∑j=0

[eα, eβ+jαe−β−jα]

=k−1∑j=0

Nα,β+jαeβ+(j+1)αe−β−jα +k∑j=1

Nα,−β−jαeβ+jαe−β−(j−1)α

=k−1∑j=0

(Nα,β+jα +Nα,−β−(j+1)α)eβ+(j+1)αe−β−jα.

By the invariance of the bilinear form,

Nα,β+jα = ([eα, eβ+jα], e−β−(j+1)α)

= −(eβ+jα, [eα, e−β−(j+1)α])= −Nα,−β−(j+1)α.

Therefore∑

j∈S[eα, eβ+jαe−β−jα] vanishes. The other statement is proved by re-placingα by−α and noticing thatS(−α, β) = S(α, β). 2

Proof of Proposition 2.2.Consider

T (λ,µ)T (ν, µ) =∑α,β

〈α, λ〉〈β, ν〉〈α,µ〉〈α,µ〉 [eαe−α, eβe−β]. (5)

Let us show that this expression is a regular function ofµ ∈ h. Since for nontrivialλ,µ it converges to zero at infinity, it then vanishes identically.

We compute the residue of (5) at〈α,µ〉 = 0:∑γ :γ 6=±α

〈α, λ〉〈γ, ν〉 − 〈γ, λ〉〈α, ν〉〈γ,µ〉 [eαe−α, eγ e−γ ], (6)

a function on the hyperplane〈α,µ〉 = 0. The sum overγ of the formβ + jα,j ∈ S(α, β) gives for〈α,µ〉 = 0,∑

j∈S

〈α, λ〉〈β, ν〉 − 〈β, λ〉〈α, ν〉〈β,µ〉 [eαe−α, eβ+jαe−β−jα] = 0, (7)

by the previous lemma. Since the sum overγ in (6) can be written as a sum of suchterms, it vanishes.

3. Hypergeometric Solutions

Let g be a simple complex Lie algebra. Choose a setf1, . . . , fr , e1, . . . , er ofChevalley generators of the Lie algebrag associated with simple rootsα1, . . . , αr .


Let λ = (m1, . . . ,mr) ∈ Nr . Let Q+ = ∑Nαi be thepositive root latticefor

g. Define a mapα: Nr → Q+ by α(λ) = ∑miαi. Let V be a tensor product

of highest weight modulesVj of g with respective highest weights3j , wherej = 1, . . . , n. Set3 = ∑n

j=13j . DenoteVλ the weight space ofV with weight3− α(λ). The hypergeometric solutions of the KZ equations inVλ, see [10], havethe formu(z) = ∫

γ (z)8(z, t)1/kω(z, t). We will describe the explicit construction.

The number of integration variables(tk)mk=1 is m = ∑mi. Let c be the unique

non-decreasing function from{1, . . . ,m} to {1, . . . , r} (i = 1, . . . , r), such that#c−1({i}) = mi . Define

8(z, t) =∏i<j

(zi − zj )(3i,3j )∏k,j

(tk − zj)−(αc(k),3j )∏k<l

(tk − tl )(αc(k),αc(l)).

Them-form ω(z, t) is a closed logarithmic differential form onCn × Cm withvalues inVλ. It has the following combinatorial description. LetP(λ, n) be theset of sequencesI = (i11, . . . , i

1s1; . . . ; in1, . . . , insn) of integers in{1, . . . , r} with

sj > 0, j = 1, . . . , n and such that, for all 16 j 6 r, j appears precisely|c−1(j)|times inI . For I ∈ P(λ, n), and a permutationσ ∈ 6m, setσ1(l) = σ(l) andσj(l) = σ(s1 + · · · + sj−1 + l), j = 2, . . . , n, 1 6 l 6 sj . Define6(I) = {σ ∈6m | c(σj (l)) = ilj for all j andl}.

Fix a highest-weight vectorvj for each representationVj , j = 1, . . . , n. ToeveryI ∈ P(λ, n) we associate a vectorfIv = fi11 · · · fi1s1v1⊗ · · · ⊗ fin1 · · · finsn vnin Vλ, and meromorphic differentialm-forms ωI,σ = ωσ1(1),...,σ1(s1)(z1) ∧ · · · ∧ωσn(1),...,σn(sn)(zn), labeled byσ ∈ 6(I), whereωi1,...,is (z) = d log(ti1 − ti2)∧ · · · ∧d log(tis−1 − tis ) ∧ d log(tis − z) is a meromorphic one form onC×Cs. Finally

ω(z, t) =∑

I∈P(λ,n)

∑σ∈6(I)

(−1)|σ |ωI,σfI v.

It obeys the equation8−1d8∧ω = K ∧ω, whereK =∑i<j �(ij) dzi−dzj

zi−zj . As aconsequence, for each horizontal family of twisted cyclesγ (z) in {z}×Cm, u obeysthe KZ equations (3) withµ = 0 (see [10]). This construction can be modified togive solutions for generalµ:

THEOREM 3.1. The integrals∫γ (z)

81kµω, 8µ = exp

(−

m∑i=1

〈αc(i), µ〉ti +n∑j=1

〈3j,µ〉zj)8, (8)

are solutions of the KZ equations(3).Proof.The proof follows from the identity

8−1µ d8µ ∧ ω =

n∑i=1

µ(i)dzi ∧ ω +K ∧ ω.


To prove this identity notice thatω is a sum of several termsωs, which can begrouped according to how the integration variablest1, . . . , tm are distributed amongthe pointsz1, . . . , zn. If the variabletk is associated to the pointzi in one termωs,then(dtk − dzi) ∧ ωs = 0. Moreover, if the termωs obeysh(i)ωs = (〈3i, h〉 −∑m′j (i)〈αj , h〉)ωs, h ∈ h, where3i is the highest weight of theith representation,

then the number of variablestk associated tozi such thatc(k) = j is m′j (i) (j =1, . . . , r). Thus we have

8−1µ d8µ ∧ ω =

n∑i=1

(〈3i,µ〉 −

∑j

m′j (i)〈αj , µ〉)

dzi ∧ ω +K ∧ ω.

The proof is complete. 2THEOREM 3.2. The hypergeometric integrals of Theorem3.1obey the Dynami-cal differential equations(4).

The proof of the Theorem is given in Section 7.An application of the above two theorems is a determinant formula.

COROLLARY 3.3. Fix a basisv1, . . . , vd of a weight spaceVλ. Suppose thatui(µ, z) = ∑d

j=1 ui,j vj , i = 1, . . . , d is a basis of the space of solutions in aneighbourhood of a generic point(µ, z) ∈ h × Cn. Let δα = trVλ(e−αeα) (α ∈1, α > 0), εij = trVλ(�ij ). Then there is a constantC = C(V1, . . . , Vn, λ, κ) 6= 0such that

det(uij ) = C exp

(n∑i=1

zi

κtrVλ(µ

(i))

)∏α>0

〈α,µ〉δα/κ∏i<j

(zi − zj )εij /κ .

4. Free Lie Algebras. Dynamical Differential Equations

4.1. THE DEFINITION OF KZ AND DYNAMICAL DIFFERENTIAL EQUATIONS

Following [10] let us fix the following data:

(1) A finite-dimensional complex vector spaceh;(2) A non-degenerate symmetric bilinear form( , ) onh;(3) Linearly independent covectors (‘simple roots’)α1, . . . , αr ∈ h∗.

We denote byb: h → h∗ the isomorphism induced by( , ), and we transfer theform ( , ) to h∗ via b. Setbij = (αi, αj ), hi = b−1(αi) ∈ h. Denote byg the Liealgebra generated byei , fi for i = 1, . . . , r andh subject to the relations:

[h, ei] = 〈αi, h〉ei, [h, fi] = −〈αi, h〉fi, [ei, fj ] = δij hi, [h, h′] = 0,

for all i, j = 1, . . . , r andh, h′ ∈ h. Thus we have constructed a Kac–Moody Liealgebra without Serre’s relations. We denote byn− (resp. byn+) the subalgebra


of g generated byfi (resp.ei) for i = 1, . . . , r. We haveg = n− ⊕ h ⊕ n+. Setb± = n± ⊕ h. These are subalgebras ofg.

Let3 ∈ h∗. Denote byM(3) the Verma module overg generated by a vectorvsubject to the relationsn+v = 0 andhv = 〈3,h〉v for all h ∈ h. Let us fix weights31, . . . ,3n ∈ h∗, and letM = M(31)⊗ · · · ⊗M(3n).

Forλ = (m1, . . . ,mr) ∈ Nr , set

(n±)λ ={x ∈ n± | [h, x] =

⟨±∑

miαi, h⟩x, for all h ∈ h

},

M(3)λ ={x ∈ M(3) |hx =

⟨3−

∑miαi, h

⟩x, for all h ∈ h

},

Mλ ={x ∈M |hx =

⟨∑3j −

∑miαi, h

⟩x, for all h ∈ h

}.

We haven± = ⊕λ(n±)λ, b± = h ⊕ (⊕λ(n±)λ), M(3) =

⊕λ M(3)λ, M =⊕

λ Mλ.Let τ : g→ g be the Lie algebra automorphism such thatτ(ei) = −fi, τ(fi) =

−ei , τ(h) = −h, for h ∈ h. Setn∗± =⊕

λ(n±)∗λ. SetM(3)∗ =⊕λ M(3)

∗λ. Define

a structure of ag-module onM(3)∗ by the rule

〈gφ, x〉 = 〈φ,−τ(g)x〉 for φ ∈M(3)∗, g ∈ g, x ∈ M(3). (9)

There is a unique bilinear formK( , ) ong such that:K coincides with( , ) onh;Kis zero onn+ andn−; h andn− ⊕ n+ are orthogonal;K(fi, ej ) = K(ej , fi) = δijfor i, j = 1, . . . , r; K is g-invariant, that isK([x, y], z) = K(x, [y, z]) for allx, y, z ∈ g.

A bilinear formS ong is defined by the ruleS(x, y) = −K(τ(x), y). The formS is symmetric,τ -invariant, andS([x, y], z) = S(x, [τ(y), z]). The subspacesn+,h, n− are pairwise orthogonal with respect toS.

For a Verma moduleM(3) with highest weight3 and generating vectorvthere is a unique bilinear formS onM(3) such that:S(v, v) = 1; S(eix, y) =S(x, fiy); S(fix, y) = S(x, eiy), for all x, y ∈ M(3) and i = 1, . . . , r. S issymmetric. The subspacesM(3)λ are pairwise orthogonal with respect toS. Theform S induces a homomorphism ofg-modulesS: M(3)→M(3)∗. The moduleM(3)/ kerS is the irreducibleg-module with highest weight3. More generally,on the space

∧p n− ⊗M(31) ⊗ · · · ⊗M(3n) for 3j ∈ h∗, a bilinear formS isdefined by the rule

S(g1 ∧ · · · ∧ gp ⊗ x1⊗ · · · ⊗ xn, g′1 ∧ · · · ∧ g′p ⊗ x′1⊗ · · · ⊗ x′n)

= detS(gi, g′j ) ·

n∏i=1

S(xi, x′i ).

This form induces a map

MS:∧p

n− ⊗M(31)⊗ · · · ⊗M(3n)


→(∧p

n− ⊗M(31)⊗ · · · ⊗M(3n))∗, (10)

S is called the contravariant form. The linear map (10) depends analytically on(bij ) and31, . . . ,3n ∈ h∗ for a fixedr. It is non-degenerate for general values ofthe parameters, see Theorem 3.7 in [10].

A Lie Bialgebra Structure onb± [10]. A Lie bialgebra is a vector spaceg witha Lie algebra structure and a Lie coalgebra structure, such that the cocommutatormapν: g→ g∧ g is a one-cocycle:xν(y) − yν(x) = ν([x, y]), for all x, y ∈ g.Here the action ofg ong∧ g is the adjoint one:a(b ∧ c) = [a, b] ∧ c+ b ∧ [a, c].The dual map to the cocommutator map,ν∗: (g∧ g)∗ → g∗, defines a Lie algebrastructure ong∗.

Let g be a Lie bialgebra. The double ofg is the Lie algebra equal tog⊕ g∗ as avector space with the bracket ong andg∗ defined by the Lie algebra structure ongandg∗, and forx ∈ g andl ∈ g∗, [l, x] = l+ x, wherex ∈ g andl ∈ g∗ are definedby the rulesl(y) = l([x, y]), m(x) = [m, l](x), for all y ∈ g, m ∈ g∗. The doubleis denoted byD(g).

Let g be the Kac–Moody algebra we defined at the beginning of the section.g isa Lie bialgebra with respect to the following cobracket. There exists a unique mapν: g→ g∧ g such thatxν(y) − yν(x) = ν([x, y]), andν(h) = 0, andν(fi) =12fi ∧hi, andν(ei) = 1

2ei ∧hi. In the previous four equalities,h ∈ h, i = 1, . . . , r,the action ofg on g∧ g is the adjoint one, see [4], Example 3.2 and [10].b− andb+ are subbialgebras. The mapν has the propertyτν+ντ = 0. Thus, ifρ: b−∗ →End(V ) is a representation of the Lie algebra(ν|b−)∗: 32b−∗ → b−∗, then−ρ ◦τ : b+∗ → End(V ) is a representation of the Lie algebra(ν|b+)∗: 32b+∗ → b+∗.

Note that the coalgebra mapν defines a Lie algebra structure onb∗±.

Comultiplication. LetM = M(31)⊗ · · · ⊗M(3n). Let v = v1⊗ · · · ⊗ vn ∈Mbe the product of the generating vectors. Set3 = ∑

3j . Let b− act onb− ⊗Mby the rulea(b ⊗ m) = [a, b] ⊗ m + b ⊗ am. For 1 6 i 6 n, anda, b ∈ g,m = x1⊗ · · · ⊗ xn ∈M, seta(i)m = x1⊗ · · · ⊗ xi−1⊗ axi ⊗ xi+1⊗ · · · ⊗ xn anda(i)(b⊗m) = [a, b] ⊗m+ b⊗ a(i)m.

There is a unique linear mapνM : M → b− ⊗M such that

νM(h · x) = h · νM(x) for anyh ∈ h andx ∈ M;νM(x) = 1

2(b−1(3− α(λ)))⊗ x + νM−(x) for x ∈Mλ.

Recall thatb−1(α(λ)) = ∑mib

−1(αi) = ∑mihi, andb−1: h∗ → h is defined

at the beginning of Section 4.1. The mapνM−: M → n− ⊗M is defined via aninductive definition.νM−(v) = 0, νM−(x) =∑n

k=1 ν(k)M−(x) where

ν(k)M−(f

(j)

i x) = f (j)i ν(k)M−(x) for k 6= j, and

ν(k)M−(f

(k)i x) = fi ⊗ h(k)i x + f (k)i ν

(k)M−(x).


In all formulae we have 16 i 6 r, and 16 j, k 6 n.

Remark. The corresponding definition ofνM−(x) in [10] should be correctedas above. Note that the two definitions coincide if we have one tensor factor, i.e.n = 1. We have the following lemma (cf. Lemma 6.15.2 in [10]).

LEMMA 4.1. For anyx, y ∈ M, a ∈ b−,

S(νM(x), a ⊗ y) ={

12S(x, ay) if a ∈ h;S(x, ay) if a ∈ n−.

(11)

HereS is defined onb−⊗M by the ruleS(a⊗x, b⊗y) = S(a, b)S(x, y), cf. (10).

The proof of Lemma 4.1 is given in Section 4.5.Note that the above lemma renders the following diagram commutative.

n− ⊗M standard

S

M

S

n−∗ ⊗M∗ν∗M−

M∗

S is an isomorphism for general values of parameters(bij ), (3k)nk=1, see [10],

sections (3.7) and (6.6). Hence,ν∗M : b−∗ ⊗M∗ → M∗ is ab−∗-module structurewith respect to the Lie algebra structureν∗: 32b−∗ → b−∗ for any values of theabove parameters.

COROLLARY 4.2. If y ∈ kerS: n− → n− thenyM ⊂ kerS: M →M.

ACTIONS OF THE DOUBLES OFb± ON m AND m∗ RESPECTIVELY

Consider the standard action ofb− onM∗, i.e. ∀a ∈ b− ∀φ ∈ M∗, 〈a · φ, · 〉 =〈φ,−a · 〉, where the action on the right-hand side is the standard action ofg onM.This map together withν∗M− defines an action ofb− ⊕ b−∗ onM∗. Lemma 6.17.1[10] asserts thatM∗ is aD(b−)-module under this action, whereD(b−) is thedouble ofb−.

For anya ∈ b−∗, the action ofν∗M defines a mapν∗M(a, ·): M∗ → M∗. Setρ(a) = −(ν∗M(a, ·))∗: M → M. The mapρ: b−∗ → End(M) gives an action ofb−∗ onM. An action ofb+∗ onM is defined byω = −ρ ◦ τ : b+∗ → End(M).The rulea ⊗ x → τ(a)x, for a ∈ b+, x ∈ M, defines an action ofb+ onM. Thisaction and the mapω define an action ofb+ ⊗ b+∗ onM. Lemma 11.3.28 [11]implies thatM is aD(b+) module.

Note that for any Kac–Moody Lie algebrag without Serre’s relations, and ag-moduleV we can define ag-module structure onV ∗ by the rule〈gφ, · 〉 =


〈φ,−g · 〉 for all g ∈ g andφ ∈ M∗. With respect to this module structure,Mbecomes aD(b−)-module andM∗ becomes aD(b+)-module.

KZ Equations and Dynamical Differential Equations inM andM∗. For a vectorspaceV denote by�(V ) ∈ V ⊗V ∗ the canonical element. Forλ ∈ Nr , set�−λ,± :=�((n±)λ) ∈ (n±)λ ⊗ (n±)∗λ, set�+λ,± := �((n±)∗λ) ∈ (n±)∗λ ⊗ (n±)λ, �0 :=(�(h)+�(h∗)) ∈ h⊗ h∗ + h∗ ⊗ h. Set

�± =∑λ

�−λ,± +�0+∑λ

�+λ,± ∈ D(b±)⊗D(b±).

Let�+,ij be the operator onM (orM∗) acting as�+ onM(3i)⊗M(3j), (M(3i)∗

⊗M(3j)∗ respectively) and as the identity on the other factors. The action of

D(b±) onM(3j) is the action decribed in the previous paragraph whenn = 1.The KZ equations with additional parameterµ ∈ h, for a functionu(µ, z) on nvariablesz = (z1, . . . , zn) taking values inM (orM∗) are

κ∂u

∂zi= µ(i)u+

∑i 6=j

�+,ijzi − zj u, i = 1, . . . , n, κ ∈ C. (12)

Let α be a positive root forg and(y(α)i ) a basis of(n+)α. Setx(α)i = τ(y(α)i ).

Then((y(α)i )∗), (x(α)i ) and((x(α)i )∗) are bases of(n+∗)α, (n−)α, (n−∗)α, respectively.Define operators1±,α on M (or M∗) via the formulae1+,α = ∑

i y(α)i (y

(α)i )∗,

1−,α = ∑i(x

(α)i )∗x(α)i . The dynamical differential equations for the function

u(µ, z) with values inM (M∗ respectively) are


zi(µ′)(i)u+

∑α>0

〈α,µ′〉〈α,µ〉1+,αu, µ′ ∈ h, κ ∈ C. (13)

4.2. PROPERTIES OF THE OPERATORS1+,α

The properties of the operator�+,ij are thoroughly described in [11]. Now we aregoing to study the operators1+,α.

LEMMA 4.3. The following diagram is commutative:

M1+,α

S

M

S

M∗−1−,α

M∗

In particular, the operators1+,α preserve the kernel of the mapS: M →M∗.


Proof. We fix α throughout this proof and will drop it from the notation of thebases. Fix a basis(uk) of kerS: (n+)α → (n+∗)α. Complete it to a basis of(n+)αby vectors(vl). Let (u∗k), (v

∗l ) be the dual basis of(n+∗)α. Moreover,

(v∗l ) =∑p

(A−1)lpS(vp, ·) and (τ(vl)∗) =

∑p

(A−1)lpS(τ(vp), ·), (14)

whereA = (alp) is a nondegenerate matrix with entriesalp = S(vl, vp).Fory ∈ (n+)α, consider the mapy: M →M via the action ofD(b+). Lety ·p

denote theD(b+) action, andyp denote the standard action ofb+. Then we have

S(y · p, q) = S(τ(y)p, q) = S(p,−yq) for anyp, q ∈ M. (15)

Consider the map(vl)∗: M → M via the action ofD(b+). For anyp, q ∈ M wehave

S((vl)∗p, q) = 〈(vl)∗p, S(q, · )〉 = 〈(ν∗M(τ(vl)∗, ·))∗p, S(q, · )〉

= 〈p, ν∗M(τ(vl)∗, S(q, · ))〉=⟨p, ν∗M

(∑j

(A−1)ljS(τ(vj ), ·), S(q, · ))

⟩=∑j

(A−1)lj 〈p, S(τ(vj )q, ·)〉

=∑j

(A−1)ljS(p, τ(vj )q). (16)

The first three equalities come from the definition of the action ofD(b+) onM, thelast two from formula (14). We combine (15), (16), and Corollary 4.2 to obtain:

S(1+,αp, q) = S

((∑k

uk(u∗k)+

∑l

vl(v∗l )

)p, q

)= S

(∑l

vl(v∗l )p, q

)= S((v∗l )p, (−vl)q)

= −∑j,l

(A−1)ljS(p, τ(vj )vlq). (17)

Now we trace the arrows in the alternative direction. Forx ∈ (n−)α, considerthe mapx: M∗ → M∗ via the action ofD(b−). Denote this action by ‘·’. Let xqdenote the standard action ofb− onM. We have

〈x · S(p, ·), q〉 = 〈S(p,−τ(τ(x)) · ), q〉 = S(p,−xq)for all x ∈ b−; p, q ∈M. (18)


Consider the map(τ(vl))∗: M∗ →M∗ via the action ofD(b−).

〈(τ(vl))∗S(p, · ), q〉 =⟨(∑

j

(A−1)ljS(τ(vj ), ·))S(p, · ), q

⟩=∑j

(A−1)lj〈S(τ(vj )p, · ), q〉

=∑j

(A−1)ljS(τ(vj )p, q). (19)

Finally combine (18), (19), and Corollary 4.2 withuk ∈ kerS to get

〈−1−,αS(p, · ), q〉=⟨−(∑

k

τ (uk)∗τ(uk)+

∑l

τ (vl)∗τ(vl)

)S(p, · ), q

⟩= −

⟨∑k

τ (uk)∗S(p,−τ(uk) · ), q

⟩−⟨∑

l

τ (vl)∗S(p,−τ(vl) · ), q

⟩=⟨−∑k

τ (uk)∗S(−τ(−τ(uk))p, · ), q

⟩+

+⟨−∑l

τ (vl)∗S(−τ(−τ(vl))p, · ), q

⟩= 0−

⟨∑j,l

(A−1)ljS(τ(vj )(vl)p, · ), q⟩

= −∑j,l

(A−1)ljS(τ(vj )(vl)p, q)

= −∑j,l

(A−1)ljS(p, τ(vl)vjq). (20)

Since the matrixA is symmetric (17) and (20) prove that the diagram is commuta-tive. 2

As a corollary of the lemma we have that1+,α naturally acts onL = M/ ker(S:M → M∗). We describe this action. Consider the Kac–Moody algebrag = g/ker(S: g → g∗). Let x 7→ x denote the canonical projectionsM → L, g → g.kerS is an ideal and the formS induces a non-degenerate Killing form ong viathe formulaK(x, y) = −S(τ(x), y), see [11].K induces a non-degenerate pairingbetween root spacesgα and g−α. Let (e(α)l ) be a basis ofgα, and let(f (α)l ) be thedual basis ofg−α with respect toK. Let 1α =∑l f

(α)l e

(α)l .


COROLLARY 4.4. The following diagram is commutative

M1+,α

D(b+)-action M

L1α

standardL.

Proof. L(3) ∼= Im{S: M → M∗} via x 7→ S(x, · ). We keep the notationfrom the lemma above. Setwl = −∑j (A

−1)lj τ (vj ). From the computation in thelemma we have

1+,αp = S(1+,αp, · ) = S(∑

l

(−∑j

(A−1)lj τ (vj )

)vlp, ·

)= S

(∑l

wlvlp, ·)=∑l

wlvlp =∑l

wl vl p.

Finally notice that the set(vl) forms a basis ofgα, and the set(wl) forms the dualbasis ofg−α with respect toK. 2COROLLARY 4.5. Fix λ ∈ Nr . Letm ∈ Mλ, and let(mj) be a basis ofMλ, andlet (m∗j ) be the dual basis ofM∗λ . Then the following decomposition holds

1+,αm =∑j

〈−1−,αm∗j ,m〉mj.

Proof.Lety ∈ b+ andx = τ(y) ∈ b−. Letp ∈M, andφ ∈M∗. As in the proofof Lemma 4.3〈y∗p, φ〉 = 〈p, x∗φ〉, whereD(b+) acts onM andD(b−) acts onM∗. Moreover〈y · p, φ〉 = 〈p,−x · φ〉, whereD(b+) acts onM andD(b−) actsonM∗. Finally noting that1+,α =∑i y

(α)i (y

(α)i )∗ and1−,α =∑i τ (y

(α)i )∗τ(y(α)i )

we have

1+,αm =∑j

〈m∗j ,1+,αm〉mj =∑j

〈−1−,αm∗j ,m〉mj . 2

4.3. AN INTEGRAL FORM OF THE DYNAMICAL DIFFERENTIAL EQUATIONS

Our aim now is to rewrite the Dynamical equations in a form related to the hyper-geometric solutions. Fixλ = (m1, . . . ,mr) ∈ Nr . LetM be a tensor product ofhighest weight modules of the Kac–Moody Lie algebra without Serre’s relationsg. Set3 =∑3j , the sum of the respective highest weights. Consider the weightspaceMλ of M with weight3 − α(λ), whereα(λ) = ∑

miαi. Fix a highest-weight vectorvj for each moduleM(3j), j = 1, . . . , n. To everyI ∈ P(λ, n) weassociate a vectorfIv = fi11

· · · fi1s1v1 ⊗ · · · ⊗ fin1 · · · finsn vn in Mλ, cf. Section 3.Note that the vectors(fIv)I∈P(λ,n) form a basis ofMλ.


n− acts onM∗λ via theD(b−) action. ThereforeU(n−) acts onM∗λ . Explicitly,x · φ( · ) = φ(−x · ) for x ∈ n−, φ ∈ M∗ (cf. Section 4.1), where the action on theleft-hand side is theD(b−) one and the action on the right-hand side is the standardone. LetV be a vector space freely generated byf1, . . . , fr . Therefore we have aninclusion of tensor algebrasT (V ) ⊂ T (n−). MoreoverT (V ) is an associativeenveloping algebra of the Lie algebran−. SinceT (V ) is a free associative algebra,T (V ) is isomorphic to the universal enveloping algebraU(n−). From now on wewill refer to the monomial basis ofT (V ) as to the monomial basis ofU(n−), andto the dual of the monomial basis ofT (V ) as to the monomial basis ofU(n−)∗.Rewrite a commutatorx ∈ n− as an element ofU(n−) in the formx = ∑

ajxjwhereaj ∈ Z, andxj ’s are elements of the monomial basis ofU(n−). Thus

〈x · φ, · 〉 =⟨φ,−

∑ajxj ·

⟩. (21)

Denotei: n− → U(n−) the inclusion monomorphism. Letσj ∈ {1, . . . , r}for j = 1, . . . , k. Let the positive root ofg,

∑nj=1 ασj , correspond to ther-tuple

λ′ ∈ Nr , i.e.α(λ′) =∑nj=1 ασj . Define an element1σ1,...,σk of (n−)∗λ′ via the rule

〈1σ1,...,σk , x〉 = 〈(fσ1 · · · fσk)∗, i(x)〉, wherex ∈ n−, and

(fσ1 · · · fσk)∗ ∈ U(n−)∗. (22)

Thus〈1σ1,...,σk , x〉 is the coefficient offσ1 · · · fσk in the decomposition ofi(x) intoa sum of monomials.

LEMMA 4.6. Let α = ∑rk=1m

′iαi be a positive root ofg. Let I ∈ P(λ, n). Set

λ′ = (m′1, . . . ,m′r ) ∈ Nr . Then we have

−1−,α(fI v)∗ =( ∑(i1,...,im′ )∈P(λ′,1)

1i1,...,im′ (eim′ · · · ei1))(fIv)

∗, (23)

where1i1,...,im′ acts according to theD(b−) action onM∗λ , the product ofe’s acts

on one tensor factor at a time(eim′ · · · ei1) =∑n

j=1 e(j)

im′ · · · e(j)

i1, andej acts via the

standard action(9) on each tensor factor.Proof.Note thatα = α(λ′). Let x ∈ (n−)λ′. If there existsi such thatm′i > mi,

then formula (21) impliesx · (fIv)∗ = 0 for everyI ∈ P(λ, n) because eachmonomialxj in theU(n−) expansion ofx has morefi ’s thanfIv.

Now let m′i 6 mi for any 1 6 i 6 r. Setm′ = ∑m′i . First consider the

casen = 1, I = (i1, . . . , im) ∈ P(λ,1). Let (x∗j ) be a basis of(n−)∗λ′ such thatx∗1 = 1i1,...,im′ , and let(xj ) be the dual basis of(n−)λ′. Formula (22) implies thatthe coefficient of the monomialfi1 · · · fim′ in theU(n−) expansion ofx1 is 1, andthe coefficient offi1 · · · fim′ in theU(n−) expansion ofxj is 0 for j > 1. Now use(21) to obtain

−x1 · (fIv)∗ = (fim′+1· · · fimv)∗ = eim′ · · · ei1(fIv)∗,

−xj · (fIv)∗ = 0 for j > 1. (24)


For any elementI ′ = (i′1, . . . , i′m′) ∈ P(λ′,1), such thatI ′ 6= (i1, . . . , im′) we haveei′m′ · · · ei′1(fIv)∗ = 0. The proof forn = 1 is finished.

Let n be arbitrary natural number.

(fIv)∗ = (fi11 · · · fi1s1v1)

∗ ⊗ · · · ⊗ (fin1 · · · finsn vn)∗.Formula (21) implies〈−x · (fIv)∗, · 〉 = 〈(fIv)∗, x · 〉 = 〈(fIv)∗,∑n

j=1 x(j) · 〉.

This and the computation forn = 1 give

−1−,α(fI v)∗ =∑

(i1,...,im′ )∈P(λ′,1)

n∑j=1

1i1,...,im′ e(j)

im′ · · · e(j)

i1(fIv)

∗. 2

For every positive rootα = ∑mαi αi of g, setλα = (mα1, . . . ,mαr ) ∈ Nr . Nowwe combine Corollary 4.5 and Lemma 4.6 to obtain the following form of theDynamical KZ equation.

LEMMA 4.7. Let u(µ, z) = ∑I∈P(λ,n) uIfI v, and letµ′ ∈ h be a direction

of differentiation. The Dynamical differential equation(13) is equivalent to theequation


zi(µ′)(i)u+

∑I∈P(λ,n)

∑α>0

〈α,µ′〉〈α,µ〉 ×

×∑

J∈P(λ,n)

⟨( ∑(i1,...,im′ )∈P(λα,1)

1i1,...,im′ (eim′ · · · ei1))(fJv)

∗, fI v⟩uIfJv. 2 (25)

4.4. A SYMMETRIZATION PROCEDURE

The definition of the hypergeometric differential form involves a symmetrizationprocedure, see Section 3. Now we will study the behavior of the operator1i1,...,im′ eim′ · · · ei1 for (ii, . . . , im′) ∈ P(λ′,1), whereα = α(λ′), under the sametype of symmetrization procedure.

Complexes[10]. For a Lie algebrag and ag-module M, denote byC•(g,M) thestandard chain complex ofg with coefficients inM. Cp(g,M) = 3pg⊗M and

d: gp ∧ · · · ∧ g1⊗ x

=p∑i=1

(−1)i−1gp ∧ · · · ∧ gi ∧ · · · ∧ g1⊗ gix+

+∑

16i<j6p(−1)i+jgp ∧ · · · ∧ gj ∧ · · · ∧ gi ∧ · · · ∧ g1 ∧ [gj , gi] ⊗ x.


Let 31, . . . ,3n ∈ h∗. SetM = M(31) ⊗ · · · ⊗ M(3n). Consider the complexC•(n−,M). We have the weight decompositionC•(n−,M) =⊕λ∈Nr C•(n−,M)λ.

In Section 4.1 we recalled a Lie algebra structure onn−∗ and an−∗-modulestructure onM∗. LetC•(n−∗,M∗) be the corresponding standard chain complex:C•(n−∗,M∗) = ⊕

λ∈Nr C•(n−∗,M∗)λ. The covariant form induces a graded ho-

momorphism of complexes,S: C•(n−,M)→ C•(n−∗,M∗), see [10].Let λ = (m1, . . . ,mr) ∈ Nr , andm = ∑

mi . Define a subgroup6λ of thesymmetric group6m via the direct product6λ = 6m1 × · · · × 6mr , where6mjpermutes the set of indices{∑j−1

p=1mp + 1, . . . ,∑j−1

p=1mp +mj }. Introduce a free

Lie algebran− on generatorsf1, . . . , fm. Define a map of Lie algebrasn− → n−by settingfi 7→∑mi

j=1 fm(i)+j , wherem(i) = m1+· · ·+mi−1. It induces a map ofcomplexesC•(n−, U(n−)⊗n)→ C•(n−, U(n−)⊗n)6λ . Setλ = (1,1, . . . ,1︸︷︷︸

m

). Let

s: C•(n−, U(n−)⊗n)λ→ C•(n−, U(n−)⊗n)6λλ (26)

be the previous map composed with the projection on theλ-component.On the other hand there is a map of Lie algebrasn− → n− defined byfj 7→

fi, for m(i) < j 6 m(i + 1). It induces the mapπλ: C•(n−, U(n−)⊗n)λ →C•(n−, U(n−)⊗n)λ. Note thats(y) equals the sum over the preimages ofy underπλ, for anyy ∈ C•(n−, U(n−)⊗n)λ. Each such preimage is uniquely described byan elementσ ∈ 6λ.EXAMPLE. Letn = 1 andI = (i1, . . . , im) ∈ P(λ,1). ConsiderfI as an elementof C0(n−, U(n−))λ. Thens(fI ) = ∑

σ fσ1 . . . fσm, where the sum is over the set{σ ∈ 6(I)} ∼= 6λ.

LEMMA 4.8. Let n = 1, I ∈ P(λ,1), and m′ 6 m. Then the maps∗:(C1(n−, U(n−))6λλ )

∗ → (C1(n−, U(n−))λ)∗ has the following property

s∗(

1

|6λ|∑σ∈6(I)

1σ1,...,σm′ ⊗ (fσm′+1. . . fσm)

∗)

= 1i1,...,im′ ⊗ (fim′+1. . . fim)

∗. (27)

Proof. Choose a basis(x∗j ) of n−∗ such thatx∗1 = 1i1,...,im′ and let(xj ) be thedual basis ofn−. Take a tensor product of the basis(xj ) with the monomial basisin U(n−) to get a basis inn− ⊗ U(n−). Let us compare the two sides of (27) onthat basis.1i1,...,im′ ⊗ (fim′+1

. . . fim)∗(x1⊗ fim′+1

. . . fim) = 1. The right-hand sideis zero on any other element of the basis.

Let i: n−⊗U(n−)→ U(n−)⊗U(n−), andi: n−⊗U(n−)→ U(n−)⊗U(n−)be the natural inclusion maps. Clearlyi ◦ s = s ◦ i. Moreover,〈1σi1,...,σim′

, s(x)〉 =〈(fσi1 · · · fσim′ )∗, (i ◦ s(x))〉 by definition forx ∈ n−. For a fixedσ ∈ 6λ we have

〈s∗(1σ1,...,σm′ ⊗ (fσm′+1. . . fσm)

∗), x1⊗ fim′+1. . . fim〉


= 〈1σi1,...,σim′ ⊗ (fσim′+1. . . fσim )

∗, s(x1⊗ fim′+1. . . fim)〉

= 〈(fσ1 . . . fσm′ )∗ ◦ i ⊗ (fσm′+1

. . . fσm)∗, s(x1 ⊗ fim′+1

. . . fim)〉= 〈(fσ1 . . . fσm′ )

∗ ⊗ (fσm′+1. . . fσm)

∗, s(i(x1)⊗ fim′+1. . . fim)〉.

The duality of the bases impliesi(x1 ⊗ fim′+1. . . fim) = (fi1 . . . fim′ + other

monomials) ⊗ fim′+1· · · fim , ands ◦ i(x1 ⊗ fim′+1

. . . fim) = (fσ1 . . . fσm′ + othermonomials) ⊗ fσm′+1

· · · fσm . Therefore,〈s∗(1σ1,...,σm′ ⊗ (fσm′+1. . . fσm)

∗), x1 ⊗fim′+1

. . . fim〉 = 1. The same way we check that the left-hand side is zero on theother basis elements. 2COROLLARY 4.9. Let π be the restriction of the projectionπλ to the subspaceC.(n−, U(n−)⊗n)6λλ of C.(n−, U(n−)⊗n)λ. Let J = (j1, . . . , jm) ∈ P(λ, n) andI = (i1, . . . , im′) ∈ P(λα,1) for a positive rootα. Then we have( ∑

τ∈6(I)1τ1,...,τm′ (eτm′ . . . eτ1)

)( ∑σ∈6(J )

(fσ1 . . . fσm)∗)

= π∗((1i1,...,im′ (eim′ . . . ei1))(fj1 · · · fjm)∗). (28)

Proof. Assumen = 1. The general case follows from this one because theoperators(eτm′ . . . eτ1) and (eim′ . . . ei1) act on one tensor factor at a time. Sinces ◦ π = |6λ|id we haveπ∗ ◦ s∗ = |6λ|id. s∗ andπ∗ are maps of complexes, i.e.they commute with the corresponding differentials. Thus, Lemma 4.8 implies afterapplying differentials and takingπ∗ from both sides

π∗(1j1,...,jm′ (fjm′+1· · · fjm)∗) =

∑σ∈6(J )

1σ1,...,σm′ (fσm′+1· · · fσm)∗.

The right-hand side of our formula is non-zero if and only ifm′ 6 m andik = jkfor 16 k 6 m′. Thus we compute

π∗(1i1,...,im′ eim′ . . . ei1(fj1 . . . fjm)∗)

= π∗(1j1,...,jm′ (fjm′+1. . . fjm)

∗)

=∑

σ∈6(J )1σ1,...,σm′ (fσm′+1

. . . fσm)∗

=∑

σ∈6(J )1σ1,...,σm′ eσm′ . . . eσ1(fσ1 . . . fσm)

∗.

Note that(σ1, . . . , σm′) ∈ 6(I). For any otherτ ∈ 6(I), we have

eτm′ . . . eτ1(fσ1 . . . fσm)∗ = 0.

Therefore we rewrite the last equality in the form (28). 2


Our last step is to show that the Dynamical equations for any Lie algebra in anyweight space follow from the Dynamical equations for any Lie algebra in a weightspace with weightλ = (1,1, . . . ,1).

Fix a finite-dimensional complex vector spaceh, a non-degenerate symmet-ric bilinear form ( · , · ) on h, and a set of linearly independent ‘simple roots’α1, . . . , αr ∈ h∗. Consider the corresponding Kac–Moody Lie algebra withoutSerre’s relations,g, defined at the beginning of Section 4.1. Recall thatb: h→ h∗denotes the isomorphism induced by the bilinear form,hi = b−1(αi), and and theform ( · , · ) is transfered toh∗ via the mapb.

Fix λ = (m1, . . . ,mr) ∈ Nr . Consider the corresponding positive rootα(λ) =∑ri=1miαi of g. Up to reordering of theα’s we can assume thatα(λ) =∑p

i=1miαiwheremi > 0 for 1 6 i 6 p andp 6 r is fixed. The corresponding coloringfunction iscλ: {1, . . . ,m = ∑

mi} → {1, . . . , p}. We use the following linearalgebraic fact when symmetrizing.

PROPOSITION 4.10.Leth be a finite-dimensional vector space with a non-dege-nerate symmetric bilinear form( · , · ), and a set of linearly independent vectors(hi)

ri=1 ⊂ h. Then there exists a finite-dimensional vector spaceh with a non-

degenerate symmetric bilinear form( · , · )1, a set of linearly independent vectors(hj )

mj=1 ⊂ h, and a monomorphismsh: h→ h such that

(a) sh(hi) = 1mi

∑mij=1 hm(i)+j , wherem(i) = m1+ · · · +mi−1, i = 1, . . . , p;

(b) (hj , sh(h′))1 = (hc(j), h′) and(h′, h′′) = (sh(h′), sh(h′′))1 for anyh′, h′′ ∈ h,j = 1, . . . ,m.

Proof.Let q = dim h. Complete the seth1, . . . , hr to a basish1, . . . , hr , hr+1,. . . , hq of h. Consider a complex linear spaceh′ = C{h1, . . . , hm, hm+1,. . . , hm+q−p}. Extend the coloring functionc: {1, . . . ,m + q − p} → {1, . . . , q}settingc(m + j) = r + j for j = 1, . . . , q − p. Define a symmetric degeneratebilinear form onh′ by the rules(hj , hk)1 = (hc(j), hc(k)) for 16 j, k 6 m+q−p.The rank of the form isq and the dimension of its kernel ism− p. There exists anextensionh of the vector spaceh′ and an extension of( · , · )1 to a non-degeneratesymmetric bilinear form onh.

Define a monomorphismsh by sh(hi) = 1mi

∑mij=1 hm(i)+j , wherem(i) = m1 +

· · · + mi−1, i = 1, . . . , q, and note thatsh(hp+j ) = hm+j for j = 1, . . . , q − p.Now checking (b) on a basis is straightforward. 2

Setαj = (hj , · )1 ∈ h∗ for j = 1, . . . ,m. Considerg, a Kac–Moody Lie alge-bra without Serre’s relations corresponding to the datah, ( · , · )1, and(αj )mj=1. Notethat 16 j 6 m implies 〈αj , sh(h′)〉 = (hj , sh(h′))1 = (hc(j), h′) = 〈αc(j), h′〉 foranyh′ ∈ h.

LetM = M(31)⊗· · ·⊗M(3n) be a tensor product of Verma modules forgwithcorresponding highest weights31, . . . ,3n ∈ h∗. Sincesh is a monomorphism,s∗h : h∗ → h∗ is a linear epimorphism. Choose highest weights31, . . . , 3n ∈ h∗


such thats∗h(3j ) = 3j for 1 6 j 6 n, and consider the corresponding tensorproduct of Verma modules forg, M = M(31)⊗ · · · ⊗ M(3n).

LEMMA 4.11. Let λ = (1,1, . . . ,1︸︷︷︸m

). Let u(µ, z) =∑K∈P (λ,n) uKfK be a hyper-

geometric solution of the Dynamical equations with values in theλ weight spaceof a g-moduleM ∼= U(n−)⊗n. Thenu(µ, z) = π(u)(sh(µ), z) is a hypergeomet-ric solution of the Dynamical equations with values in theλ weight space of ag-moduleM ∼= U(n−)⊗n, i.e.u =∑I∈P(λ,n) uIfI .

Proof. Note that by definitionP (λ, n) = ⋃I∈P(λ,n){K ∈ 6(I)}. From the

definition of the hypergeometric differential form, see Section 3, it follows thatuI =∑k∈6(I) uK . Therefore

π(u) = π

( ∑I∈P(λ,n)

∑K∈6(I)

ukfK

)=

∑I∈P(λ,n)

( ∑K∈6(I)

uk

)fI =

∑I∈P(λ,n)

uI fI . (29)

Fix a pointµ ∈ h and a direction of differentiationµ′ ∈ h. Denoteµ = sh(µ) andµ′ = sh(µ′). Since

∂µ′ exp

(−

m∑j=1

〈αc(j), µ〉tj +n∑l=1

〈3l,µ〉)

=(−

m∑j=1

〈αc(j), µ′〉tj +n∑l=1

〈3l,µ′〉)

exp

(−

m∑j=1

〈αc(j), µ〉tj +n∑l=1

〈3l,µ〉)

=(−

m∑j=1

〈αj , µ′〉tj +n∑l=1

〈3l, µ′〉)

exp

(−

m∑j=1

〈αj , µ〉tj +n∑l=1

〈3l, µ〉)

= ∂µ′ exp

(−

m∑j=1

〈αj , µ〉tj +n∑l=1

〈3l, µ〉), (30)

we haveπ(∂µ′u(µ, z)) = ∂µ′π(u)(µ, z). If I = (I1, . . . , In) ∈ P(λ, n) andK =(K1, . . . ,Kn) ∈ 6(I), whereKj = (kj1, . . . , kjsj ) andIj = (ij1, . . . , ijsj ), then

µ′(j)fK =⟨3j −

sj∑l=1

αkjl, µ′

⟩fK =

⟨3j −

sj∑l=1

αijl, µ′

⟩fK,

π

(µ′(j)

∑K∈6(I)

uKfK

)=⟨3j −

sj∑l=1

αijl, µ′

⟩( ∑K∈6(I)

uKπ(fK)

)

=⟨3j −

sj∑l=1

αijl, µ′

⟩( ∑K∈6(I)

uK

)fI


=⟨3j −

sj∑l=1

αijl, µ′

⟩uIfI = µ′(j)fI . (31)

Combine formulae (29) and (31) to obtain

π

(n∑j=1

zj µ′(j)u

)=

n∑j=1

zjµ′(j)u. (32)

Let α =∑ri=1m

′iαj be a positive root forg. Lemma 4.6 gives a necessary con-

dition for a non-zero action of1−,α onM∗λ . Namelym′i 6 mi for all i = 1, . . . , r.Analogously, for a positive rootα = ∑m

j=1pj αj of g a necessary condition fora non-zero action of1−,α on M∗

λis pj = 0,1 for j = 1, . . . ,m. Call all such

α’s (α’s) λ-admissible (λ-admissible). Sinces∗h(αj ) = αc(j), s∗h maps the set ofλ-admissible roots ofg onto the set ofλ-admissible roots ofg.

Let α =∑m′iαi be aλ-admissible root forg andm′ =∑m′i. For anyα, suchthats∗h(α) = α, we have〈α, µ′〉/〈α, µ〉 = 〈α,µ′〉/〈α,µ〉. Consider

π

( ∑α, s∗h(α)=α

〈α, µ′〉〈α, µ〉1+,αu

),

where the sum is overλ-admissible roots. Corollary 4.5 applied to the basis(fK)K∈P (λ,n) of Mλ gives

〈α,µ〉〈α,µ′〉π

( ∑α, s∗h(α)=α

〈α, µ′〉〈α, µ〉1+,αu

)

= π( ∑K∈P (λ,n)

⟨−

∑α, s∗h(α)=α

1−,α(fK)∗, u⟩fK

)

=∑

I∈P(λ,n)

⟨(−

∑α, s∗h(α)=α

1−,α)( ∑

K∈6(I)(fK)

∗), u

⟩fI . (33)

Lemma 4.6 asserts that

−∑

α, s∗h(α)=α1−,α =

∑α, s∗h(α)=α

( ∑(l1,...,lm′ )∈P (λα,1)

1l1,...,lm′ (elm′ . . . el1)

).

Rearrange the summation using that sum over(l1, . . . , lm′) ∈ P (λα,1) such thats∗h(α) = α equals the sum over(p1, . . . , pm′) ∈ 6(J ) such thatJ = (j1, . . . , lm′) ∈P(λα,1). Combine such rearrangement with Lemma 4.6 and Corollary 4.9 to sim-plify formula (33).

〈α,µ〉〈α,µ′〉π

( ∑α, s∗h(α)=α

〈α, µ′〉〈α, µ〉1+,αu

)


=∑

I∈P(λ,n)

⟨ ∑J∈P(λα,1)

( ∑(p1,...,pm′ )∈6(J )

1p1,...,pm′ (epm′ . . . ep1)

)×

×( ∑K∈6(I)

(fK)∗), u

⟩fI

=∑

I∈P(λ,n)

⟨ ∑J∈P(λα,1)

π∗(1j1,...,jm′ (ejm′ . . . ej1)(fI )∗), u

⟩fI

=∑

I∈P(λ,n)

⟨ ∑J∈P(λα,1)

1j1,...,jm′ (ejm′ . . . ej1)(fI )∗, π(u)

⟩fI

=∑

I∈P(λ,n)〈−1−,α(fI )∗, u〉fI = 1+,αu. (34)

Finally (32) and (34) imply

∂µ′u = π(∂µ′u) = π

((n∑j=1

zj µ′(j) +

∑α>0

〈α, µ′〉〈α, µ〉1+,α

)u

)

=(

n∑j=1

zjµ′(j) +

∑α>0

〈α,µ′〉〈α,µ〉1+,α

)u. 2 (35)

4.5. THE PROOF OF LEMMA4.1

Recall that the linear mapνM : M → b− ⊗M has the following propertyνM(x) =12(b−1(3−α(λ)))⊗x+νM−(x), wherex ∈ Mλ, νM−(x) ∈ n−⊗M, andb−1: h∗ →

h is defined at the beginning of Section 4.1. Leta ∈ h, x ∈Mλ. SinceS(b⊗x, a⊗y) = S(a, b)S(x, y) for anyb ∈ g, y ∈ M, andh is orthogonal ton− with respectto S, and( · , · ) coincides withS on h we have

S(νM−(x), a ⊗ y) = S(

12(b−1(3− α(λ))), a)S(x, y) = 1

2S(x, ay). (36)

This proves the first equality in the lemma. To prove the second part for a monomialx, we use double induction by the number of tensor factors and the number off ′sin x. We useνM−(x) instead ofνM(x) because of the orthogonality mentionedabove.

LetM = M(31) be a highest weight module ofg with a highest vectorv, anda ∈ n−, y ∈M. SinceS(νM−(v)) = 0, we haveS(νM−(v), a⊗y) = 0= S(v, ay).The inductive step is as follows. AssumeS(νM−(x), a ⊗ y) = S(x, ay). Then

S(νM−(fix), a ⊗ y)= S(fi ⊗ hix, a ⊗ y)+ S(fiνM−(x), a ⊗ y)= S(fi, a)S(hix, y) + S(νM−(x), ei (a ⊗ y))


= S(fi, a)S(hix, y) + S(νM−(x), [ei , a] ⊗ y)+ S(νM−(x), a ⊗ eiy)= S(fi, a)S(hix, y) + S(νM−(x), [ei , a] ⊗ y)+ S(x, aeiy)= S(fi, a)S(hix, y) + S(νM−(x), [ei , a] ⊗ y)++ S(x, eiay)− S(x, [ei , a]y)= S(fi, a)S(hix, y) + S(νM−(x), [ei , a] ⊗ y)++ S(fix, ay) − S(x, [ei , a]y). (37)

If a = fi, then (37) and the propertiesS(x, hy) = S(hx, y), S(νM−(x), h⊗y) = 0for h ∈ h imply

S(νM−(fix), a ⊗ y) = S(hix, y) + S(νM−(x), hi ⊗ y)+ S(fix, ay) − S(x, hiy)= S(fix, y).

If a is orthogonal tofi with respect toS, then (37) and the inductive hypothesisgive

S(νM−(fix), a ⊗ y) = 0+ S(x, [ei , a]y) + S(fix, ay) − S(x, [ei , a]y)= S(fix, a).

Thus the statement is proved for one tensor factor.Assume thatS(νM−(fix), a⊗ y) = S(x, ay) for a moduleM, which is a tensor

product of up ton− 1 tensor factors(n > 2).Let M = M(31) ⊗ · · · ⊗ M(3n). Recall thatνM−(x) = ∑n

k=1 ν(k)M (x)−,

whereν(k)M (f(j)

i x)− = f(j)

i ν(k)M (x)− for k 6= j , andν(k)M (f

(k)i x)− = fi ⊗ h(k)i x +

f(k)i ν

(k)M (x)−. The following commutation relations will be useful.S(f (j)i ν

(k)M−(x),

a⊗y) = S(ν(k)M−(x), a⊗e(j)i y), for j 6= k, andS(f (j)i ν(j)

M (x)−, a⊗y) = S(ν(j)M−(x),[ei, a]⊗y)+S(ν(j)M−(x), a⊗e(j)i y). Both equalities are corollaries of the Lemma 4.1for one tensor factor, and the definition ofS, e.g.S(x1⊗ · · · ⊗ xj−1⊗ νM−(xj )⊗xj+1⊗· · ·⊗xn, a⊗y1⊗· · ·⊗yj⊗· · ·⊗yn) = S(νM−(xj ), a⊗yj )∏k 6=j S(xk, yk).In all formulae 16 i 6 r, and 16 j, k 6 n, and the upper script indicates thetensor factor where the action is applied. Leta ∈ n−. The base for the induction isexactly as forn = 1. The inductive step is as follows.

S(νM−(f(j)

i x), a ⊗ y)=∑k 6=j

S(ν(k)M−(f

(j)

i x), a ⊗ y)+ S(ν(j)M−(f (j)i x), a ⊗ y)

=∑k 6=j

S(f(j)

i ν(k)M−(x), a ⊗ y)+

+ S(fi ⊗ h(j)i x, a ⊗ y)+ S(f (j)i ν(j)

M (x)−, a ⊗ y)= S(fi, a)S(h(j)i x, y) + S(ν(j)M−(x), [ei , a] ⊗ y)+

+n∑k=1

S(ν(k)M−(x), a ⊗ e(j)i y). (38)


The result for one tensor factor gives∑k

S(ν(k)M−(x), a ⊗ e(j)i y) =

∑k

S(x, a(k)e(j)

i y)

=∑k

S(x, e(j)

i a(k)y)− S(x, [ei , a](j)y)

= S(f(j)

i x, ay) − S(x, [ei , a](j)y). (39)

If a = fi, then (38), (39) and the propertiesS(x, h(k)y) = S(h(k)x, y), S(ν(k)M−(x),h⊗ y) = 0 for h ∈ h, k = 1, . . . , n imply

S(νM−(f(j)

i x), a ⊗ y) = S(h(j)

i x, y) + 0+ S(f (j)i x, ay) − S(x, h(j)i y)= S(f

(j)

i x, ay).

If a is orthogonal tofi with respect toS, then (38), (39) and the result for onetensor factor give

S(νM−(f(j)

i x), a ⊗ y)= 0+ S(x, [ei , a](j)y)+ S(f (j)i x, ay) − S(x, [ei , a](j)y)= S(f (j)i x, ay).

This finishes the inductive argument. The lemma is proved. 2

5. Flags, Orlik–Solomon Algebra, Hypergeomertic Differential Forms

In this section we will formulate results from [10] which define a map between thecomplex of hypergeometric differential forms and the complexC•(n−∗,M∗) for asuitable Lie algebran− and an−-moduleM.

5.1. COMPLEXES

Let W be an affine complexm-dimensional space and letC be a configurationof hyperplanes inW . Define Abelian groupsAk(C,Z), 0 6 k 6 m, as follows.A0(C,Z) = Z. Fork > 1, Ak(C) is generated byk-tuples(H1, . . . , Hk),Hi ∈ C,subject to the relations:(H1, . . . , Hk) = 0 if H1, . . . , Hk are not in general position (codimH1 ∩ · · · ∩

Hk 6= k);(Hσ(1), . . . , Hσ(k)) = (−1)|σ |(H1, . . . , Hk) for any permutationσ ∈ 6k ;∑k+1

i=1(−1)i(H1, . . . , Hi , . . . , Hk+1) = 0 for any (k + 1)-tupleH1, . . . , Hk+1

which is not in general position and such thatH1 ∩ · · · ∩Hk 6= 0.The direct sumA•(C,Z) =⊕m

k=0 Ak(C,Z) is a graded skew commutative al-gebra with respect to the multiplication(H1, . . . , Hk) · (H ′1, . . . , H ′l ) =


(H1, . . . , Hk,H′1, . . . , H

′l ). A•(C,Z) is called theOrlik–Solomon algebraof the

configurationC.

Flags. For 0 6 k 6 m, denote by Flagk(C) the set of all flagsL0 ⊃ L1 ⊃· · · ⊃ Lk, whereLi is an edge ofC of codimensioni. Denote byFlag

k(C) the

free Abelian group on Flagk(C) and byFlk(C,Z) the quotient ofFlagk(C) by the

following relations.For everyi, 0< i < k, and a flag with a gap,F = (L0 ⊃ · · · ⊃ Li−1 ⊃ Li+1 ⊃

Lk), whereLj is an edge of codimensionj , we set∑

F⊃F F = 0 in Flk(C,Z),where the summation is over all flagsF = (L0 ⊃ Lk) ∈ Flagk(C) such thatLj = Lj for all j 6= i.

To define the relation betweenAk(C,Z) andFlk(C,Z)we define the followingmap. For(H1, . . . , Hk) in the general position,Hi ∈ C, defineF(H1, . . . , Hk) =(H1 ⊃ H12⊃ · · · ⊃ H12...k) ∈ Flagk(C), whereH12...i = H1∩H2∩ · · · ∩Hi. For aflagF ∈ Flagk(C), define a functionalδF ∈ Flk(C,Z)∗ asδF (F ′) = 1 if F ′ = FandδF (F ′) = 0 otherwise. For(H1, . . . , Hk) in general position, define a map

ϕk(H1, . . . , Hk) =∑σ∈6k

(−1)|σ |δF(Hσ1,...,Hσk ). (40)

Thus we have a homomorphismϕk: Ap(C,Z) → Flp(C,Z)∗. The followingstatements are from [10]. All groupsFlp(C,Z) are free overZ. Ap(C,Z) andFlp(C,Z) are dual and the mapϕk is an isomorphism.

SetAk(C) = Ak(C,Z)⊗Z C andFlk(C) = Flk(C,Z)⊗Z C for all k.From now on we assume that the configurationC is weighted, that is, to any

hyperplaneH ∈ C its weight, a numbera(H) ∈ C, is assigned. Define the quasi-classical weight of any edgeL of C as the sum of the weights of all hyperplanesthat contain the edge.

Say that ak-tuple H = (H1, . . . , Hk), Hi ∈ C, is adjacent to a flagF if thereexistsσ ∈ 6k such thatF = F(Hσ1, . . . , Hσk). This permutationσ is unique.Denote it byσ(H , F ).

Define a symmetric bilinear formSk onFlk(C). ForF,F ′ ∈ Flagk(C), set

Sk(F, F ′) = 1

k

∑(−1)σ(H ,F )σ (H,F

′)a(H1) . . . a(Hk), (41)

where the summation is over allH = (H1, . . . , Hk) adjacent to bothF andF ′.The formSk is called thequasiclassical contravariant formof the configuration

C. It defines a bilinear symmetric form onFlk(C). See [10].

Flag Complex. Define a differential d:Flk → Flk+1 by d(L0 ⊃ · · · ⊃ Lk) =∑Lk+1(L0 ⊃ · · · ⊃ Lk ⊃ Lk+1), where the sum is taken over all edgesLk+1 of

codimensionk + 1 such thatLk ⊃ Lk+1. From the definition of the groupsFlk itfollows that d2 = 0.


A Complex(A•, d(a)). Setω = ω(a) = ∑H∈C a(H)H, ω(a) ∈ A1. Define

a differential d= d(a): Ak → Ak+1 by the rule dx = ω(a) · x. It is clear thatd2 = 0.

For anyk, the quasiclassical bilinear form onC defines a homomorphism

Sk: Flk → (F lk)∗ ' Ak, (42)

whereSk(F ) = (−1)k(k−1)/2S(F, ·).LEMMA 5.1. S• defines a map of complexesS• = S•(a): (F l•(C),d)→ (A•(C),d(a)).

Note. There is a misprint in [10] in the definition ofSk where the factor(−1)k(k−1)/2 is missing.

Proof. For any edgeL, set S(L) = ∑H∈C, L⊂H a(H)H , S(L) ∈ A1. It is

easy to see that the homomorphismSk is defined bySk(L0 ⊃ · · · ⊃ Lk) =(−1)k(k−1)/2S(L1) · S(L2) · · · S(Lk). In other words

Sk(L0 ⊃ · · · ⊃ Lk) = (−1)k(k−1)/2∑

a(H1) . . . a(Hk)(H1, . . . , Hk),

where the sum is over allk-tuples (H1, . . . , Hk) such thatHi ⊃ Li for all i.Therefore, we have

Sk+1d(L0 ⊃ · · · ⊃ Lk)= Sk+1

( ∑Lk+1,Lk+1⊂Lk

(L0 ⊃ · · · ⊃ Lk ⊃ Lk+1)

)= (−1)

(k+1)k2

∑a(H1) . . . a(Hk)a(Hk+1)(H1, . . . , Hk,Hk+1)

=((−1)

k(k−1)2

∑a(H1) . . . a(Hk)(H1, . . . , Hk)

)· (−1)k

∑a(Hk+1)Hk+1

=((−1)

k(k−1)2

∑a(H1) . . . a(Hk)(H1, . . . , Hk)

)· (−1)kω(a)

= Sk(L0 ⊃ · · · ⊃ Lk) · (−1)kω(a) = ω(a) · Sk(L0 ⊃ · · · ⊃ Lk)= d(a)Sk(L0 ⊃ · · · ⊃ Lk).

The second, third and fourth sum are over allHi, such thatHi ⊃ Li, for 16 i 6 kandHk+1∩Lk 6= 0. Note thatHk+1∩Lk = 0 implies(H1, . . . , Hk,Hk+1) = 0, andthus the fourth equality is justified. The sixth one comes from the skew symmetryin A•(C). 2

Recall that we have a weighted configuration of hyperplanes in a complexm-dimensional spaceW , anda = {a(H) | H ∈ C} are the weights. Fix an affineequationlH = 0 for each hyperplaneH ∈ C. SetY = W −⋃H∈C H . Considerthe trivial line bundleL(a) overY with an integrable connection d(a): O → �1


given by d+�(a) = d+∑H∈C a(H)d loglH , where d is the de Rham differential.Denote by�•(L(a)) the complex ofY -sections of the homomorphic de Rhamcomplex ofL(a).

To anyH ∈ C assign the one-formi(H) = d loglH ∈ �1(L(a)). This con-struction defines a monomorphismi(a): (A•(C),d(a))→ (�•(L(a)),d(a)). Theimage of this monomorphism is called thecomplex of the hypergeometric dif-ferential forms of weight a. It is denoted by(A•(C, a),d(a)). The image of thehomomorphismi(a)S: (F l•(C),d)→ (�•(L(a)),d(a)) is called thecomplex ofthe flag hypergeometric differential forms of weight a. It is denoted by(F l•(C, a),d(a)). For further details see [10].

5.2. DISCRIMINANTAL CONFIGURATIONS

Let W be an affine complex space of dimensionm. Let z1, . . . , zn be pairwisedistinct complex numbers. Denote byCm a configuration inW consisting of hyper-planesHkl: tk − tl = 0; 1 6 k < l 6 m. SoC1 = ∅, andY (Cm) is the spaceof m-tuples of ordered distinct points inC. Denote byCn;m(z) a configuration inW consisting of hyperplanesHj

k : tk − zj = 0, 1 6 k 6 m, 1 6 j 6 n, andHkl, 16 k < l 6 m. Thus,Y (Cn;m(z)) = p−1(z) wherep: Y (Cn+m)→ Y (Cn) isthe projection on the firstn coordinates. DefineC0;m = Cm.

Edges and Flags ofCn,m. For every non-empty subsetJ = {j1, . . . , jk} ⊂ [m]setLJ = Hj1j2∩Hj2j3∩· · ·∩Hjk−1jk ∈ Ck−1

n;m .L is an edge of codimensionk−1. Inparticular setLJ = W , for k = 1. Fori ∈ [n] defineLiJ = Hi

j1∩Hi

j2∩ · · · ∩Hi

jk∈

Ckn;m.LiJ is an edge of codimensionk. SetLi∅ = W . Given non-intersecting subsets

J1, . . . , Jk; I1, . . . , In ⊂ [m], defineLJ1,...,Jk;I1,...,In = (⋂kj=1LJj ) ∩ (

⋂ni=1L

iIi).

Multiplication of Flags. Given two subsetsJ ⊂ [m] and I ⊂ [n], denote byCJ ;I ⊂ Cm;n the subset consisting of all hyperplanesHj1j2 with j1, j2 ∈ J andHi

j

with j ∈ J, i ∈ I . Given subsetsJ, J ′ ⊂ [m]; I, I ′ ⊂ [n] such thatJ ∩ J ′ = ∅;I ∩ I ′ = ∅, define maps◦: Flagk(CI ;J ) × Flagl(CI ′;J ′) → Flagk+l(CI∪I ′,J∪J ′)as follows. ForF = F(H1, . . . , Hk) ∈ Flagk(CI ;J ), F = F(H ′1, . . . , H

′l ) ∈

Flagl(CI ′;J ′), setF ◦F ′ = (H1, . . . , Hk,H′1, . . . , H

′l ). The following lemma, [10],

Lemma 5.7.2, takes place.

LEMMA 5.2. The above map correctly defines the mapFlk(CI ;J )⊗Fll(CI ′;J ′)→Flk+l(CI∪I ′,J∪J ′). Moreover, for allx ∈ Flk(CI ;J ), y ∈ Fll(CI ′;J ′)we havex◦y =(−1)kly ◦ x.


5.3. TWO MAPS OF COMPLEXES

Let g be a Kac–Moody Lie algebra without Serre’s relations. LetM = M(31) ⊗· · ·⊗M(3n) be a tensor product of Verma modules with weights31, . . . ,3n ∈ h∗.Setλ = (1,1, . . . ,1︸︷︷︸

m

). In this case the number of generators(fj )r1 of n− equalsm,

i.e. r = m. Two maps of complexesψ• andη• are described in [10]:

ψp: Cp(n−,M)λ→ Flm−p,ηp = ϕ−1 ◦ (ψ∗p)−1: Cp(n−∗,M∗)λ→ Am−p, (43)

whereϕ is the map (40).

Note. The mapsψp define isomorphism of complexes. Theorem 6.6 [10] im-plies that the maps(−1)p(−1)(m−p)(m−p−1)/2ηp define isomorphism of complexes.The sign is due to Lemma 5.1 and the fact that the contravariant formS in this paperis minus the contravariant form in [10], see formula (10) and [10], formula (6.2.3).

We will recall the explicit description ofψ• under the above assumption onλ. Let g ∈ n−. A length l = l(g) of a commutatorg is given via an inductivedefinition. Setl(fj ) = 1 for j = 1, . . . ,m. If g = [g1, g2] and l1 = l(g1), l2 =l(g2), then setl(g) = l1+ l2. Sol(g) = ‘the number off ’s in g’.

To every commutatorg assign abracket signb(g) ∈ Z/2Z as follows. Setb(fj ) = 0; b([g1, g2]) = b(g1)+ b(g2)+ l(g1)mod 2.

To every commutatorg assign a flagFl(g) ∈ Fll(g)−1(C0;|g|) as follows. SetFl(fj ) = �. If g = [g2, g1], setFl(g) equal toFl(g1) ◦ Fl(g2) completed by theedgeL|g|.

Finally, for a commutatorg setF(g) = (−1)b(g)F l(g) ∈ Fll(g)−1(C0;|g|). ForI = (i1, . . . , il) ⊂ {1, . . . ,m} and 16 i 6 n, setfI = fil . . . fi1 ∈ U(n−)andF i(fI ) = F(H i

i1, . . . , H i

il) ∈ Fll(C{i};I ). Let z ∈ Cp(n−, U(n−)⊗n)λ and

z = gp ∧ gp−1 ∧ · · · g1 ⊗ fIn ⊗ fIn−1 ⊗ · · · ⊗ fI1, where allgi are commutators,li = l(gi). Let {fi1, . . . , fim} be the list offi ’s in z read from right to left. Defineσ(z) ∈ 6m by σ(z)(j) = ij . Set

ψp(z)

= (−1)|σ(z)|+∑pi=1(i−1)(li−1)F 1(fI1) ◦ · · · ◦ Fn(fIn) ◦ F(g1) ◦ · · · ◦ F(gp). (44)

Note. There is a correction of the sign in the definition ofψ compared with[10].

EXAMPLES. Letn = 1.ψ(fm . . . f1) = F(H 11 , . . . , H

1m), andη((fσ1 . . . fσm)

∗) =(−1)|σ |Hσ1,σ2 ◦ · · · ◦ Hσm−1,σm ◦ H 1

σm. Compose the inclusion mapi(a):

(A•(C),d(a))→ (�•(L(a)),d(a)) (see Section 5.1) with the mapη to get

i(a) ◦ η((fσ1 . . . fσm)∗)

= (−1)|σ |d ln(tσ1 − tσ2) ∧ · · · ∧ d ln(tσm−1 − tσm) ∧ d ln(tσm − z1).


Let I ∈ P(λ, n) andI = (i11, . . . , i1s1, . . . , in1, . . . , insn). Sinceλ = (1,1, . . . ,1),I ∈ 6m. Let Ij = (i

jsj , . . . , i

j

1) for 1 6 j 6 n. We havei(a) ◦ η((fIn)∗ ⊗· · · ⊗ (fI1)∗) = (−1)|I |ωI . Thereforei(a) ◦ η(∑I∈P(λ,n)(fI )

∗fI ) = ω(z, t), seeSection 3.

Let I = (i11, . . . , i1s1, . . . , in1, . . . , insn) ∈ P(λ, n), and 16 k 6 sj . Define

fI ;ijk = fIn ⊗ · · · ⊗ fIj+1 ⊗ fijk+1

. . . fijsj

⊗ fIj−1 ⊗ · · · ⊗ fI1,θI ;ijk = ωi11,...,i

1s1∧ · · · ∧ ω

ij−11 ,...,i

j−1sj−1∧ ω

ijk+1,...,is

jj

∧ ωij+11 ,...,i

j+1sj+1∧ · · ·

∧ωin1 ,...,insn ∧ [d ln(tij1− t

ij2) ∧ · · · ∧ d ln(t

ijk−1− t

ijk)], (45)

HI ;ijk = H 1

i11,...,i1s1◦ · · · ◦Hj−1

ij−11 ,...,i

j−1sj−1

◦Hj

ij

k+1,...,isjj

◦Hj+1

ij+11 ,...,i

j+1sj+1

◦ · · ·

◦Hnin1 ,...,i

nsn◦ [H

ij1 ,i

j2◦ · · · ◦H

ijk−1,i

jk],

whereHp

i1,...,il= Hi1,i2 ◦ · · · ◦Hil−1,il ◦Hp

il.

LEMMA 5.3. Let I ∈ P(λ, n). Letεjk = k((sj − k)+ sj+1+ · · · + sn). Then

i(a) ◦ η(1σij1,...,σ

ijk

⊗ (fI ;ijk )

∗) = (−1)|I |+εjk θI ;ijk . (46)

Proof.The statement of the lemma is equivalent to the equation:

1σij1,...,σ

ijk

⊗ (fI ;ijk )

∗ = (−1)|I |+εjk η−1(H

I,ijk). (47)

It is sufficient to compute the two sides on elements of typeg ⊗ fI,i

jk

whereg is acommutator of lengthk on f

ij

1, . . . , f

ijk. Let σ ∈ 6k andf

ijσ1, . . . , f

ijσk

be the list

of fij ’s enteringg from right to left. The left hand side and the right-hand side of(47) evaluated ong ⊗ f

I,ijk

give

1σij1,...,σ

ijk

⊗ (fI ;ijk )

∗(g ⊗ fI ;ijk ) = 1σ

ij1,...,σ

ijk

(g),

(−1)|I |+εjk η−1(H

I,ijk)(g ⊗ f

I ;ijk ) = (−1)|I |+εjk ϕ(H

I,ijk)(ψ(g ⊗ f

I ;ijk )), (48)

respectively. See formula (43). Use the definition ofψ to obtain

ψ(g ⊗ fI ;ijk ) = (−1)|τ |F(H 1

i1s1, . . . , H 1

i11) ◦ · · · ◦ F(Hj

i1sj, . . . , H

j

ijk+1

) ◦ · · ·◦F(Hn

insn, . . . , Hn

in1) ◦ F(g),

whereτ=(

1 . . . . . . . . . . . . m

i1s1 · · · i11 · · · ijsj · · · i

j

k+1 · · · insn · · · in1 ijσ1· · · ijσk

).

I as an element of6m has the form

I =(

1 . . . . . . . . . m

i11 · · · i1s1 · · · ij

1 · · · ijsj · · · in1 · · · insn

).


Thus

|I | = (|τ | + Sjk + k((sj − k)+ sj+1 + · · · + sn)+ |σ |)mod 2, (49)

where

Sj

k =n∑

l=1, l 6=j

sl(sl − 1)

2+ (sj − k)(sj − k − 1)

2.

Note thatHp

i1,...,il= (−1)l(l−1)/2H

p

il◦ Hil−1,il ◦ · · · ◦Hi1,i2. Use the definition ofϕ,

(40), to compute

ϕ(Hp

i1,...,il) = (−1)l(l−1)/2ϕ(H

p

il◦Hil−1,il ◦ · · · ◦Hi1,i2)

= (−1)l(l−1)/2δF(Hpil,...,H

pi1) + otherδ-summands. (50)

ϕ(HI,i

jk) = (−1)S

jk ϕ(H 1

i1s1◦Hi1s1−1,i

1s1◦ · · · ◦Hi11,i12 ◦ · · ·

◦Hj

ijsj

◦Hij

sj−1,ijsj

◦ · · · ◦Hij

k+1,ij

k+2◦ · · · ◦Hn

insn◦Hinsn−1,i

nsn◦ · · ·

◦Hin1 ,in2 ◦ [Hij1 ,ij2 ◦ · · · ◦Hijk−1,ijk]). (51)

We use formulae (49), (50), (51) to simplify (48).

(−1)|I |+εjk η−1(H

I,ijk)(g ⊗ f

I ;ijk )

= (−1)|σ |ϕ(Hij

1,ij

2◦ · · · ◦H

ij

k−1,ijk)(F (g)). (52)

The proof of Lemma 5.3 is finished modulo the following result. 2LEMMA 5.4. Let ηI, ψI be the combinatorial maps(43), (44)defined on the setof distinct indicesI, I = {i1, . . . , ik} ⊂ {1, . . . ,m}. ThenηI(1i1,...,ik ) = Hi1,i2 ◦· · · ◦Hik−1,ik , for anyk = 2, . . . ,m.

Proof.Induction byk. Fork = 2,g = [fi1, fi2] forms a base of the commutatorsof length 2 on fi1 and fi2. 1i1,i2(g) = 1. Since b(g) = 1 andF(g) = (−1)b(g)F (Hi1,i2) and σ = ( 1 2

2 1

), we haveη−1(Hi1,i2)([fi1, fi2]) =

ϕ(Hi1,i2)(ψ(g)) = δF(Hi1,i2)((−1)|σ |+b(g)F (Hi1,i2)) = 1.Let 2< k 6 m. Assume that for anyj , 26 j < k, and 16 s1 < · · · < sj 6 k

we haveη(1is1,...,isj) = His1,is2 ◦ · · · ◦Hisj−1 ,isj

. Let g be a commutator of lengthkonfi1, . . . , fik . Theng = [g1, g2] with l(g1) = l1, andl(g2) = l2, andl1+ l2 = k.Let σ ∈ 6k be such thatfiσ1

, . . . , fiσk is the list offi ’s in g read from right to left.In order to evaluateη−1(Hi1,i2 ◦ · · · ◦Hik−1,ik )(g) = ϕ(Hi1,i2 ◦ · · · ◦Hik−1,ik )(ψ(g))

remark that

ψ(g) = (−1)|σ |+b(g)(F l(g2) ◦ Fl(g1), L|g|)


= (−1)|σ |+l(g1)(F (g2) ◦ F(g1), Li1,...,ik ),

η−1(Hi1,i2 ◦ · · · ◦Hik−1,ik )(ψ(g))

=∑τ∈6k−1

(−1)|τ |δF(Hiτ1 ,iτ1+1,...,Hiτk−1 ,iτk−1+1)(ψ(g)). (53)

Since a link corresponding to a hyperplaneHj,j+1 connects only neighbouringindices in a flag of a typeF(Hiτ1,iτ1+1, . . . , Hiτk−1,iτk−1+1

) we have

F(Hiτ1,iτ1+1, . . . , Hiτk−1,iτk−1+1)

= (�, . . . , (ti1 = · · · = tiτk−1; tiτk−1+1 = · · · = tik ),

Li1,...,ik = (ti1 = · · · = tik )). (54)

Let δF(Hiτ1 ,iτ1+1,...,Hiτk− ,iτk−1+1)(ψ(g)) 6= 0. Since(−1)|σ |+b(g)ψ(g) = (F l(g2) ◦

Fl(g1), L|g|) = (�, . . . , L|g2| ∩ L|g1|, L|g|) formula (53) implies eitherL|g1| =(ti1 = · · · = tiτk−1

); L|g2| = (tiτk−1+1 = · · · = tik ), or L|g2| = (ti1 = · · · = tiτk−1);

L|g1| = (tiτk−1+1 = · · · = tik ). Without loss of generality we will assume that thesecond case takes place, i.e.L|g2| = (ti1 = ti2 = · · · = tiτk−1

); L|g2| = (tiτk−1+1 =· · · = tik−1 = tik ). Compare the lengths of the flags to conclude thatτk−1 = l2. Inorder to have non-zero multiples in the product

δF(Hiτ1,iτ1+1,...,Hiτk−1 ,iτk−1+1)(F l(g2) ◦ Fl(g1), L|g|)

= δF(Hiτ1,iτ1+1,...,Hiτl2−1 ,iτl2−1+1)(F l(g2))δF(Hiτl2 ,iτl2+1,...,Hiτk−2 ,iτk−2+1)

(F l(g1)) (55)

we need(τ1, . . . , τl2−1) to be a permutation of the set(1, . . . , l2− 1) and(τl2, . . .,τk−2) to be a permutation of the set(l2 + 1, . . . , k − 1). Setτ ′ = ( 1 · · · l2 − 1

τ1 · · · τl2−1

)andτ ′′ = ( 1 · · · l1− 1

τl2 − l2 · · · τk−2 − l2). Then(−1)|τ | = (−1)|τ ′|+|τ ′′|+l1−1, b(g) + l1 =

b(g1)+ b(g2) mod 2, and

η−1(Hi1,i2 ◦ · · · ◦Hik−1,ik )(ψ(g))

= (−1)|σ |+b(g)+l1−1∑

τ ′∈6l2−1

(−1)|τ′|δF(Hi

τ ′1,iτ ′1+1

,...,Hiτ ′l2−1

,iτ ′l2−1+1

)(F l(g2))×

×∑

τ ′′∈6l1−1

(−1)|τ′′|δF(Hi

τ ′′1+l2,iτ ′′1+l2+1

,...,Hiτ ′′l1−1+l2

,iτ ′′l1−1+l2+1

)(F l(g1))

= (−1)|σ |−1η−1(Hi1,i2 ◦ · · · ◦Hil2−1,il2)(F (g2))×

×η−1(Hil2+1,il2+2 ◦ · · · ◦Hik−1,ik )(F (g1))

= (−1)η−1(Hi1,i2 ◦ · · · ◦Hil2−1,il2)(ψ(g2))×

×η−1(Hil2+1,il2+2 ◦ · · · ◦Hik−1,ik )(ψ(g1)). (56)

The last equality holds becauseσ = ( 1 · · · k

σ1 · · · σk

) = ( 1 · · · l2σ1 · · · σl2

)×( l2+ 1 · · · lkσl2+1 · · · σlk

).

Using the inductive hypothesis rewrite (56) as

η−1(Hi1,i2 ◦ · · · ◦Hik−1,ik )(ψ(g)) = (−1)1i1,...,il2(g2)1il2+1,...,ik (g1)

= 1i1,...,ik ([g1, g2]). 2 (57)


6. Derivation of the Dynamical Differential Equation

In this sectiong will be a Kac–Mody Lie algebra without Serre’s relations,λ =(1,1, . . . ,1), r = m. We will work in a weight spaceMλ of the moduleM =M(31) ⊗ · · · ⊗ M(3n). We will differentiate the hypergeometric formω(z, t),express the result in terms of the complexC•(n−∗,M∗), and derive the Dynamicaldifferential equation in the form (25).

The integrand of a hypergeometric solution have the following form, see Sec-tion 3.

81/κµ ω = exp

(1

κ

(−

m∑i=1

〈αc(i), µ〉ti +n∑j=1

〈3j,µ〉zj))81/κ,

where

8(z, t) =∏i<j

(zi − zj )(3i,3j )∏k,j

(tk − zj)−(αc(k),3j )∏k<l

(tk − tl )(αc(k),αc(l)).

Fix µ′ ∈ h and let∂µ′ be the partial derivative with respect to the parameterµ inthe direction ofµ′. Then

κ∂µ′(81/κµ ω)

=(−

m∑i=1

〈αc(i), µ′〉ti +n∑j=1

〈3j,µ′〉zj

)81/κµ

∑I∈P(λ,n)

(−1)|I |ωIfIv. (58)

Let I = (i11, . . . , i1s1; . . . ; in1, . . . , insn) ∈ P(λ, n). MoreoverI ∈ 6m because of theform of λ. Sincer = m we havec(i) = i. Sett

ij

sj+1= zj andαik;j = αijk + αijk−1

+· · · + α

ij1. Rearrange the following expression:(−

m∑i=1

〈αi, µ′〉ti +n∑j=1

〈3j,µ′〉zj

)ωI

=(−

n∑j=1

sj∑k=1

〈αijk, µ′〉t

ijk+

n∑j=1

〈3j,µ′〉zj

)ωI

=(−

n∑j=1

sj∑k=1

〈αijk, µ′〉(t

ijk− t

ijk+1+ t

ijk+1− t

ijk+2+ · · · + t

ijsj

− zj + zj )+

+n∑j=1

〈3j,µ′〉zj

)ωI

=(−

n∑j=1

sj∑k=1

〈αik;j , µ′〉(tijk − tijk+1)

)ωI+


+(

n∑j=1

zj 〈3j − αij1 − · · · − αijsj , µ′〉)ωI . (59)

LEMMA 6.1. Letu(µ, z) = ∫γ (z)

81/κµ ω. Then

κ∂µ′u−n∑j=1

zjµ′(j)u

=∑

I∈P(λ,n)(−1)|I |

∫γ (z)

81/κµ

(−

n∑j=1

sj∑k=1

〈αik;j , µ′〉(tijk − tijk+1)

)ωIfIv. (60)

Proof.Combine formulae (58), (59) with the factµ′(j)fI v = 〈3j − αij1 − · · · −αijsj

, µ′〉fIv to obtain the result. 2LEMMA 6.2. Define an operatorL byLu = κ∂µ′u−∑n

j=1 zjµ′(j)u. Then

Lu =∑α>0

〈α,µ′〉〈α,µ〉1+,αu.

Proof.Let I ∈ P(λ, n) and 16 k 6 sj .(tijk− t

ij

k+1)ωI

= ωi11,...,i1s1 ∧ · · · ∧ (tijk − tijk−1)ω

ij1 ,...,i

jsj

∧ · · · ∧ ωin1 ,...,insn= ωi11,...,i1s1 ∧ · · · ∧ [d ln(t

ij

1− t

ij

2) ∧ · · · ∧ d ln(t

ij

k−1− t

ijk) ∧ d(t

ijk− t

ij

k+1)∧

∧d ln(tijk+1− t

ijk+2) ∧ · · · ∧ d ln(t

ijsj

− zj )] ∧ · · · ∧ ωin1 ,...,insn= (−1)ε

jk ωi11,...,i

1s1∧ · · · ∧ ω

ijk+1,...,i

jsj

∧ · · · ∧ ωin1 ,...,insn∧∧ [d ln(t

ij

1− t

ij

2) ∧ · · · ∧ d ln(t

ij

k−1− t

ijk)] ∧ d(t

ijk− t

ij

k+1)

= (−1)εjk θI ;ijk ∧ d(t

ijk− t

ijk+1) = (−1)ε

jk+(m−1)d(t

ijk− t

ijk+1) ∧ θ

I ;ijk , (61)

whereεjk = k(sj+1+ · · · + sn + sj − k). It is clear that dθI ;ijk = 0. Thus

κdt (81/κµ θ

I ;ijk ) =(−

m∑i=1

〈αi, µ〉dti +8−1(dt8)

)81/κµ θ

I ;ijk . (62)

Rearrange as in formula (59) and simplify to get(n∑j=1

sj∑k=1

〈αil;j , µ〉d(tijl − tijl+1)

)∧ θ

I ;ijk

= (8−1(dt8)) ∧ θI ;ijk − κ8−1/κµ dt (8

1/κµ θ

I ;ijk ),〈αik;j , µ〉d(tijk − tijk+1)) ∧ θ

I ;ijk= (8−1(dt8)) ∧ θI ;ijk − κ8

−1/κµ dt (8

1/κµ θ

I ;ijk ). (63)


Sinceγ (z) is a cycle, formulae (61), (63) and Lemma 5.3 allow us to rewriteLu as

Lu =∑

I∈P(λ,n)(−1)|I |

∫γ (z)

(−1)εjk ×

×(−

n∑j=1

sj∑k=1

〈αik;j , µ′〉〈αik;j , µ〉

(81/κµ 8−1(dt8)) ∧ θI ;ijk − κdt (8

1/κµ θ

I ;ijk )

)

=∑

I∈P(λ,n)

∫γ (z)

81/κµ

(−

n∑j=1

sj∑k=1


(−1)m−1 dt8

8∧

∧ (i(a) ◦ η(1ij1 ,...,i

jk⊗ (f

I,ijk)∗)

). (64)

SinceθI,i

jk

is closed, its differential in the complex of the hypergeometric differen-tial forms reduces to multiplication byi(a)(�(a) = dt8/8. Taking into accountformula (43) and the note after it we have

Lu =∑

I∈P(λ,n)fI

∫γ (z)

81/κµ

(−

n∑j=1

sj∑k=1


(−1)m−1d(a)×

×(i(a) ◦ η(1ij1 ,...,i

jk⊗ (f

I,ijk)∗)

)

=∑

I∈P(λ,n)fI

∫γ (z)

81/κµ

(n∑j=1

sj∑k=1


(i(a) ◦ η(d1ij1 ,...,i

jk⊗ (f

I,ijk)∗)

)

=∑

I∈P(λ,n)fI

∫γ (z)

81/κµ

(n∑j=1

sj∑k=1


(i(a) ◦ η(1ij1 ,...,i

jk(fI,i

jk)∗)

)

=∑

I∈P(λ,n)fI

∫γ (z)

81/κµ

(n∑j=1

sj∑k=1


×

×(i(a) ◦ η(1ij

1 ,...,ijkeijk. . . e

ij

1(fI )

∗)

). (65)

Note thatαik;j , k = 1, . . . , sj describe allλ-admissible roots such that1(j)+,α(fI )∗ 6=

0, see Section 4.4. Therefore

Lu =∑

I∈P(λ,n)fI

∫γ (z)

81/κµ

(i(a) ◦ η

(∑α>0

〈α,µ′〉〈α,µ〉 ×

×∑

(i1,...,im′ )∈P(λα,1)1i1,...,im′ eim′ . . . ei1(fI )

∗))


=∑

I∈P(λ,n)fI

∫γ (z)

81/κµ

(i(a) ◦ η

(∑α>0

〈α,µ′〉〈α,µ〉 −1−,α(fI )

∗))

=∑

K,I∈P(λ,n)fI

∫γ (z)

81/κµ

(i(a) ◦ η

(∑α>0

〈α,µ′〉〈α,µ〉〈−1−,α(fI )

∗), fK〉(fK)∗

)

=∑α>0

〈α,µ′〉〈α,µ〉

∑K,I∈P(λ,n)

〈−1−,α(fI )∗, fK〉uKfI , (66)

Lu =∑α>0

〈α,µ′〉〈α,µ〉〈−1−,α(fI )

∗, u〉fI =∑α>0

〈α,µ′〉〈α,µ〉1+,αu. 2 (67)

The statement of Lemma 6.2 is equivalent to the Dynamical differential equationin the direction ofµ′ for a functionu with values in the(1,1, . . . ,1) weight spaceof ag moduleM. The Symmetrization Lemma 4.11 deduces the general case fromthis one.

7. Main Theorems

In this section we conclude the proofs of the theorems from Section 3 in the set-ting of Kac–Moody Lie algebras without Serre’s relations. Then we deduce thecorresponding results for any simple Lie algebra.

Let g be a Kac–Moody Lie algebra without Serre’s relations. Letλ ∈ Nr . LetM = M(31) ⊗ · · · ⊗ M(3n) be a tensor product of Verma modules forg withhighest weights31, . . . ,3n ∈ h∗. Let u(µ, z) = ∑

I∈P(λ,n) uIfI be a hypergeo-metric integral with values in the weight spaceMλ as described in Section 3, i.e.uI =

∫γ (z)

81/κµ (

∑σ∈6(I)(−1)|σ |ωI,σ ).

THEOREM 7.1. The functionu(µ, z) solves the KZ equations(12) in Mλ.Proof. The proof given in Section 3 holds, because all relations we used are

proved in [10] in the general setting described above. 2THEOREM 7.2. The functionu(µ, z) solves the dynamical differential equations(13) in Mλ.

Proof. Lemma 4.11 reduces the case of a general weight spaceMλ to the caseof a weight spaceMλ, whereλ = (1,1, . . . ,1︸︷︷︸

m

). Lemma 6.2 derives the theorem in

that case. 2Proof of Theorem 3.2.Combine Corollary 4.4 and Theorem 7.2 to derive the

dynamical differential equations for any Kac–Moody Lie algebra. In particular wehave it for a simple Lie algebra. 2


Finally we will prove a determinant formula which establishes a basis of solu-tions for the system of KZ and dynamical differential equations in a weight spaceMλ. From that formula we will derive the compatibility of the system of KZ andDynamical differential equations.

Fix λ ∈ Nr . Fix a basis(fIv)I∈P(λ,n) of the weight spaceMλ. Assume that a set(γI (z))I∈P(λ,n) of horizontal families of twisted cycles in{z} × Cm is given.

DenoteuIJ =∫γI (z)

81/κµ (

∑σ∈S(J )(−1)|σ |ωJ,σ ).

PROPOSITION 7.3.Let δα = trMλ(1+,α) for a positive rootα of g. Denoteεij =

trMλ(�ij,+). Then we have

(a) For any horizontal families of twisted cycles(γI (z))I∈P(λ,n) in {z} ×Cm, thereexists a constantC = C(31, . . . ,3n, λ, κ) such that

det(uIJ ) = C exp

(n∑i=1

zi

κtrMλ

(µ(i))

)∏α>0

〈α,µ〉δα/κ∏i<j

(zi − zj )εij /κ . (68)

In the first product only finite number of factors are different from1, i.e.δα 6= 0if and only if0< α 6 λ.

(b) For generic values of the parameters(3j )mj=1, (αi)

ri=1, κ in a neighbourhood

of a generic point(µ, z) ∈ h × Cn we can choose cycles(γI (z))I∈P(λ,n) suchthat the constantC from (a) is non-zero. Moreover the set of functions{uI =∑

J∈P(λ,n) uIJ fJ }I∈P(λ,n) form a fundamental system of solutions for the systemof KZ and dynamical differential equations.

Proof. Part (a) is a corollary of Theorems 7.1 and 7.2. We will prove part (b)for values of the parameters such that all numbers(αi, αj )/κ, −(αi,3k)/κ havepositive real parts for 16 i, j 6 r, 16 k 6 n, and for a point(µ, z) suchz ∈ Rn,z1 < z2 < · · · < zn and〈αi, µ〉/κ > 0 for anyi = 1, . . . , r. For generic values of(3j)

mj=1, (αi)

ri=1, z, µ, κ (b) holds by analytic continuation.

The caseλ = (1,1, . . . ,1︸︷︷︸m

). Setf0(t) =∑mj=1〈αcλ(j), µ〉tj . LetI = (i11, . . . , i1s1;

. . . ; in1 , . . . , insn) ∈ P(λ, n). SetγI (z) = {t ∈ Rn : zj < tij1< · · · < t

ijsj

<

zj+1 for all j = 1, . . . , n}, wherezn+1 = ∞. Note that{γI (z)}I∈P(λ,n) is the set ofall domains for the configuration of hyperplanesHij : ti− tj = 0,Hk

i : ti−zk = 0,1 6 i < j 6 m, 1 6 k 6 n which are either bounded, or the limit off0 onthem is+∞ when‖t‖ →∞. In [13] a linearly independent set of hypergeometricdifferentialn-forms, calledβnbc differentialn-forms, associated to those domainsis defined. An explicit non-vanishing formula for the corresponding determinant isgiven in [8], Theorem 6.2, see also [6]. Sinceλ = (1,1, . . . ,1) the space of hyper-geometricn-forms is isomorphic to the spaceC0(n−∗,M∗)λ, see Section 5.3. Thelatter has basis(fIv∗)I∈P(λ,n) which gives the basis(ωI = i(a) ◦ η(fIv∗))I∈P(λ,n)of the space of hypergeometricn-forms. Since this basis and theβnbc set have the


same cardinality the non-zero determinant formula for the integrals ofβnbc formsover the domains{γI (z)}I∈P(λ,n) implies a non-zero determinant formula for theintegrals of(ωI )I∈P(λ,n) over the same domains. Since the determinant is non-zeroat one point(µ, z), it is non-zero at any point(µ, z) under the above conditions onthe parameters.

The case of genericλ ∈ Nr . ConsiderC0(n−∗,M∗)λ andC0(n−, M∗)(1,1,...,1) asin Section 4.4. A basis for the6λ-symmetric hypergeometric differential forms isgiven by(ωI )I∈P(λ,n), whereωI =∑J∈6(I) ωJ andωJ = i(a) ◦ η(f ∗J ).

P((1,1, . . . ,1︸︷︷︸m

), n) =⋃

I∈P(λ,n)S(I )

a disjoint union. Thus the set(ωI )I∈P(λ,n) consists of linearly independent formsin the space of all hypergeometric forms. The integral pairing described in theprevious case is non-degenerate. Therefore there there exists a subset of the set(γJ (z))J∈P((1,1,...,1),n) indexed by the setP(λ, n) such that the corresponding deter-minant is non-zero. 2COROLLARY 7.4. The system consisting of the union of KZ and Dynamic differ-ential equations for any Kac–Moody Lie algebra with(or without) Serre’s relationsis a compatible system of differential equations.

Remark.An algebraic proof of the compatibility of the system of KZ equationsis given in [10].

Proof.Let us write the differential operators which determine the KZ equations(12) and the dynamical equations (13) in the form

KZ:∂

∂zj+ Bj, Dynamical:

∂

∂µ′+ Cµ′,

wherej = 1, . . . , n andµ′ ∈ h. (69)

The operatorsBj andCµ′ are linear for anyj = 1, . . . , n, µ′ ∈ h. In order toprove the compatibility of the system of KZ and Dynamical differential equationswe need to check

[ ∂∂zj+ Bj, ∂

∂zk+ Bk] = 0,

[ ∂∂zj+ Bj, ∂

∂µ′+ Cµ′ ] = 0,

and

[ ∂∂µ′+ Cµ′, ∂

∂µ′′+ Cµ′′ ] = 0.


First consider the case of a Kac–Moody Lie algebra without Serre’s relations,g, acting on a tensor product of highest weight modulesM. We have[

∂

∂zj+ Bj, ∂

∂µ′+ Cµ′

]=(∂

∂zjCµ′

)−(∂

∂µ′Bj

)+ [Bj ,Cµ′ ]. (70)

The result is a linear operator with meromorphic coefficients depending on para-meters{z, µ, (αi)ri=1, (3j)

nj=1, κ}. Analogously, the commutators[

∂

∂zj+ Bj, ∂

∂zk+ Bk

]and

[∂

∂µ′+ Cµ′, ∂

∂µ′′+ Cµ′′

]are linear operators with meromorphic coefficients depending on the above set ofparameters, wherej, k = 1, . . . , n, µ′, µ′′ ∈ h.

It is enough to show the commutativity of the above operators for generic val-ues of the parameters. Then the commutators will be zero for any values of theparameters by analytic continuation.

Take such parameters{z, µ, (αi)ri=1, (3j )nj=1, κ} that the set of hypergeometric

solutions of the system of KZ and dynamical differential equations forms a basisofM. According to Proposition 7.3(b) this is a generic choice of parameters. Sincethe KZ and the Dynamical differential operators act as zero on the set of hypergeo-metric solutions, their commutators also act as zero on the same set. Therefore thecommutators act as zero on theg-moduleM.

Finally, consider a Kac–Moody Lie algebra with Serre’s relationsg =g/ ker(S: g→ g∗) which acts onL = M/ ker(S: M → M∗). Corollary 4.4 and[10], Corollary 7.2.11 show that the Dynamical and the KZ operators forg corre-spond to the the the Dynamical and the KZ operators forg under this factorization.Then the commutativity of the operators ong implies that they are commutative ong as well. 2

References

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2. Chalykh, O. A., Feigin, M. V. and Veselov, A. P.: New integrable generalizations of Calogero–Moser quantum problem,J. Math. Phys.39 (1998), 695–703.

3. Chalykh, O. A. and Veselov, A. P.: Commutative rings of partial differential operators and Liealgebras,Comm. Math. Phys.126(1990), 597–611.

4. Drinfeld, V.: Quantum groups, in:Proc. ICM (Berkeley, 1986), Vol. 1, Amer. Math. Soc.,Providence, RI, 1987, pp. 798–820.

5. Duistermaat, J. J. and Grünbaum, F. A.: Differential operators in the spectral parameter,Comm.Math. Phys.103(1986), 177–240.

6. Douai, A. and Terao, H.: The determinant of a hypergeometric period matrix,Invent. Math.128(1997), 417–436.


7. Harnad, J. and Kasman A. (eds.):The Bispectral Problem(Montréal, 1997), CRM Proc. LectureNotes 14, Amer. Math. Soc., Providence, RI, 1998.

8. Markov, Y., Tarasov, V. and Varchenko, A.: The determinant of a hypergeometric period matrix,Houston J. Math.24(2) (1998), 197–219.

9. Orlik, P. and Solomon, L.: Combinatorics and topology of complements of hyperplanes,Invent.Math.56 (1980), 167–189.

10. Schechtman, V. and Varchenko, A.: Arrangements of hyperplanes and Lie algebra homology,Invent. Math.106(1991), 139–194.

11. Varchenko, A.:Multidimensional Hypergeometric Functions and Representation Theory of LieAlgebras and Quantum groups, Adv. Ser. Math. Phys. 21, World Scientific, Singapore, 1995.

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179

A Riemann–Hilbert Problem for Propagation ofElectromagnetic Waves in an Inhomogeneous,Dispersive�Waveguide

DMITRY SHEPELSKYMathematical Division, Institute for Low Temperature Physics, 47 Lenin Avenue, 310164, Kharkov,Ukraine. e-mail: [email protected]

(Received: 10 January 2000; accepted: 30 May 2000)

Abstract. We consider the inverse scattering problem for a model of electromagnetic wave propaga-tion in a rectangular waveguide filled with dispersive�material. The waveguide is inhomogeneous inthe longitudinal direction but homogeneous in the transverse directions. Dispersive properties of thematerial are described by a single-resonance Lorentz model. By reformulating the scattering problemin the frequency domain as a Riemann–Hilbert problem, we prove that the constitutive parameters ofthe inhomogeneous waveguide are reconstructed uniquely from the scattering data.

Mathematics Subject Classifications (2000):34L25, 34A55, 78A50.

Key words: inverse problem, scattering, waveguide.

1. Introduction

Recently, increasing attention has been paid to wave propagation, scattering, andguidance in complex media, such as bi-isotropic materials [7, 8, 15], uniaxial bian-isotropic chiral materials [11, 23], chiral-omega materials [12, 16, 22], etc. (for theclassification of bi-anisotropic media, see, e.g., [21]). Due to the advancement ofmaterial sciences, it is possible now to manufacture these materials by putting smallmetal elements in a host dielectric medium. These materials possess additionaldegrees of freedom (i.e. in the form of additional parameters in the constitutiverelations), which may be used to provide solutions to current engineering problems.

A class of new complex materials, called� (omega) media, was introduced in[16]. The microstructure of these materials consists of small metal elements in theshape of an� (a half-loop with two extended arms) embedded in a dielectric. Pos-sible applications for�materials are microwave phase-shifters, scanning antennae,perfectly matched layers, antireflection coatings (see, e.g., [12, 14, 16, 22, 20]).

Transient electromagnetic wave propagation in waveguides filled with homo-geneous as well as inhomogeneous materials has been studied extensively in thepast few years, see, e.g., [2, 6, 10, 19]. The inverse problem of determining the

180 DMITRY SHEPELSKY

constitutive parameters of waveguides filled with homogeneous� materials wasaddressed in [9] and [13].

In the present paper, the propagation of transient electromagnetic waves ina metallic rectangular waveguide filled with inhomogeneous� material is con-sidered. The waveguide is placed along thez direction, its inner dimensions are0 6 x 6 a and 06 y 6 b, and the� material inhomogeneous in the longitudinaldirectionz occupies the region 06 z 6 L.

It is assumed that all the loops of the� inclusions have their extended armsparallel to each other (along the directiony) and all the normals to the plane of theloops are also parallel (to the directionx). The constitutive relations for a generalbi-anisotropic medium are (cf. [21])

D = εE+ ξH, B = ζE+ µH, (1)

whereε andµ are the permittivity and the permeability tensors, respectively, andξ , ζ are tensors describing the crosscoupling between the electric and magneticfields. In the case of the� material considered in the present paper, these tensorshave the following forms (cf. [13, 9]):

ε =(εh 0 00 εd 00 0 εh

), µ =

(µd 0 00 µh 00 0 µh

),

ξ =( 0 0 0−i� 0 0

0 0 0

), ζ =

(0 i� 00 0 00 0 0

). (2)

Since�materials are highly dispersive [9] (the resonance character of the para-meters is due to that of a single� particle), the frequency dependence of materialparameters must be modelled whenever the analysis is expected to be performedover a broad frequency band. It is assumed that the resonant components,εd , µd ,and�, are inhomogeneous, whereas the non-resonant components,εh andµh, areconstants (and known when considering the inverse problem). The dispersion inthe frequency domain (the time dependence of all fields is expiωt) is described bya single-resonance Lorentz model [18] with a fixed resonance frequency,ω0, butvarying (inz) amplitudes:

εd(z, ω) = εh + εa(z)

ω20 − ω2

,

µd(z, ω) = µh + µa(z)ω2

ω20 − ω2

,

�(z, ω) = �a(z)ω

ω20 − ω2

. (3)

The homogeneous parts of the waveguide,z < 0 andz > L, are assumed to bevacuum regions, withε = ε0I , µ = µ0I , and� = 0, whereI is the unitary matrix,andε0 andµ0 are the vacuum permittivity and permeability, respectively.

A RIEMANN–HILBERT PROBLEM FOR PROPAGATION OF ELECTROMAGNETIC WAVES 181

We consider the scattering matrix consisting of the reflection and transmissioncoefficients forTEm0 modes propagating in the waveguide. We are particularlyinterested in the inverse problem, when the scattering matrix (as a function of thefrequencyω) is given, and the material parametersεa, µa, and�a (as functions ofz, z ∈ [0, L]) are to be reconstructed.

Our approach to the scattering problem is based on its reformulation as aRiemann–Hilbert problem relative to a contour in theω plane, where the jump ma-trix across the contour is constructed from the scattering matrix. Such an approachfor studying inverse scattering problems is applied in [3] and [4] for stratifiedbi-isotropic non-dispersive materials. The inverse problem for a dispersive chiralslab is studied in [5]. In the present paper, this approach allows us to prove thatthe dispersive parameters of the inhomogeneous� waveguide,εa(z), µa(z), and�a(z), are determined uniquely by the scattering matrix relative to the basic modein the waveguide,TE10.

2. The Scattering Problem

We assume thatεa(z), µa(z), and�a(z) are positive absolutely continuous func-tions on the intervalz ∈ [0, L], µh > µa(z), andεa(z)µa(z) > �2

a(z).Since the principal axes of the medium coincide with the principal directions of

the rectangular waveguide, Maxwell’s equations

∇ × H = iωD, ∇ × E = −iωB,

considered together with the constitutive relations (1) allow the propagation ofTEm0 andT E0n modes,m,n = 0,1, . . .. Particularly, for theTEm0 modes, with∂/∂y = 0, one obtains the following equations forE2,H1, andH3:

∂E2

∂x= −iωµhH3,

∂E2

∂z= iωµdH1− ω�E2,

∂H1

∂z− ∂H3

∂x= iωεdE2+ ω�H1. (4)

The boundary conditions for theT Em0 modes are

E2

∣∣x=0,a = H1

∣∣x=0,a =

∂H3

∂x

∣∣∣∣x=0,a

= 0. (5)

In the vacuum regions of the waveguide,z < 0 andz > L, the fields of theTEm0

modes are

E2(x, z, ω) = Clm sinλmxe±γmz,

H1(x, z, ω) = ±Clmγm

iωµ0sinλmxe±γmz,

H3(x, z, ω) = −Clmλm

iωµ0cosλmxe±γmz, (6)


wherel = 0, L,

λm = πm

a, γm(ω) = i

√ω2ε0µ0− λ2

m, m = 0,1,2, . . . .

The continuity of the tangential fieldsE2 andH1 at z = 0 andL implies, for−∞ < z <∞,

E2(x, z, ω) = sinλmx · e(z, ω),H1(x, z, ω) = sinλmx · h(z, ω),

wheree(z, ω) andh(z, ω) satisfy the system of differential equations

dU

dz= W(z,ω)U, −∞ < z <∞. (7)

HereU(z, ω) =(e(z, ω)

h(z, ω)

),

W =( −ω� iωµd

iωεd + λ2m

iωµhω�

). (8)

In the vacuum regions,z 6∈ [0, L],

W(z,ω) ≡ Wb(ω) =(

0 iωµ0

iωε0+ λ2m

iωµ00

).

Introduce the matrixTb(ω) diagonalizingWb(ω):

Tb(ω) =1 − iωµ0

γm

1iωµ0

γm

, Tb(ω)Wb(ω)T−1b (ω) =

(−γm(ω) 00 γm(ω)

).

The propagating solutions of (7),U+(z, ω), andU−(z, ω), −∞ < z < ∞, aredetermined by the boundary conditionsU−(0, ω) = U+(L,ω) = I . In the emptyparts of the waveguide, they are written as follows:

U−(z, ω) = T −1b (ω)

(e−γm(ω)z 0

0 eγm(ω)z

)Tb(ω), for z 6 0,

U+(z, ω) = T −1b (ω)

(e−γm(ω)(z−L) 0

0 eγm(ω)(z−L)

)Tb(ω), for z > L. (9)

The scattering matrixS(ω) relatesU−(z, ω) andU+(z, ω) through

U−(z, ω) · S(ω) = U+(z, ω), −∞ < z <∞. (10)


The scattering matrix is expressed in terms of the reflection and transmission coef-ficients for a particular mode. Namely, the reflection coefficient for the wave propa-gating in the positive direction ofz, rp, can be expressed asrp(ω) = S21(ω)/S11(ω),whereS(ω) = Tb(ω)S(ω)T −1

b (ω). For the wave propagating in the opposite direc-tion, rn(ω) = −S12(ω)/S11(ω). The transmission coefficients aretp(ω) = tn(ω) =1/S11(ω).

In the inverse problem, the 2× 2 scattering matrixS as a function ofω issupposed to be known and is used to reconstruct the unknown material parameters(as functions ofz).

THEOREM 1. The2× 2 scattering matrixS(ω) given in a finite frequency bandand corresponding to theT E10 mode uniquely determines the inhomogeneousparametersεa(z), µa(z), and�a(z), z ∈ [0, L], of the dispersive� waveguide.

Our approach to the inverse problem is based on the reformulation of the scat-tering problem (10) as a Riemann–Hilbert problem. We seek for an invertiblematrix-valued function piecewice holomorphic in theω-plane (z being consideredas a parameter) which, on the one hand, has good behavior near the poles (finiteor infinite) ofW(z,ω), and, on the other hand, is constructed from the properlychosen solutions of (10).

Nearω = ∞,W(z,ω) is written as

W(z,ω) = iωW∞1 (z)+W∞0 (z)+O(1/ω), (11)

where

W∞1 (z) =(

0 µ1(z)

εh 0

), W∞0 (z) = �a(z)

(1 00 −1

),

µ1(z) = µh − µa(z). The transformation matrixT∞(z) is defined asT∞(z) =T 1∞(z), where

T =(−1 1

1 1

), 1∞(z) =

((εh/µ1)

1/4 00 (εh/µ1)

−1/4

)(z),

henceT∞(z)W∞1 (z)T−1∞ (z) = D∞(z) with

D∞(z) =√εhµ1(z)

(−1 00 1

).

The transformationF∞(z, ω) = T∞(z)U(z, ω), considered forz ∈ [0, L], givesthe differential equation

dF∞dz

(z, ω)

= {iωD∞(z)+A∞(z)+ R∞(z, ω)}F∞(z, ω), 06 z 6 L, (12)


where

A∞(z) =(

1

4

d

dz

(ln

εh

µ1(z)

)+�a(z)

)(0 −1−1 0

), (13)

R∞(z, ω) = O(1/ω) asω → ∞. Notice thatA∞(z) is off-diagonal, thereforeA∞(z) ∈ ran adD∞(z) for eachz ∈ [0, L], where(adA)B ≡ [A,B] = AB−BA.

We seek solutions of (12),F±∞(z, ω), for |ω| > R, whereR is big enough, andsuch that:

(a) F±∞(z, ω) are holomorphic in the half-neighborhoods ofω = ∞: {|ω| >R,± Imω > 0};

(b) F±∞(z, ω)E−1∞ (z, ω)→ I2 asω→∞,

whereE∞(z, ω) = exp{−iω ∫ LzD∞(t)dt};

(c) F±∞(z, ω)∣∣z=0,L are triangular matrices,(F±∞(z, ω))jj (L, ω) = 1, j = 1,2.

Such solutions are determined byF±∞(z, ω) = F±∞(z, ω)E∞(z, ω) whereF±∞(z, ω) are the solutions of the system of Fredholm equations

(F m∞)jk(z, ω)

= δjk +z∫

Q(j,k,m)

exp

{iω

∫ z

t

[(D∞)jj (s)− (D∞)kk(s)]ds}×

× [(A∞(t)+ R∞(t, ω)) Fm∞(t, ω)]jkdt, (14)

wherej, k ∈ {1,2},

Q(j, k,m) ={

0, m(j − k) > 0,L, j = k orm(j − k) < 0.

PROPOSITION 1.The system of integral equations(14),m = +,−, has a uniquesolutionF m∞ satisfying conditions(a) and(b) for a sufficiently largeR.

Proof of the proposition is similar to the case with a constant main termD∞,see, e.g., [25]. Note that the off-diagonal structure ofA∞ is of importance here.

On the other hand, the construction of (14) yields triangular boundary values ofFm∞(z, ω) at z = 0 andz = L:

F−∞(0, ω) = V −,l∞ (ω), F−∞(L,ω) = V −,u∞ (ω),

F+∞(0, ω) = V +,u∞ (ω), F+∞(L,ω) = V +,l∞ (ω), (15)

whereV ±,l∞ (ω) are lower-triangular,V ±,u∞ (ω) are upper-triangular, and the diagonalelements ofV +,l∞ (ω) andV −,u∞ (ω) are equal to 1.


PROPOSITION 2. Triangular matricesV m,l∞ (ω) and V m,u∞ (ω), m = +,−, for|ω| = R, ω ∈ (−∞,−R), and ω ∈ (R,∞) are uniquely determined by ascattering matrixS(ω) given in any(real) frequency band(outsideω = ±ω0).

Proof.RelatingF±∞(z, ω) andT∞(z)U+(z, ω), one gets

F±∞(z, ω) = T∞(z)U+(z, ω)T −1∞ (L)F±∞(L,ω). (16)

Settingz = 0 in (16) and using the scattering relation (10), one obtains

F±∞(0, ω) = T∞(0)U−(0, ω)S(ω)T −1∞ (L)F±∞(L,ω)

= T∞(0)S(ω)T −1∞ (L)F±∞(L,ω),

or, in view of (15),

V −,l∞ (ω) = S∞(ω)V −,u∞ (ω),

V +,u∞ (ω) = S∞(ω)V +,l∞ (ω), (17)

whereS∞(ω) = T∞(0)S(ω)T −1∞ (L).

Relations (17) may be viewed as triangular factorizations of the matrixS∞(ω)(for each fixedω), which allows a unique reconstruction ofV ±,l∞ (ω) andV ±,u∞ (ω)

providedS∞(ω) is known. The scattering matrixS(ω) is analytically continuablein C from the given frequency band. The constant factors,T∞(0) andT −1∞ (L), areuniquely determined by the asymptotics ofS(ω) asω→∞. Indeed, relations (17)and the large-ω behavior of the triangular factorsF±∞(L,ω) andF±∞(0, ω) implythat, asω→∞,

S∞(ω) = E∞(0, ω)(I +O(1/ω))

=(

eiωκ 00 e−iωκ

)(I +O(1/ω)),

whereκ = ∫ L0 √εhµ1(z)dz. Therefore,

S(ω) = T −1∞ (0)S∞(ω)T∞(L) = 1−1

∞ (0)T−1S∞(ω)T 1∞(L)

=(σ0 00 σ−1

0

)(cosωκ −i sinωκ−i sinωκ cosωκ

)(σ−1L 00 σL

)(I +O

(1

ω

)),

whereσ0 = (µ1(0)/εh)1/4 andσL = (µ1(L)/εh)1/4, that allows us to determine

µ1(0), µ1(L), andκ.For anω close toω0,W(z,ω) can be written as

W(z,ω) = 1

ω − ω0W1

1 (z)+W10 (z)+O(ω− ω0),

where

W11 (z) =

1

2

(ω0�a(z) −iω2

0µa(z)

−iεa(z) −ω0�a(z)

), W1

0 (z) =( 3

4�a(z) B(z)

C(z) −34�a(z)

),


B(z) = iω0

(µh − 5

4µa(z)

), C(z) = iω0εh − i

4ω0εa(z)+ λ2

m

iω0µh.

The diagonalization ofW11 (z) is performed by usingT1(z) = E(z)T1(z), where

T1(z) =(γ2(z) −1−γ1(z) 1

), E(z) = diag{e1(z), e2(z)},

γ1 = �a + iQiω0µa

, γ2 = �a − iQiω0µa

, Q =√εaµa −�2

a,

e1(z) = exp

{∫ L

z

1

γ2− γ1

(3

4�a(γ2+ γ1)+ Bγ2γ1− C + dγ2

dt

)dt

},

e2(z) = exp

{∫ L

z

1

γ2− γ1

(−3

4�a(γ2+ γ1)− Bγ2γ1+ C − dγ1

dt

)dt

}.

ThenT1(z)W11 (z)T

−11 (z) = D1(z), where

D1(z) = i

2ω0Q(z)

(−1 00 1

).

The transformed differential equation forF1(z, ω) = T1(z)U(z, ω) is

dF1

dz(z, ω) =

{1

ω − ω0D1(z)+A1(z)+ R1(z, ω)

}F1(z, ω),

06 z 6 L, (18)

where

A1(z) =(

0 β1(z)

β2(z) 0

), R1(z, ω) = O(ω− ω0) asω→ ω0,

β1 = 1

γ2− γ1

(3

2�aγ2+ Bγ 2

2 − C +dγ2

dt

)e1e−12 ,

β2 = 1

γ2− γ1

(−3

2�aγ1− Bγ 2

1 − Cdγ1

dt

)e−1

1 e2.

The solutions of Equation (18) for|ω − ω0| < δ, F+1 (z, ω) andF−1 (z, ω), areconstructed via the solutionsF±1 (z, ω) of the Fredholm integral equations similarto (14):

(F m1 )jk(z, ω)

= δjk +z∫

Q(j,k,m)

exp

{1

ω − ω0

∫ z

t

[(D1)jj (s)− (D1)kk(s)]ds}×

×[(A1(t)+ R1(t, ω)) F

m1 (t, ω)

]jk

dt. (19)


One has

F±1 (z, ω) = F±1 (z, ω)E1(z, ω),

where

E1(z, ω) = exp

{− 1

ω− ω0

∫ L

z

D1(t)dt

}.

FunctionsF±1 (z, ω) are holomorphic in the half-disks{ω : |ω−ω0| < δ, ± Imω >

0} for sufficiently smallδ > 0, andF±1 (z, ω)→ I asω→ ω0,± Imω > 0.Arguing as in the case of the infinite pole, we obtain the relations between

F±1 (z, ω) andU+(z, ω)

F±1 (z, ω) = T1(z)U+(z, ω)T −1

1 (L)F±1 (L,ω) (20)

and another set of triangular factorizations for the scattering matrix:

V−,u1 (ω) = S1(ω)V

−,l1 (ω),

V+,l1 (ω) = S1(ω)V

+,u1 (ω), (21)

whereS1(ω) = T1(0)S(ω)T−11 (L). The triangular factors in (21) are the boundary

values ofF±1 (z, ω) at z = 0 andz = L:

F−1 (0, ω) = V −,u∞ (ω), F−1 (L,ω) = V −,l∞ (ω),

F+1 (0, ω) = V +,l∞ (ω), F+1 (L,ω) = V +,u∞ (ω), (22)

whereV ±,l1 (ω) are lower-triangular,V ±,u1 (ω) are upper-triangular, and the diagonalelements ofV −,l1 (ω) andV +,u1 (ω) are equal to 1. The constant factorsT1(0) andT −1

1 (L) (or, in other words, the constantsγ1,2(0), γ1,2(L), e1,2(0), and∫ L

0 Q(z)dz)are determined by the asymptotics ofS(ω) asω→ ω0.

The considerations near−ω0 are literally the same, giving

D2(z) = − i2ω0Q(z)

(−1 00 1

), T2(z) = E(z)T2(z),

T2(z) =(γ2(z) −1−γ1(z) 1

), E2(z, ω) = exp

{− 1

ω + ω0

∫ L

z

D2(t)dt

},

V−,l2 (ω) = S2(ω)V

−,u2 (ω),

V+,u2 (ω) = S2(ω)V

+,l2 (ω), (23)

whereS2(ω) = T2(0)S(ω)T−12 (L).


Considering a neighborhood ofω = 0, we notice that that the functionP(z, ω) =KU+(z, ω)K−1, whereK =

( 1iω

0

0 1

), is holomorphic nearω = 0, detP(z, ω) ≡

1, andP(z, ω) = P(z,0)(I +O(ω)) asω→ 0, where

P(z,0) =(

coshλm(z− L) µhλm

sinhλm(z− L)λmµh

sinhλm(z− L) coshλm(z− L)).

Now we are able to construct a piecewise holomorphic functionm(z, ω) (z isa parameter) relative to a contour in theω-plane, such that its jumps across thecontour are essentially determined by the scattering matrix. The contour0 consistsof the circles|ω| = R, |ω− ω0| = δ, |ω + ω0| = δ, and the real axis Imω = 0.

Set

m(z, ω) =

F±∞(z, ω), |ω| > R, ± Imω > 0,T∞(z)T −1

1 (z)F±1 (z, ω), |ω − ω0| < δ, ± Imω > 0,T∞(z)T −1

2 (z)F±2 (z, ω), |ω + ω0| < δ, ± Imω > 0,T∞(z)U+(z, ω)K−1, |ω| < R, |ω ± ω0| > δ.

(24)

The contour0 divides the complex plane into two open sets,Q+ andQ−, beingthe positively oriented boundary ofQ+. Denote bym±(z, η), η ∈ 0, the boundaryvalues ofm(z, ω) asω→ η,ω ∈ Q±. Then (24) together with (16) and (20) yields

m+(z, η) = m−(z, η) · V (z, η), η ∈ 0, (25)

where

V (z, ω)

=

KT −1∞ (L)V +,l∞ (ω)E−1∞ (z, ω), |ω| = R, Imω > 0,

E∞(z, ω)(V −,u∞ (ω)

)−1T∞(L)K−1, |ω| = R, Imω < 0,

E∞(z, ω)(V −,u∞ (ω)

)−1V +,l∞ (ω)E−1∞ (z, ω), Imω = 0, |ω| > R,

KT −11 (L)V

+,u1 (ω)E−1

1 (z, ω), |ω− ω0| = δ, Imω > 0,

E1(z, ω)(V−,l1 (ω)

)−1T1(L)K

−1, |ω− ω0| = δ, Imω < 0,

E1(z, ω)(V−,l1 (ω)

)−1V+,u1 (ω)E−1

1 (z, ω), Imω = 0, |ω− ω0| < δ,KT −1

2 (L)V+,l2 (ω)E−1

2 (z, ω), |ω+ ω0| = δ, Imω > 0,

E2(z, ω)(V−,u2 (ω)

)−1T2(L)K

−1, |ω+ ω0| = δ, Imω < 0,

E2(z, ω)(V−,u2 (ω)

)−1V+,l2 (ω)E−1

2 (z, ω), Imω = 0, |ω+ ω0| < δ,I, Imω = 0, |ω| < R,

|ω ± ω0| > δ.

(26)

From the definition ofm(z, ω) it follows that

m(z, ω) = I +O

(1

ω

)asω→∞, (27)


m(z, ω) = T∞(z)K−1P(z, ω) = T(iω 00 1

)1∞(z)P (z, ω)

= T

(iω 00 1

)P (z, ω), (28)

whereP (z, ω) is holomorphic and invertible nearω = 0.Note thatEk(z, ω), k = ∞,1,2, are expressed in terms of two combinations

of the (unknown) material parameters,∫ Lz

√εhµ1(t) dt and

∫ LzQ(t)dt . In order to

construct a family of the Riemann–Hilbert problems with jumps independent ofthe unknown functions, two auxiliary real parametersJ1 andJ2 are introduced. Ifone defines

E∞(J1, J2;ω) = diag{e−iωJ1,eiωJ1

},

E1(J1, J2;ω) = diag{e−

iω02(ω−ω0)

J2,eiω0

2(ω−ω0)J2},

E2(J1, J2;ω) = diag{e

iω02(ω+ω0)

J2,e−iω0

2(ω+ω0)J2},

then

Ek(z, ω) = Ek(J1, J2;ω)∣∣∣∣J1=J1(z)J2=J2(z)

, k = 1,2,∞, (29)

where

J1(z) = −∫ L

z

√εhµ1(t)dt,

J2(z) = −∫ L

z

Q(t)dt. (30)

A family of the Riemann–Hilbert problems parametrized byJ1 andJ2 is givenas follows: find a 2× 2 matrix functionG(J1, J2;ω) that satisfies the followingconditions:

G(·, ·;ω) is piecewise holomorphic relative to the contour0; (31)

G(·, ·;ω) = I +O

(1

ω

)asω→∞; (32)

G(·, ·;ω) is invertible forω 6= 0; (33)

(−i/ω 00 1

)(−1 11 1

)G(·, ·;ω) is holomorphic and invertible

in a neighborhood ofω = 0; (34)


G+(J1, J2; η) = G−(J1, J2; η)V (J1, J2; η), η ∈ 0, (35)

where

G±(·, ·; η) = limω→ηω∈Q±

G(·, ·;ω),

andV (J1, J2; η) is constructed via (26) withEk(z, ω) replaced byEk(J1, J2;ω),k = 1,2,∞.

By the Liouville theorem, the solution of the Riemann–Hilbert problem (31)–(35) is unique for eachJ1 andJ2 (if G1 andG2 are two solutions, thenG1G

−12 is an

entire function,G1G−12 (ω)→ I asω→∞, thereforeG1G

−12 ≡ I ). This solution

is related tom(z, ω) as follows:

G(J1, J2;ω)∣∣J1=J1(z)J2=J2(z)

= m(z, ω). (36)

3. Reconstruction of Material Parameters

The fact that the solution of the Riemann–Hilbert problem (31)–(35) is unique to-gether with relation (36) allows us to develop the procedure of unique simultaneousreconstruction of the material parameters and, hence, to prove Theorem 1.

First, the jump matrixV (J1, J2; η), η ∈ 0, is constructed from the scatteringmatrix S(ω) related to someT Em0 mode, for instance, the basic modeT E10. Theconstruction involves analytic continuation ofS(ω) on the related parts of0, eval-uation ofS(ω) asω→∞ andω→ ±ω0, and triangular factorizations (17), (21),and (23).

Second, the Riemann–Hilbert problems (31)–(35) are solved for eachJ1 andJ2, givingG(J1, J2;ω). Evaluating the solution of the Riemann–Hilbert problematω = 0 andω = ω0 gives, in view of (36) and (24),

limω→0

1

2

(−i/ω 00 1

)(−1 11 1

)G(J1, J2;ω)

∣∣∣∣J1=J1(z)J2=J2(z)

= 1∞(z)P (z,0), (37)

T −1G(J1, J2;ω0)∣∣J1=J1(z)J2=J2(z)

= 1∞(z)T −11 (z). (38)

Let us denote

G(0)(J1, J2) = limω→0

1

2

(−i/ω 00 1

)(−1 11 1

)G(J1, J2;ω),

M1(J1, J2) = G(0)22 (J1, J2)

G(0)11 (J1, J2)

, M2(J1, J2) = G21(J1, J2;ω0)

G11(J1, J2;ω0),

M3(J1, J2) = G22(J1, J2;ω0)

G12(J1, J2;ω0).


Then, in view of (37) and (38),√εhµ1(z) = εhM1(J1, J2)

∣∣J1=J1(z)J2=J2(z)

,

Q(z) = ω0

2(M2(J1, J2)−M3(J1, J2))(µhM

−11 (J1, J2)− εhM1(J1, J2))

∣∣J1=J1(z)J2=J2(z)

,

µa(z) = µh − εhM21(J1, J2)

∣∣J1=J1(z)J2=J2(z)

,

�a(z) = iω

2µa(z)M

−11 (J1, J2)(M2(J1, J2)+M3(J1, J2))

∣∣J1=J1(z)J2=J2(z)

,

εa(z) = −ω20µa(z)M

−21 (J1, J2)M2(J1, J2)M3(J1, J2)

∣∣J1=J1(z)J2=J2(z)

. (39)

Since, by the definition ofJ1(z) andJ2(z), dJ1/dz = √εhµ1(z) and dJ2/dz =Q(z), we arrive at the system of differential equations for determiningJ1(z) andJ2(z):

dJ1

dz= εhM1(J1, J2), J1(L) = 0,

dJ2

dz= ω0

2(M2(J1, J2)−M3(J1, J2))(µhM

−11 (J1, J2)− εhM1(J1, J2)),

J2(L) = 0.

Finally, substitutingJ1(z) andJ2(z) into (39) givesµa(z),�a(z), andεa(z).

4. Conclusion

When implementing any inversion method, one faces the uniqueness problem,i.e., the question about the amount of information which is necessary and suffi-cient to achieve, in principle, a unique reconstruction. Ill-posed nature of mostinverse problems requires searching for a minimum information which determinesthe problem uniquely (overdetermining an inverse problem may increase its ill-posedness).

The connection of the inverse scattering problems and the Riemann–Hilbertproblems was first established by Shabat [17] and rigorously developed further in[1, 24, 25]. The present paper illustrates usefulness of the method of Riemann–Hilbert problem for studying scattering problems relevant to wave propagation incomplex inhomogeneous dispersive media. We believe that the Riemann–Hilbertapproach to the inverse scattering problem is an efficient tool for achieving abetter understanding of the relations between the scattering data and the materialparameters to be reconstructed.


Acknowledgements

The author is grateful for the hospitality at the Laboratory of Mathematical Physicsand Geometry, University Paris-7 (where the work was finalized) and to the Em-bassy of France in Ukraine (MAE) for financial support.

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195

Second-Order Covariant Tensor Decompositionin Curved Spacetime

GIANLUCA GEMELLIDipartimento di Matematica, Università di Torino, Via Carlo Alberto n.10, I-10123 Turin, Italy.e-mail: [email protected]

(Received: 8 September 1998; in final form: 22 February 2000)

Abstract. Local four-dimensional tensor decomposition formulae for generic vectors and 2-tensorsin spacetime, in terms of scalar and antisymmetric covariant tensor potentials, are studied within theframework of tensor distributions. Earlier first-order decompositions are extended to include the caseof four-dimensional symmetric 2-tensors and new second-order decompositions are introduced.

Mathematics Subject Classifications (2000):53B50, 35Q75.

Key words: Laplace operator in a curved spacetime, covariant potentials for tensor fields.

1. Introduction

A classic decomposition theorem (see, e.g., [22], p. 26, [10], p. 43, [18], p. 49)states that any regular vector field in the ordinary three-dimensional space can bewritten as the sum of a gradient plus a divergence-free component.

A similar decomposition can be considered for skew-symmetric 2-tensors[9, 13].

Several decomposition theorems are known for the Weyl tensor and, more gen-erally, for a generic antisymmetric tensor in a curved spacetime [1, 13, 19].

Covariant decompositions of symmetric 2-tensors into their irreducible partscan be introduced on a spacelike three-dimensional manifold; such decomposi-tions proved to be useful tools for the study of the Cauchy problem for Einsteingravitational equations (see, e.g., [8, 29, 21]).

A systematic study of this problem should therefore include the missing pieces:decompositions of four-dimensional vectors and symmetric 2-tensors in curvedspacetime. Moreover, since the decompositions mentioned above are first-order,i.e. they involve first derivatives of the potentials, it is also interesting to considersecond-order decompositions obtained by iteration.

Here we attempt to reach these goals and to revisit the whole matter within theframework of tensor distributions.

In Section 2, the main definitions about vector distributions and general tensordistributions are introduced.

196 GIANLUCA GEMELLI

In Section 3, the theory of the Laplace operator within the framework of tensordistributions in a curved spacetime is recalled and a useful result about first-ordercovariant decomposition of a divergence-free 4-vector (Theorem 1) is readily ob-tained. The same result was proved in the dual framework ofC∞ functions in [13]with the help of spinor analogues of tensor equations.

In Section 4, the generalized Clebsch Theorem (Theorem 2) is proved. Such atheorem rigorously establishes a first-order covariant decomposition formula for ageneric (nondivergence-free) 4-vector, a formula which is usually taken for grantedas the generalization of the ordinary three-dimensional theorem. The 3+1 splittingof the four-dimensional formula, however, permits us to understand the relationsbetween the two formulations and to determine under which additional hypothesisthe ordinary three-dimensional formula is obtained exactly as a corollary from thefour-dimensional one (Corollaries 2.1, 2.2 and 2.3).

In Section 5, a first-order covariant decomposition formula for a skew-symme-tric 2-tensor in a curved spacetime is proved (Theorem 3). Such a formula wasoriginally introduced in flat spacetime [9]; the curved spacetime generalizationwas again proved within the framework ofC∞ functions and with the help ofspinor techniques in [13]. The splitting of the formula is examined, which permitsus to relate the electric and magnetic components of the skew-symmetric tensorto its vector potentials. The particular case when the tensor is divergence-free isconsidered (Corollaries 3.1 and 3.3). Moreover, a new second-order covariant de-composition formula is introduced (Corollary 3.2) as a consequence of Theorems 2and 3.

In Section 6, a four-dimensional covariant first-order decomposition formulafor a symmetric 2-tensor is introduced (Theorem 4), which generalizes the usualthree-dimensional transverse decomposition on a spacelike hypersurface. A par-tially second-order decomposition formula is also given (Corollary 4.3). Again therelations between the two formulations are studied by means of the splitting of thespacetime (Corollaries 4.1 and 4.2).

In Section 7, the case of flat spacetime is examined. Here a second-order de-composition formula for a generic divergence-free symmetric 2-tensor in terms of adouble 2-form potential is introduced (Theorem 5). In the particular case where thetensor under investigation is trace-free, the same property is held by the trace of thepotential (Corollary 5.1). We thus obtain a complete second-order decompositionformula for a generic symmetric 2-tensor (Corollary 5.2). The splitting of such aformula permits us to establish similar three-dimensional results (Corollaries 5.3and 5.4). Conversely, the second-order decomposition of a double 2-form in termsof a divergence-free symmetric potential, and the corresponding three-dimensionalresults are obtained (Theorem 6 and relative corollaries).

In Section 8, some applications of the theory of potentials to electromagnetismare studied. A general theorem of the existence and uniqueness of the solutionsof the Maxwell equations system is given as a consequence of Theorem 1 (Corol-

SECOND-ORDER COVARIANT TENSOR DECOMPOSITION 197

lary 1.1). The 4-vector potentials of electromagnetism are introduced and theLorentz gauge is defined within the framework of Theorem 3 (Corollary 3.4).

2. Tensor Distributions in a Curved Spacetime

Let V4 be the spacetime of general relativity, i.e. a four-dimensional orientedpseudo-Riemannian manifold of classC3, whose metricg is of classC2 and ofnormal hyperbolic type, with the signature− + + +. Greek indices run from 0 to3. Units are chosen in order to have the speed of light in empty spacec = 1.

The Riemann curvature tensorRαβρσ is defined by the following Ricci formula:

(∇β∇α −∇α∇β)V σ = RαβρσV ρ (1)

which holds for any regular vector fieldV . The symmetric Ricci tensor, trace ofthe Riemann tensor, is defined byRβρ = Rαβρα.

Let η = √|g|ε denote the unit volume 4-form (Ricci antisymmetric tensor),whereε is the Levi-Civita indicator;η is used to define the dual∗F of a given anti-symmetric 2-tensorF : (∗F)αβ = (1/2)ηαβµνFµν . The dual operator is involutive.Useful properties ofη are

(1/2)ηαρλµηβρσν = gαβgµ[σ gν]λ + gλβgα[σ gν]µ + gµβgλ[σ gν]α,

(1/2)ηρσλµηρσαβ = gµαgλβ − gλαgµβ. (2)

Our investigations are of a local nature. For a pointx′ ∈ V4, let us consider an openneighbourhood� ⊂ V4 with compact closure, homeomorphic to an open sphereof R4, such that for allx ∈ �, there exists a unique arc of geodesic`(x, x′) joiningx andx′. Forx ∈ �, let E+x [resp.E−x ] denote the image in� of the set of futurepointing timelike paths originating atx [resp. ending atx]. A setA is called past-compact [resp. future-compact] ifA∩E−x [resp.A∩E+x ] is compact for anyx ∈ �.If a set is compact, then it is also both past-compact and future-compact.

A p-tensor-distributionT on� is a continuous (in an appropriate sense) linearform on the space of regular [sayC∞(�), although this assumption could be re-laxed to someCk, depending on the applications] test-p-tensorsU with compactsupportK ⊂ � (for complete details see, e.g., [3, 5, 6, 14, 17]). In particular, atest-p-tensor goes smoothly to zero on the border∂K of its support.

If � is the domain of a local chart, in this chart a generic tensor distribution on� has components which are scalar distributions on� [14, 17].

If f is a function integrable in�, there is a corresponding distributionf D, de-fined, on the generic test functionϕ, by the following Riemann 4-volume integral:

f D(ϕ) =∫�

f ϕ. (3)


Tensor distributions corresponding to generic integrable p-tensors are definedsimilarly. For example, ifV is an ordinary 4-vector, locally integrable, there is acorresponding 4-vector distributionVD, defined, on the generic test-4-vectorU , by

V D(U) =∫�

VαUα. (4)

Tensor distributions which correspond to integrable tensors are called integrable;they are also said to be equivalent to the corresponding tensor. A tensor-distributionwhich is not integrable is called singular. An important example of singular distri-bution is the Dirac mensure distributionδ6 associated to a regular hypersurface6 ⊂ � (see, e.g., [14, 15, 16, 17]).

Since the space of tensor distributions includes integrable and singular tensordistributions, it is an extension of the space of ordinary integrable tensors.

The support of a tensor distribution on� is the smallest closed setS in� outsidewhich T is identically zero (i.e. it is zero for all test tensors with support outsideS). For example, if a tensor distribution is equivalent to a tensor, its support is thesupport of the tensor. For the singular distributionδ6, the support is6.

The covariant derivative of a tensor distribution is, in any case, defined by

(∇T )(U) = −T (DivU), (5)

where(DivU)α1...αp = ∇βUβα1...αp . With this definition, the classical propertiesof the covariant derivative also hold for tensor distributions, including, for exam-ple, (1).

As for the covariant derivative of the singular distributionδ6, it is possible toprove (see, e.g., [14, 17]) that there is a singular distributionδ′6, with support on6, such that∇αδ6 = `αδ′6 , where`α is the gradient vector normal to6.

In the following, we will consider differential operators on distributions andtensor distributions with a past-compact or future-compact support in�. In partic-ular, we will often have to handle with divergence-free (4-)vector distributions withpast-compact or future compact support. It is therefore useful to consider them herein some detail.

Divergence-free vector distributions include integrable and singular ones. Anintegrable divergence-free vector distributionV D is equivalent to an ordinary reg-ular divergence-free tensorV and has the same support. In fact, ifV has a locallyintegrable covariant derivative, then for any test functionϕ, by definition of thederivative in the sense of distributions, we have(

Div(V )D)(ϕ) = (DivV )D(ϕ)−

∫�

Div(ϕV ) (6)

which, by the Green theorem, implies that Div(V )D is null for all test functionsϕ if V is divergence-free and regular, i.e. continuous in�. In particular,V mustgo to zero in a regular way on the border of its support (otherwise, the integral atthe right-hand side would produce a term involving the jump ofV ). It is therefore


easy to construct examples of such vector distributions: it suffices, for example, toconsider the divergenceV α = ∇βF αβ of a skew-symmetric potentialF of classC1.The support of∇F , then, is also that of the vector distributionV D be it compact,past-compact, future-compact, or other. The interesting point is that more generalskew-symmetric potentials exist for any divergence-free vector distribution, be itintegrable or not (see Theorem 1 in the following section).

Let us now construct an example of singular divergence-free vector distribution.Consider the product (in the sense of distributions) of the Dirac mensure distribu-tion δ6 associated with a given regular hypersurface6 ⊂ � and of a regular andintegrable vectorV with supportS ⊂ �. Such a vector distribution is singular andits support is6 ∩ S, which thus can be supposed to be compact, past-compact orfuture-compact, with a suitable choice ofS. By definition, for any test vectorU ,one has

(δ6V )(U) = δ6(V αUα). (7)

Since the ordinary rules of derivation of a product also hold for distributions andtensor distributions, we have

Div(δ6V ) = (`αV α)δ′6 + (DivV )δ6. (8)

Thus, the considered singular vector distribution is divergence-free if the vectorV

is both divergence-free and tangent to6. It is not difficult to construct an exampleof such a vector field. Again, it suffices, for example, to considerV α = ∇βF αβwith F tangent to6.

Therefore, as one would expect, the space of divergence-free vector distribu-tions with past-compact [future-compact] support is larger than that of divergence-free regular vectors with past-compact [future-compact] support.

3. The Laplace Operator in a Curved Spacetime

For a tensorT of orderp, the (generalized) Laplace operator is defined by [14]:

(1T )α1...αp = ∇µ∇µTα1...αp +p∑k=0

RαkµTα1...µ...αp +

+p∑

k=1, k 6=lRαkραlσ Tα1...

ρ...σ...αp , (9)

where in the second term at the right-hand side,µ is at thekth place, while in thethird term,ρ andσ ar at thekth andlth place, respectively.

For example, for a scalaru, a vectorV and 2-tensorT , we have, respectively,

1u = ∇ρ∇ρu,(1V )α = ∇ρ∇ρVα + RαµVµ, (10)

(1T )αβ = ∇ρ∇ρTαβ + RαµTµβ + RµβTαµ + 2RαµβνT

µν.


The following equivalent definitions for vectors and 2-tensors, in terms of commu-tators, are sometimes useful:

(1V )α = ∇ρ∇ρVα + (∇α∇ρ −∇ρ∇α)V ρ,

(1T )αβ = ∇ρ∇ρTαβ + (∇α∇σ −∇σ∇α)T σ β + (∇β∇σ − ∇σ∇β)Tασ . (11)

For example, from (11) it is not difficult to see that1 commutes with the opera-tions of index contraction and of adjoint of a skew-symmetric 2-tensor [∗(1F) =1(∗F)].

We have the following lemma:

LEMMA 1. For any tensor distributionv with a past-compact[ future compact]support, there is only one tensor distributionu with a past-compact[ future com-pact] support which is solution of1u = v.

This lemma holds since1 is a linear hyperbolic and self-adjoint operator (see[3, 5, 14]). More generally, the same result holds for a generic hyperbolic linearself-adjoint operatorL, with coefficients of classC0(�).

We have a dual version of Lemma 1 which holds within the framework of reg-ularC∞ ordinary functions, in a fixed local chart in the domain�. One thus mayeither work with tensor distributions or with regular functions and tensor fields.In the following, we will state our results within the framework of distributions,letting it be understood that a dual point of view also holds true.

A useful theorem is the following, which we are going to prove with the help ofLemma 1.

THEOREM 1. LetV,W be two given divergence-free vector distributions(∇αV α

= ∇αWα = 0) with past-compact[ future-compact] support. Then there exists aunique antisymmetric tensor distributionFαβ with past-compact[ future-compact]support such that

Vβ = ∇αF αβ; Wβ = ∇α(∗F)αβ, (12)

where(∗F)αβ = (F∗)αβ = (1/2)ηαβµνFµν is the dual ofF , which is antisymmet-ric too.

Proof. Recall that for any antisymmetric 2-tensorFαβ , we have the identity∇α∇βF αβ = 0 (which follows from direct calculation and the use of the identityon Christoffel symbols:0αβα = ∂β log

√|g| or, equivalently, from (1) and thesymmetry of the Ricci tensor), so the compatibility condition which assures thatV

andW are divergence-free is automatically satisfied as soon as we find a suitableF solution of (12).

Consider∇α(∗F)αβ ; from (2), we have the identity:

ηβρσγ∇α(∗F)αβ = (1/2)ηρβγ σ ηαβ

µν∇αFµν= ∇ρF σγ +∇σF γρ +∇γ F ρσ


and similarly forηβρσγ∇αF αβ , replacingF with (∗F). Consequently, we have

ηβρσγ∇ρ∇α(∗F)αβ = ∇ρ∇ρF σγ +∇ρ∇σF γρ +∇ρ∇γ F ρσ .

Using the Ricci formula for the inversion of iterated covariant derivatives and theantisymmetry ofF , we equivalently have

ηβρσγ∇ρ∇α(∗F)αβ = (1F)σγ +∇σ∇ρF γρ +∇γ∇ρF ρσ

and similarly forηβρσγ∇ρ∇αF αβ , replacingF with (∗F).Now suppose that (12) holds. In this case we have

(1F)σγ = ηνρσγ∇ρWν +∇σV γ −∇γ V σ (13)

or, equivalently, in terms of(∗F):[1(∗F)]λδ = ηλδσγ∇σVγ + ∇λWδ −∇δWγ . (14)

In case (12) holds, the equivalent relations (13) and (14) are identities. As anequation for an unknownF , however, (13) [or, equivalently, (14)] has a uniquesolution, according to Lemma 1. Let us prove that this solution necessarily alsosatisfies (12).

LetDβ = ∇α(∗F)αβ −Wβ and we have

ηβρσγDβ = ∇ρF σγ +∇γ F σρ +∇σF γρ − ηβρσγWβ

and from (13)ηβρσγ∇ρDβ = 0. By saturation withησγ λµ, we thus have∇[λDµ] =0. Consequently,D is a solution of the equation(1D)µ−∇µ∇λDλ = 0 and sinceD is divergence-free, also of the equation(1D)µ = 0. This equation, according toLemma 1, has a unique solution: the null one. We conclude thatDβ = 0 and thus∇α(∗F)αβ = Wβ . Similarly from the equivalent equation (14), we can show thatwe necessarily have∇αF αβ − V β = 0 and our lemma is proved. 2

The dual result within the framework ofC∞ functions was proved in [13], withthe help of spinor analogues of tensor equations.

An easy example of an application of Theorem 1 is the existence and uniquenesstheorem for the solution of the Maxwell equations (see Section 8).

4. Decomposition of a Vector

The classic decomposition of a vector into a gradient plus a divergence-free com-ponent has many applications in hydrodynamics and electromagnetism.

This result is often called theClebsch theorem(see, e.g., [7, 10]), and in thefollowing we will also adopt this name. However, we have to remark that, when theresult is stated within the framework of global analysis on a Riemannian manifold,the preferred denomination is the Helmholtz decomposition theorem (see, e.g.,[4, 27]). Moreover, the name of Clebsh is usually used to name the transformation


of the velocity field of a hydrodynamical system which permits us to automaticallysatisfy the Euler equation (see, e.g., [11]), and its extensions to general Hamiltoniansystems, which permits us to represent a generic 1-form in terms of the canonicalcoordinates (see, e.g., [2, 12, 23, 24]).

The proof of the classic Clebsch theorem in any case involves the ordinaryelliptic Laplace operator.

It is not difficult then to extend the classical three-dimensional decomposi-tion formula to the case of the curved spacetimeV4, by means of the generalizedLaplace hyperbolic operator.

THEOREM 2. Let V and W be two vector-distribution with a past-compact[ future-compact] support and letW be divergence-free. Then there exist a uniquedistributionφ and a unique antisymmetric tensor distributionF , with past-compact[ future-compact] support, such that

Vα = ∇αφ + ∇βF βα (15)

and that

Wα = ∇β(∗F)βα. (16)

Proof. Let φ be the solution of the equation1φ = ∇αV α. Then the vectorVα−∇αφ is divergence-free. From Theorem 1, then, there exists an antisymmetricFαβ such thatWβ = ∇α(∗F)αβ and thatVα − ∇αφ = ∇βF βα. Thus (16) and (15)hold and our theorem is proved. 2

As a corollary, we have the ordinary three-dimensional Clebsch theorem, as weare going to show.

We first need to split the spacetime and define the three-dimensional space.Let the latin indices run from 1 to 3 and let us consider a local coordinate chart

adapted to some given reference frame, i.e. such thatg00 = −1,g0i = g0i = 0. Wesay that a vector is ‘spatial’ (with respect to the chosen reference) if its componentsof index 0 are null.

Let us introduce the ‘magnetic’ part spatial vectorHj = (1/2)ηikjFik and the‘electric’ partEj = (1/2)ηikj (∗F)ik of a generic antisymmetric 2-tensorF , whereηijk = η0

ijk is the three-dimensional spatial volume element. We have

F ik = ηikjH j , (∗F)ik = ηikjEj ,F 0i = Ei, (∗F)0i = −Hi.

(17)

Now let us consider the hypothesis of Theorem 2. The split version of (15) and (16)is the following:

Vi = ∇iφ − ηkj i∇kHj +∇0Ei, Wi = −ηkj i∇kEj −∇0Hi,

V0 = ∇0φ +∇kEk, W0 = −∇kH k.(18)


In other words, we have proved the following corollary:

COROLLARY 2.1. In a given local chart adapted to a reference frame, letV be ageneric spatial vector distribution with a past-compact[ future-compact] support.Then there are two spatial vector distributions with past-compact[ future-compact]supportH andE such that

Vi = ∇iφ − (CurlH)i + ∇0Ei, (19)

where(CurlH)i = ηi jk∇jHk.HereH andE are not unique, unless one introduces the supplementary condi-

tions suggested by (18):

Wi = −(CurlE)i −∇0Hi, (20)

whereW is an arbitrarily fixed spatial vector. Moreover, one still can choose arbi-trary valuesv andw to assign to∇0φ+∇kEk and−∇kH k respectively, to play therole which was ofV0 andW0 (which are now null) in (18).

We see that (19) contains the somewhat unexpected term∇0Ei ; we have theordinary form of the Clebsch Theorem if and only if such a term is null. Corollary2.1 is in fact a four-dimensional generalization of the ordinary three-dimensionalClebsch theorem and reduces to the more familiar form in the case where the metricis independent of time.

COROLLARY 2.2. In the hypothesis of Corollary2.1, let the metric tensor beindependent fromx0. Then, ifV is also independent fromx0, Equation(19) reducesto the following form:

Vi = ∇iφ − (CurlH)i. (21)

In other words,V can be split as the sum of a gradient plus a divergence-freecomponent.

Proof. Consider (19). By hypothesis, we have∂0gik = 0 and in our chart we,moreover, haveg00 = −1, g0i = 0. For the metric connection, this implies00α

σ =0 and for the Riemann tensorRα0ρ

σ = 0. In this situation, from (9) one has∇01 =1∇0, and, thus, from (13) we have that ifV andW are independent fromx0, thesame happens toEi. Therefore∇0Ei = ∂0Ei = 0 and the corollary is proved. 2

Actually, Corollary 2.1 is also a curved-space generalization of the ordinarythree-dimensional Clebsch theorem, which reduces to the more familiar form inthe case of flat spacetime:

COROLLARY 2.3. In the hypothesis of Corollary2.1, let the spacetime be flat.Then there is a spatial vector distributionH such that

Vi = ∂iφ − (curlH)i, (22)

where(curlH)i = εijk∇jHk.


Proof.To prove it, we have to show that, for any given spatialV , we can find aspatialW to satisfy (19)–(20) with, moreover,E = 0.

Let V be the given spatial vector andW a still generic spatial vector such that∂kW

k = 0. Let us denote by a dot the partial derivative with respect tox0 (timederivative). From (18), we have

Vi = ∂iφ − εkj i∂kHj + Ei , Wi = −εkj i∂kEj − Hi,v = φ + ∂kEk, w = −∂kH k = 0,

(23)

where we have introduced the two arbitrary scalarsv andw. These two scalars giveus some more degrees of freedom to use to obtainEi = 0. We thus can write

Vi = ∂iφ − εkj i∂kHj + Ei= ∂iφ + εkj i∂kWj − 1Ei + ∂i∂kEk + Ei , (24)

where we have introduced the three-dimensional ordinary elliptic Laplace operator1 = ∂k∂k. We then have

Vi = εkj i∂kWj − 1Ei + Ei + ∂iv. (25)

Since we have1E = −E + 1E, we obtain the following identity:

1Ei = −Vi + εkj i∂kWj + ∂iv. (26)

Then, if we can chooseW andv such that

−Vi + εkj i∂kWj + ∂iv = 0, (27)

we necessarily haveEi = 0 as reqiured.If (27) should hold, we would consequently have

∂ [kWj ] = (1/2)εkji(Vi − ∂iv) (28)

and, by derivation,

1Wj = εkji∂kVi + ∂j w. (29)

Now let us simply choosew = 0 and consider as our auxiliary spatial vectorW

the solution of the equation above. If we denote byDi the right-hand side of (26),we consequently have

1Di − ∂i∂kDk = 0. (30)

Now, since∂kDk = 1v − ∂kV k, it suffices to choose ourv as the solution ofequation1v = ∂kV k to haveDi as the solution of the three-dimensional homoge-neous Laplace equation1Di = 0 and, consequently,Di = 0. Thus, by (26),Ei isalso null as it is in turn a solution of the four-dimensional homogeneous Laplaceequation1Ei = 0. This completes the proof of our corollary. 2


5. Decomposition of a Skew-Symmetric 2-Tensor

For a skew-symmetric 2-tensorFαβ , one can introduce the following covariant de-composition formula where there appear first-order derivatives of vector potentials[9, 13]:

Fαβ = ∇αVβ −∇βVα + ηαβµν∇µWν. (31)

This again is a generalization of the Clebsch theorem for vectors, since if we denoteby

Hαβ = 2∇[αVβ], Kαβ = ηαβµν∇µWν, (32)

the two components of the decomposition ofF , thenH is irrotational andK isdivergence-free in the following sense:

ηαβγ δ∇γHαβ = 0, ∇αKβα = 0. (33)

Decomposition (31) was first introduced in [9] in flat spacetime. This decompo-sition formula is sometimes calledClebsch representationor Clebsch transfor-mation (see, e.g., [25, 20]). We furthermore remark that, within the frameworkof global analysis on a Riemannian manifold, the decomposition of differentialforms into the sum of exact, co-exact and harmonic components is usuallycalledHodge decomposition(for an extensive survey, see [26]). The Hodge decomposi-tion generalizes the Helmholtz decomposition and again one has to solve an ellipticboundary problem. This kind of decomposition also has notable applications to thecontinuum theory of defects, as recently shown in [28].

Let us first show that in a curved spacetime, Equations (33) are identicallysatisfied as a consequence of definitions (32). For example, we have

∇αKβα = ηαβµν∇α∇µWν = (∗R)νβγ νWγ ,

where

(∗R)νβγ ν = (1/2)ηνβλδRλδγ ν = (R∗)γ ννβand(∗R)νβγ ν is null as this is equivalent toR[αβρ]σ = 0. By the same argument,we thus haveηαβγ δ∇γHαβ = 0.

Let us then prove (31) within the framework of tensor distributions.

THEOREM 3. LetF be a given skew-symmetric2-tensor distribution with a past-compact[ future-compact] support andv,w two given distributions with a past-compact[ future-compact] support. Then there exist two unique vector distributionsV andW , with a past-compact[ future-compact] support, such that(31)holds andthat, moreover,

∇αV α = v, ∇αWα = w. (34)


Proof. If (31) holds, we obtain, by differentiation, the identity

∇αF αβ = ∇ρ∇ρVβ − ∇α∇βV α = (1V )β −∇β∇αV α.

Thus, if also (34) holds, we have

(1V )β = ∇αF αβ +∇βv. (35)

Similarly, from (31) and (34), we have the following identity forW :

(1W)β = −∇α(∗F)αβ +∇βw. (36)

In any case, (35) and (36) have, as differential equations for the unknownV andW ,a unique solution, according to Lemma 1. Then, letV andW be the correspondingsolutions. Let us prove that they necessarily satisfy (31) and (34). Let us introducethe following skew-symmetric tensor:

Dαβ = Fαβ − 2∇[αVβ] + ηαβµν∇µWν.

By (35), we have∇α(∗D)αβ = 0. By (36), we have∇αDαβ = 0 or, in terms of

∗D, ηαβµν∇α(∗D)µν = 0. Consequently, we have

(1/2)ηβρσγ ηαβµν∇α(∗D)µν = ∇ρ(∗D)σγ +∇σ (∗D)γρ +∇γ (∗D)ρσ = 0

thus, by differentiation, we have

∇ρ∇ρ(∗D)σγ +∇ρ∇σ (∗D)γρ +∇ρ∇γ (∗D)ρσ = 0

or, equivalently,[1(∗D)]σγ = 0. This last equation has a unique solution (the nullone), according to Lemma 1. We conclude that∗D = 0 andD = 0 and the theoremis proved. 2COROLLARY 3.1. In the hypothesis of Theorem3 if, moreover,F is divergence-free, then there is a vector potentialW such that we have

Fαβ = ηαβµν∇µWν. (37)

Proof.To prove this corollary we have to show that it is possible to choosev orw such thatV is necessarily null. Actually, from (31) we have

∇αF αβ = (1V )β −∇β∇αV α = (1V )β −∇βv = 0.

It therefore suffices to choosev = 0 to haveV = 0, as required. 2An example of the application of Theorem 3 is given by the definition of 4-

vector potential and of the Lorentz gauge in electromagnetism (see Section 8).Note that from the splitting of (31) we have

Fi0 = Ei = ∇iV0−∇0Vi + (CurlW)i,

(∗F)i0 = Hi = −∇iW0+∇0Wi + (CurlV )i,(38)


which permits us identify the components of the 4-vector potentials ofF withthe scalar and 3-vector potentials of the electric and magnetic partsEi andHi

according to the generalized Clebsch Theorem (see Corollary 2.1).Decomposition formula (31) is first-order, since it involves first covariant deriv-

atives of the vector potentials only. However, Theorem 2 allows us to again in-troduce potentials for the vector potentials we have just met. We thus can provea second-order decomposition formula, in terms of second derivatives of skew-symmetric tensor potentials.

COROLLARY 3.2. LetF be a given skew-symmetric2-tensor-distribution and letV and W be two given divergence-free vector distributions with a past-compact[ future-compact] support. Then there exist two unique skew-symmetric tensor di-stributionsM andN , with a past-compact[ future-compact] support, such that

Fαβ = ∇α∇µMµβ −∇β∇µMµ

α + ηαβµν∇µ∇σNσν (39)

and that

Vβ = ∇α(∗M)αβ, Wβ = ∇α(∗N)αβ. (40)

Proof. For any choice ofv,w, consider the two vector potentialsV andW ofTheorem 2. Now let us introduce their decomposition according to Theorem 1 andassociated to the givenV andW so as to additionally satisfy (40), and let us denotetheir skew-symmetric potentialsM andN , respectively:

Vα = ∇αφ + ∇µMµα, Wα = ∇αψ +∇µNµ

α, (41)

where

v = ∇αV α = 1φ, w = ∇αWα = 1ψ.By substituting (41) into (31), the gradients disappear, independently of the choiceof v andw, and we obtain (39), as required. 2

We also have the analogue of Corollary 3.1 whenF is a divergence-free field.

COROLLARY 3.3. In the hypothesis of Theorem3, if F is divergence-free, thenthere is a skew-symmetric potentialN such that

Fαβ = ηαβµν∇µ∇σNσν. (42)

Proof.To prove this corollary, we can make use of (39) and try to show that wenecessarily have∇µMµ

α = 0. Actually we find∇αF αβ = 1(∇µMµβ) = 0 and

thus the result is proved. 2Further substitution of decomposition (31) into (15) does not lead to any new

interesting second-order decomposition. It simply turns out to lead to the statement


that any vector is equal to the Laplace operator of some other vector, which is trivialas a consequence of Lemma 1.

6. Decomposition of a Symmetric 2-Tensor

We are now going to introduce a covariant four-dimensional first-order decomposi-tion of a symmetric 2-tensor distributionT . Such a decomposition generalizes theusual transverse decomposition on a spacelike hypersurface [8, 29]. Here we aregoing to make use of a different operator than the generalized Laplace operator.Also, for the transverse decomposition on a spacelike hypersurface one has tointroduce a formally similar operator different than the ordinary three-dimensionalelliptic Laplace operator. The difference is that the operator we are going to usehere is hyperbolic like the generalized Laplace operator, while that of the three-dimensional case is elliptic like the ordinary one.

THEOREM 4. Let Tαβ be a generic symmetric2-tensor distribution with past-compact[ future-compact] support. Then there exist a unique vector-distributionTα and a unique symmetric divergence-free2-tensor distributionKαβ , with a past-compact[ future-compact] support, such that the following decomposition formulaholds:

Tαβ = ∇αTβ +∇βTα +Kαβ. (43)

Proof. For (43) to hold,Tβ must be a solution of the following differentialequation:

∇ν∇νTβ +∇ν∇βT ν = ∇νT νβ. (44)

If such a solution is found, then it suffices to defineKαβ = Tαβ − 2∇(αTβ) to havethe thesis of our theorem. Actually, Equation (44) admits a unique solution withinthe framework of tensor distributions with a past-compact [future-compact] support(or in that ofC∞ functions), since the differential operatorgαβ∇ν∇ν +∇α∇β thereinvolved is hyperbolic and self-adjoint. 2

This decomposition is a generalization of the transverse decomposition on aspacelike hypersurface [8, 29]. IfT ik is a generic symmetric spatial 2-tensor dis-tribution, then there is, in fact, a unique spatial vector distributionT i and a uniquesymmetric spatial and divergence-free 2-tensor distributionKik such that the fol-lowing decomposition holds:

T ik = ∇ iT k +∇kT i +Kik, ∇kKik = 0, (45)

and that the following equation is satisfied:

∇kT ik = ∇k∇kT i + ∇k∇ iT k. (46)


T i is well-defined as the unique solution of Equation (46), since the differentialoperator involved there is elliptic and self-adjoint [8, 29]. If the domain ofT ik isa single ordinary three-dimensional Riemann manifold, then decomposition (46)coincides with the ordinary transverse decomposition considered in [8, 29].

Let us now look for the relationship between our four-dimensional decomposi-tion (43) and the ordinary three-dimensional one. Consider coordinates adapted toa reference frame, like in the hypothesis of Corollary 2.1. The splitting of decom-position (43) is the following:

T 00 = 2∇0T 0+K00,

T 0i = ∇0T i +∇ iT 0+K0i , (47)

T ik = ∇ iT k +∇kT i +Kik,

while that of system (44) is the following:

∇0T00+∇kT 0k = 2∇0∇0T 0+∇k(∇kT 0+∇0T k),

∇0T0i +∇kT ik = ∇k∇kT i +∇k∇ iT k +∇0(∇0T i +∇ iT 0).

(48)

We have just proved the following corollary.

COROLLARY 4.1. In the hypothesis of Theorem 4, in a given local chart adaptedto a reference frame, the spatial2-tensor distributionT ik admits the ordinarytransverse decomposition(45), where the spatial vectorT i is the solution of(46),if and only if we have∇0K

i0 = 0.

The proof of this corollary easily follows from substitution of (47) into (48).Even more trivially, if we have∇0K

i0 = 0, then we necessarily also have∇kKik =0 (sinceK is divergence-free), thus both (45) and (46) are automatically satisfied.

We then have that the ordinary transverse decomposition on a three-dimensionalmanifold6 directly follows from (43) when the metric is independent of time.

COROLLARY 4.2. In a given local chart adapted to a reference frame, let themetric tensor be independent ofx0. Then, ifT is a symmetric2-tensor distributionwhich is also independent ofx0, then the spatial2-tensor distributionT ik admitsthe ordinary transverse decomposition(45), where the spatial vectorT i is thesolution of(46).

The proof follows from Corollary 4.1, by the same argument as that containedin the proof of Corollary 2.2. In this case we have∇0K

0i = ∂0K0i = 0.

Let us now look for second-order potentials of a symmetric 2-tensor.First of all, the potential vectorT appearing in decomposition (43) can in turn

be decomposed according to Theorem 2:

Tα = (1/2)∇αφ + ∇µFµα (49)


so to obtain the following second-order decomposition formula:

Tαβ = ∇α∇βφ + ∇α∇µFµβ +∇β∇µFµα +Kαβ. (50)

This proves the following corollary:

COROLLARY 4.3. In the hypothesis of Theorem4, if W is a given divergence-free vector distribution with a past-compact[ future-compact] support, then thereexist a unique distributionφ, a unique antisymmetric2-tensor distributionF and aunique symmetric divergence-free2-tensor distributionKαβ , with a past-compact[ future-compact] support, such thatWα = ∇β(∗F)βα and that(50) holds.

The splitting of (50) in turn gives the following decomposition for the spatialcomponent ofT in terms of second derivatives of the electric and magnetic partsof the second-order potentialF :

T ik = ∇ i∇kφ +∇ i∇0Ek +∇k∇0E

i−−∇ i (CurlH)k −∇k(CurlH)i +Kik,

(51)

while, for the remaining components, we have

T 00 = (∇0)2φ − 2∇0∇kEk,T 0i = ∇0∇ iφ − (∇0)2Ei −∇0(CurlH)i −∇ i∇kEk +K0i .

(52)

The second-order decomposition (51) can also be obtained by replacingT i in (45)with its decomposition (19) according to the generalized Clebsch theorem (seeCorollary 2.1). Obviously, in the hypothesis of Corollaries 2.2 and 4.2, the term∇ i∇0E

k +∇k∇0Ei disappears.

In decomposition (43), the form of the transverse componentK is generallyunknown; thus the relative second-order decomposition (50) is not complete, likein the preceeding section cases.

We can, however, prescribe the form ofK, at least in the particular case of a flatspacetime, as we are going to show in the next section. In this case, in fact, we canintroduce a double 2-formH as a second-order potential forK.

7. Decomposition of Symmetric 2-Tensors and Double 2-Forms inFlat Spacetime

We recall that a 4-tensorH is called a double 2-form (or a symmetric doublebivector) if it has the same algebraic properties of a curvature tensor, i.e.

Hαβρσ = H[αβ]ρσ = Hαβ[ρσ ] = Hρσαβ, H[αβρ]σ = 0. (53)

THEOREM 5. Let the spacetime be flat, letK be a given divergence-free symmet-ric 2-tensor distribution with a past-compact[ future-compact] support. Then there


is a unique double2-form distributionH with a past-compact[ future-compact]support such that∂[αHβρ]σν = 0 and that

Kαβ = ∂µ∂νHµανβ. (54)

Proof. First, let us note that∂[αHβρ]σν = 0 can be equivalently written as∂µ(∗H∗)µανβ = 0, where(∗H∗)µανβ = (1/4)εµαλδενβρσHλδρσ . Now let us sup-pose that (54) holds; we equivalently have

Kαβ = (1/4)εµαλγ ενβρσ ∂µ∂ν(∗H∗)λγρσ . (55)

We thus obtain

εφαψεεγβλµKαβ = ∂[γ ∂ [φ(∗H∗)εψ]µλ] (56)

and, consequently,

εφαψεεγβλµ∂ε∂µKαβ = [12(∗H∗)]ψφλγ ++1[∂φ∂ε(∗H∗)εφλγ + ∂γ ∂µ(∗H∗)ψφµλ] + (57)

+1[∂λ∂µ(∗H∗)ψφγµ + ∂ψ∂ε(∗H∗)φελγ ].Then, if we also suppose that∂µ(∗H∗)µανβ = 0, we simply have the followingidentity:

[12(∗H∗)]ψφλγ = εφαψεεγβλµ∂ε∂µKαβ. (58)

As a differential equation for the unknown∗H∗, however, Equation (58) cer-tainly admits a unique solution as a consequence of Lemma 1. The equivalent dualequation forH is

(12H)ψφλγ = ∂γ ∂φKψλ + ∂λ∂ψKφγ − ∂λ∂φKψγ − ∂γ ∂ψKφλ. (59)

Let thenHψφλγ be the solution of (59) or, equivalently, the dual of the solution of(58). It is easy to check thatHψφλγ necessarily has all the required algebraic anddifferential properties. Now letDαβ = Kαβ − ∂µ∂νHµ

ανβ . By definition, we have

∂αDαβ = 0 andD = Dα

α = K + ∂µ∂νHµν , whereHµν = Hαµνα andK = Kαα .

Moreover, from (59) we have

(12H)φλ + ∂λ∂φK = −1Kφλ. (60)

Consequently, sinceKαβ is divergence-free,

12(∂φ∂λHφλ +K) = 0 (61)

and we can conclude thatD = 0. Furthermore,Dαβ is by definition the solution ofεφαψεεγβλµ∂ε∂µDαβ = 0 or, equivalently,

∂γ ∂φDψλ + ∂λ∂ψDφγ − ∂λ∂φDψγ − ∂γ ∂ψDφλ = 0. (62)


From (62) we then have

1Dψλ = ∂λ∂ψD + ∂λ∂γDψγ − ∂ψ∂γDγλ = 0. (63)

We conclude thatDαβ = 0 and our theorem is proved. 2Moreover, we have the following corollary, whose proof easily follows

from (60):

COROLLARY 5.1. In the hypothesis of Theorem5, if Kαβ is trace-free, then wehave(1H)αβ = −Kαβ andH = 0.

It is, in general, not possible to directly extend such a result to the case of curvedspacetime, since in such a case one looses the possibility of rendering explicit theproperty ofK being divergence-free by introducing the double 2-form potentialH :one in fact has that∇α∇µHαµβν is not necessarily null.

Decompositions (54) and (50) lead to the following formula:

Tαβ = ∂α∂βφ + ∂α∂µFµβ + ∂β∂µFµα + ∂µ∂νHµανβ. (64)

We have just proved the following corollary.

COROLLARY 5.2. In the hypothesis of Theorem4, if the spacetime is flat andW isa given divergence-free vector distribution with a past-compact[ future-compact]support, then there exist a unique distributionφ, a unique antisymmetric2-tensordistributionF and a unique double2-form distributionHαβρσ , with a past-compact[ future-compact] support, such thatWα = ∂β(∗F)βα and that(64) holds.

From the splitting of (54) we have, for the spatial components ofK,

Kik = ∂m∂nHmink + ∂n(H 0ink + H ni0k)+ H 0i0k. (65)

From (59) we have that in the particular case whereKαβ is independent ofx0,the same applies for the double 2-form potential. We thus have the followingcorollaries:

COROLLARY 5.3. In a given local chart adapted to a reference frame in a flatspacetime, letK be a divergence-free symmetric2-tensor distribution, independentof x0. Then the spatial2-tensor distributionKik is such that∂iKik = 0 and admitsthe following decomposition:

Kik = ∂m∂nHmink. (66)

COROLLARY 5.4. In a given local chart adapted to a reference frame in a flatspacetime, letT be a generic symmetric2-tensor distribution, independent ofx0.Then the spatial2-tensor distributionT ik admits the following decomposition:

T ik = ∂iT k + ∂kT i + ∂m∂nHmink. (67)


Apparently there seem to be no obvious ways to directly prove formulae (66)and (67) in three dimensions: one cannot introduce the adjoint of a double 2-formby means of the three-dimensional Levi–Civita indicatorεijk only.

We can also prove the converse of Theorem 5.

THEOREM 6. Let the spacetime be flat, letH be a given double2-form dis-tribution with past-compact[ future-compact] support such that∂[αHβρ]σν = 0.Then there is a unique divergence-free symmetric2-tensor distribution with a past-compact[ future-compact] support such that

Hψφλγ = ∂γ ∂φKψλ + ∂λ∂ψKφγ − ∂λ∂φKψγ − ∂γ ∂ψKφλ. (68)

Proof.LetHψγ = Hαψγα andH = Hαα. Suppose (68) holds, then by contrac-tion, one has

1Kψγ = −Hψγ − ∂γ ∂ψK, (69)

whereK = Kαα. Consequently,

1K = −(1/2)H. (70)

Thus, (69) and (70) actually are necessary conditions for (68) to hold. However, wecan solve (70) with respect to the unknown scalar distributionK and subsequently(69) with respect to the unknown tensor distributionKψγ . The results are compat-ible in the sense thatK is the trace of the resultingKψγ . Moreover, from (69) and(70) one has

1∂ψKψγ = −∂ψHψγ + (1/2)∂γH, (71)

which is null by contraction of∂[αHβρ]νσ = 0. ThusKψγ is also divergence-free.We now have to show that (69) and (70) are also sufficient. To see it, simply let

Dψφλγ = Hψφλγ − ∂γ ∂φKψλ − ∂λ∂ψKφγ + ∂λ∂φKψγ + ∂γ ∂ψKφλ. (72)

From (69) we therefore have

1Dψφλγ = 1Hψφλγ + ∂γ ∂φHψλ + ∂λ∂ψHφγ − ∂λ∂φHψγ − ∂γ ∂ψHφλ

= gαβ(∂α∂βHψφλγ + ∂γ ∂φHαψλβ − ∂γ ∂ψHαφλβ +

+ ∂λ∂ψHαφγβ − ∂λ∂φHαψγβ) (73)

and, consequently, from∂[αHβρ]νσ = 0, we find

1Dψφλγ = gαβ∂α∂ [βHλγ ]ψφ = 0, (74)

i.e.D = 0 and our theorem is proved. 2


From (70), we also immediately have the following corollary:

COROLLARY 6.1. In the hypothesis of Theorem6, if the double-traceH ofHαβρσis null, then the potentialKαβ is trace-free and(1K)αβ = −Hαβ.

Finally, if the double 2-form distributionH is independent ofx0, the same hap-pens to its 2-tensor-distribution potentialK, which leads to the following corollary.

COROLLARY 6.2. In the hypothesis of Theorem6, if Hαβρσ is independent ofx0,one has

Hijkn = ∂n∂jKik + ∂k∂iKjn − ∂k∂jKin − ∂n∂iKjk, (75)

where∂nKin = 0.

8. An Application to Electromagnetism

As an example of the application of Theorem 1, we have the existence and unique-ness theorem for weak solutions of the Maxwell equations when the electric densitycurrent vector is replaced by a generic vector-distribution. Such vector distributioncould also be singular to describe, for example, regularly discontinuous currents orconcentrated charges.

COROLLARY 1.1. Let the electric density current4-vector distributionJ α (with∇αJ α = 0) have past-compact[ future-compact] support. Then there is a uniqueelectromagnetic field tensor distribution with a past-compact[ future-compact]supportFαβ such that

∇αF αβ = J β, ∇α(∗F)αβ = 0. (76)

The proof directly follows from Theorem 1 provided we setV α = J α andWα = 0.Moreover, as an example of the application of Theorem 3, we have the following

definition of the 4-vector potential:

COROLLARY 3.4. LetFαβ be a solution of(76), then, for any couple of scalar-distributionsv andw, there are two vector-distributionsV andW such that(31)holds,V satisfies the following equation:

(1V )β = Jβ +∇βv (77)

andW is determined by

(1W)β = ∇βw. (78)

In other words, the Maxwell equations (76) are equivalent to (77) for any choiceof v, whereV is the 4-vector distribution potential of the electromagnetic fieldF .


The expression (31) ofF actually also involves the inessential vectorW , which isdetermined by (78), i.e. by the choice ofw, but which does not appear in (77). Itsuffices to choosew = 0 to haveW = 0 and, from (31),

Fαβ = ∇αVβ −∇βVα. (79)

The freedom of choice of the scalarv is instead the gauge freedom of electromag-netism. The choicev = 0 defines what we can call the Lorentz gauge.

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217

Asymptotic Completeness for a RenormalizedNonrelativistic Hamiltonian in QuantumField Theory: The Nelson Model

ZIED AMMARICentre de Mathématiques, UMR 7640 CNRS, École Polytechnique, 91128 Palaiseau Cedex, France.e-mail: [email protected]

(Received: 10 January 2000; in final form: 11 July 2000)

Abstract. Scattering theory for the Nelson model is studied. We show Rosen estimates and we provethe existence of a ground state for the Nelson Hamiltonian. Also we prove that it has a locally finitepure point spectrum outside its thresholds. We study the asymptotic fields and the existence of thewave operators. Finally we show asymptotic completeness for the Nelson Hamiltonian.

Mathematics Subject Classifications (2000):81U10, 81T10.

Key words: quantum field theory, Mourre theory, scattering theory, asymptotic completeness.

Table of Contents

1 Introduction 2182 Presentation of the Model 219

2.1 Basic definitions and notations 2192.2 Technical estimates 2222.3 The Nelson model 227

3 Construction of the Nelson Hamiltonian 2283.1 Dressing transformation 2293.2 Removal of the ultraviolet cutoff 231

4 Higher Order Estimates 2344.1 Rosen estimates 2354.2 Number-energy estimates 2454.3 Commutator estimates 246

5 Spectral Theory for the Nelson Hamiltonian 2505.1 HVZ theorem 2515.2 Mourre estimate 252

6 Construction of the Wave Operators 2606.1 Asymptotic fields 2606.2 Wave operators 266

7 Propagation Estimates 2688 Asymptotic Completeness 275

Appendix 281

218 ZIED AMMARI

1. Introduction

Recently there has been a renewed interest in quantum field theory models thatdescribe a system of nonrelativistic particles interacting with a bosonic field. Themain physical example is a nonrelativistic atom interacting with photons. For thismodel the existence of a ground state was established in [BFS], [AH]. The absenceof excited states was also shown in [BFS] using a renormalization group analysisand in [BFSS], [DJ] using a positive commutator method.

Another result related to the present work is [DG2] where for a model of aconfined atom interacting with massive bosons the asymptotic completeness ofwave operators was proved.

In all these works, the model contains an ultraviolet cutoff which switches offthe interaction between the nucleons and the bosonic field above a certain mo-mentum scale. This can be justified physically by the fact that nucleons interactingwith bosons of very high energy will become relativistic and in such a situation themodel will anyhow loose its validity.

With a cutoff these models are free of ultraviolet divergences and, hence, canbe easily constructed rigorously by rather elementary methods.

However the presence of a cutoff implies that the interaction term is now nonlocal and that quantitative results depend on the choice of the cutoff scale. There-fore it would be more satisfactory to remove the ultraviolet cutoff from the modelunder consideration.

When the interaction term is linear in the field variables, the removal of theultraviolet cutoff was done long time ago by Nelson [Ne]. This was probably thefirst model which was rigorously constructed using a renormalization procedure. Itconsists in considering cutoff HamiltoniansHκ , whereκ is some ultraviolet cutoffparameter and applying a cutoff-dependent unitary transformationUκ . After sub-stracting a divergent self-energy termEκ , the sequence of HamiltoniansUκ(Hκ −Eκ)U

∗κ converges in norm resolvent sense to a HamiltonianH∞ whenκ → ∞

while Uκ converges strongly to a unitary transformationU∞ (in other words, nochange of representation is necessary). The HamiltonianH := U ∗∞H∞U∞ is calledthe Nelson Hamiltonian. After Nelson’s paper, the Nelson model was studied byCannon [Ca] and Fröhlich [Fr].

In this paper we consider the Nelson model for a confined atom and massivebosons and study its spectral and scattering theory. Our main result is the as-ymptotic completeness of the wave operators, which implies the unitarity of theS matrix. The strategy and the proofs of our paper follow closely those of [DG2],which is devoted to a similar model with an ultraviolet cutoff. Nevertheless thereare new difficulties coming from the fact that the Nelson Hamiltonian is onlydefined as the resolvent limit of the cutoff Hamiltonians.

Let us now describe the content of the paper. In Section 2 we recall classi-cal notations related to Fock spaces, introduce some definitions and prove someextensions of Glimm–Jaffe’sNτ estimates.

A RENORMALIZED NONRELATIVISTIC HAMILTONIAN IN QFT 219

In Section 3 we recall the construction of the Nelson Hamiltonian following[Ne]. In Section 4 we prove the so-calledhigher order estimates, following an ideaof Rosen [Ro]. In Section 5 we study the spectral theory for the confined Nel-son Hamiltonian. We prove an HVZ theorem and a positive commutator estimate.Section 6 is devoted to the scattering theory for the Nelson model. We show the ex-istence of asymptotic fields and that the CCR representation they define is of Focktype, using an argument from [DG3]. In Section 7 we prove various propagationestimates for the Nelson Hamiltonian. Finally the asymptotic completeness of thewave operators is shown in Section 8.

2. Presentation of the Model

In this section we define the Nelson model. We start with a review of the basicconstruction and the main notations related to bosonic Fock spaces. For a moredetailed exposition we refer the reader to [Be], [BR], [BSZ]. In Subsection 2.2we give some technical estimates obtained by adaptation ofNτ -estimates [GJ].Finally in Subsection 2.3 we introduce the formal Hamiltonian of the interactingsystem ofP confined nonrelativistic particles (nucleons) with a relativistic scalarfield (mesons). In order to give sense to the formal Hamiltonian, we put a high-momentum cutoff in the interaction and we show that the cutoff Hamiltonian is awell defined selfadjoint operator.

2.1. BASIC DEFINITIONS AND NOTATIONS

Let h be a complex Hilbert space. Let⊗nsh denote the symmetricn-fold tensorpower ofh. We introduce thebosonic Fock spaceby 0(h) :=⊕n>0⊗ns h.⊗0

sh :=C, identified as subspace of0(h), represents the space of zero-particle states. Wedenote by� the vector(1,0, . . .) usually calledvacuum vectorand by0fin(h) thesubspace of finite particle states, which is the subspace of finite sum of vectors in⊗nsh. Among the main operators acting on0(h), we will first recall the definitionsof the most familiar asnumber operator N given in its spectral decompositionN |⊗ns h := n1. Creation operators, which are unbounded operators densely defined

onD(N12 ), are given by

a∗(h)|⊗nsh :=√(n+ 1)Sn+1h⊗ 1⊗nsh,

whereSn denotes the orthogonal projection from⊗nh into ⊗nsh. The annihilationoperatora(h) is the adjoint ofa∗(h). We will use the notationa] for a or a∗.

We define thefield operatorby

φ(h) := 1√2(a∗(h)+ a(h)), h ∈ h.

φ(h) is essentially selfadjoint on0fin(h). We still denote byφ(h) its closure. Byfunctional calculus we get unitary operators calledWeyl operators, defined asW(h):=eiφ(h). We recall a useful differentiation estimate forW(h):

220 ZIED AMMARI

Let 06 ε 6 1,

‖(W(h1)−W(h2))u‖6 Cε ‖h1− h2‖ε((‖h1‖2 + ‖h2‖2) ε2‖u‖ + ‖(N + 1)

ε2u‖), (2.1)

lims→0

sup‖h‖6c

s−1‖(W(sh)− 1− isφ(h))(N + 1)−12−ε‖ = 0, ε > 0. (2.2)

Let h be a Hilbert space. Letf : h→ h be an (unbounded) operator. We denoteby d0(f) the amplification off to the whole space0(h)

d0(f)|⊗ns h :=n∑j=1

1⊗(j−1) ⊗ f⊗ 1⊗(n−j).

Let hi , i = 1,2 be two Hilbert spaces. Letg : h1→ h2 be a bounded operator. Wedefine the operator0(g) by

0(g) : h1→ h2,

0(g)|⊗ns h1 := g⊗(n).

A less familiar operator is d0(f, g), wheref, g are two operators onh1 into h2. It isdefined as in [DG2]:

d0(f, g) : 0(h1)→ 0(h2),

d0(f, g)|⊗ns h1 :=n∑j=1

f⊗(j−1) ⊗ g⊗ f⊗(n−j) .

We notice that d0(f, f) = N0(f) and if h1 = h2, we have d0(1, g) = d0(g). If‖f‖ 6 1, the following inequality holds

‖N− 12 d0(f, g)u‖ 6 ‖d0(g∗g) 1

2u‖. (2.3)

Let i1 (resp. i2) be the injection ofh1 (resp. h2) into h1 ⊕ h2. There exists a unitarytransformationU identifying0(h1)⊗ 0(h2) with 0(h1⊕ h2), defined as follows

Uu⊗ v :=√(p + q)!p!q! Sp+q0(i1)u⊗ 0(i2)v, u ∈ ⊗ps h1, v ∈ ⊗qs h2.

This transformation has the following properties:

(i) U�⊗� = �.(ii) Let h1 ∈ h1, h2 ∈ h2

a](h1⊕ h2)U = U(a](h1)⊗ 1+ 1⊗ a](h2)),

φ(h1⊕ h2)U = U(φ(h1)⊗ 1+ 1⊗ φ(h2)).

(iii) Let fi : hi → hi , i = 1,2 be two operators


d0(f1⊕ f2)U = U(d0(f1)⊗ 1+ 1⊗ d0(f2)),

U 0(f1)⊗ 0(f2) = 0(f1⊕ f2).

We define thescattering identification operatorI

I : 0fin(h)⊗ 0fin(h)→ 0fin(h),

Iu⊗ v :=√(p + q)!p!q! Sp+q u⊗ v, u ∈ ⊗ps h, v ∈ ⊗qs h.

We can also defineI by the following formula:

I

p∏i=1

a∗(hi)�⊗q∏i=1

a∗(gi)� :=q∏i=1

a∗(gi)p∏i=1

a∗(hi)�, hi, gi ∈ h.

Let π be the following map

π : h⊕ h→ h,

(h0, h∞)→ h0+ h∞.Then we can expressI as followingI = 0(π)U . We notice thatI is unboundedsince‖π‖ = √2.

Let i = (i0, i∞) be a pair of maps fromh to h. We define

I (i) : 0fin(h)⊗ 0fin(h)→ 0fin(h),

I (i) := I 0(i0)⊗ 0(i∞).Let i = (i0, i∞), j = (j0, j∞) be two pairs of maps fromh to h. We define

dI (i, j) : 0fin(h)⊗ 0fin(h)→ 0fin(h),

dI (i, j) := I (d0(i0, j0)⊗ 0(i∞)+ 0(i0)⊗ d0(i0, j∞)).

If i0i∗0 + i∞i∗∞ 6 1 we have the estimates

‖(N0 +N∞)− 12 dI ∗(i, j)u‖ 6 ‖d0(j0j

∗0 + j∞j ∗∞)

12u‖, (2.4)

|(u2|dI ∗(i, j)u1)|6 ‖d0(|j0|) 1

2 ⊗ 1 u2‖ ‖d0(|j0|) 12u1‖+ (2.5)

+‖1⊗ d0(|j∞|) 12u2‖ ‖d0(|j∞|) 1

2u1‖.For other properties and equivalent definitions of these operators, we refer thereader to [DG3].

222 ZIED AMMARI

Let K be an auxiliary Hilbert space. Letv ∈ B(K,K ⊗ h). We define anextended creation operator:

a∗(v) :K ⊗ 0fin(h)→K ⊗ 0fin(h),

a∗(v)|K⊗⊗ns h :=√(n+ 1) (1K ⊗ Sn+1)v ⊗ 1⊗nsh.

a∗(v) is closable densely defined operator since its adjointa(v) is densely defined.We define the field operatorφ(v) as in the scalar case. Whenh = L2(Rd,dk),v ∈ B(K,K ⊗ h) can be represented as a functionk→ v(k) ∈ B(K), such thatfor x ∈K, v(k)x := vx(k), k-a.e. and

K ×K 3 (x, y)→∫(v(k)∗ v(k)x| y)K dk = (vx|vy)K⊗h

is a continuous quadratic form. A stronger condition isv ∈ L2(Rd,B(K)), i.e.:∫‖v(k)‖2B(K) dk < ∞.

Assume that[v∗1(k), v2(k′)] = 0, ∀k, k′. Then:

[a(v1), a∗(v2)] = v∗1 v2⊗ 10(h),

[φ(v1), φ(v2)] = iIm(v∗1 v2) ⊗ 10(h),

[φ(v1),W(v2)] = Im(v∗1 v2)⊗ W(v2).

2.2. TECHNICAL ESTIMATES

In this subsection we will collect some technical estimates which are adaptation ofGlimm–Jaffe’sNτ -estimates.

We recall the symbolic annihilation and creation operators in the case of a Fockspace constructed over the space of square integrable functionsh := L2(Rd,dk).Let9 ∈ 0fin(h):

(a(k)9)(n)(k1, . . . , kn) := (n+ 1)12 9(n+1)(k, k1, . . . , kn),

(a∗(k)9)(n)(k1, . . . , kn)

:= n−12

n∑j=1

δ(k − kj )9(n−1)(k1, . . . , kj , . . . , kn),

where kj means thatkj is omitted. LetS(Rd) be the Schwartz space. We candefine the monomiala(k1) . . . a(ks) as an operator from0fin(S(Rd)) into S(Rds)⊗0fin(S(Rd)).

LetK be an auxiliary Hilbert space. Letw be unbounded operator fromK⊗h⊗sintoK⊗h⊗r , with a domain containingK⊗S(Rds). A Wick monomialwith symbolw is the following sesquilinear form onK ⊗ 0fin(S(Rd))

Wr,s := a∗(k′1) . . . a∗(k′r ) w a(k1) . . . a(ks).


Letω be a positive regular function satisfying:

ω ∈ C∞(Rd),|∂αk ω(k)| 6 cα(1+ |k|)Nα for α ∈ Nd,ω(k) > m > 0.

We setNτ := d0(ωτ ).

LEMMA 2.1. Let r be a positive integer andτ := (τi)1...r be a sequence of real

numbers. Forψ ∈ D(∏ri=1N

12τi ):∥∥∥∥∥

r∏i=1

N12τi ψ

∥∥∥∥∥2

=r∑j=1

∫P τr,j (k1, . . . , kj )

∥∥∥∥∥j∏i=1

a(ki)ψ

∥∥∥∥∥2

dk1 . . .dkj , (2.6)

whereP τr,j is a sum of homogeneous functions in the variablesω(ki) of degree∑ri=1 τi and satisfying

P τr,j (k1, . . . , kj ) =∑

(is )1...r∈Sr,j

ω(ki1)τ1 . . . ω(kir )

τr , (2.7)

whereSr,j is the set, constructed by induction, of surjective mapsi from {1, . . . , r}into {1, . . . , j}, such thatis 6 s and (is)1...r−1 is in Sr−1,j or in Sr−1,j−1 andSj,j+1 = S1,0 = ∅.

Proof.∏j

i=1 a(ki) can be defined as operator on0fin(S(Rd)) into S(Rdj ) ⊗0fin(S(Rd)). Forψ ∈ 0fin(S(Rd)),

(k1, . . . , kj )→∥∥∥∥∥

j∏i=1

a(ki)ψ

∥∥∥∥∥2

is a function inS(Rdj ). This implies that the right-hand side of (2.6) is well defined

for ψ ∈ 0fin(S(Rd)). The hypothesis onω imply that0fin(S(Rd)) is a core forN12τ

and for∏ri=1N

12τi . If the lemma holds forψ ∈ 0fin(S(Rd)) then it can be extended to

ψ ∈ D(∏ri=1N

12τi ). In fact since0fin(S(Rd)) is a core for

∏ri=1N

12τi , then we can ex-

tend∏j

i=1 a(ki) to bounded operator fromD(∏ri=1N

12τi ) intoL2(Rdj , (P τr,j )

12 dk)⊗

0(h). Let us prove the lemma forψ ∈ 0fin(S(R3)) by induction inr.For r = 1,

‖N 12τ1ψ‖2 = (ψ |Nτ1ψ)

=∫ω(k)τ1‖a(k)ψ‖2 dk.

We see thatP τ11,1(k) = ω(k)τ1. (2.6), (2.7) are satisfied forr = 1. Assumethat (2.6), (2.7) hold forr. Using the fact thatNτ preserves0fin(S(Rd)) and the

224 ZIED AMMARI

induction hypothesis, we have∥∥∥∥∥r+1∏i=1

N12τi ψ

∥∥∥∥∥2

=r∑j=1

∫P τr,j (k1, . . . , kj )

∥∥∥∥∥j∏i=1

a(ki)N12τr+1ψ

∥∥∥∥∥2

dk

=r∑j=1

∫P τr,j

∥∥∥∥∥(Nτr+1 +

j∑i=1

ω(ki)τr+1

) 12 j∏i=1

a(ki)ψ

∥∥∥∥∥2

dk

=r∑j=1

∫P τr,j

{j∑i=1

ω(ki)τr+1

∥∥∥∥∥j∏i=1

a(ki)ψ

∥∥∥∥∥2

+∥∥∥∥∥N 1

2τr+1

j∏i=1

a(ki)ψ

∥∥∥∥∥2}

dk

=r+1∑j=1

∫ [P τr,j

j∑i=1

ω(ki)τr+1 + P τr,j−1ω(kj )

τr+1

] ∥∥∥∥∥j∏i=1

a(ki)ψ

∥∥∥∥∥2

dk,

wherePr,r+1 = Pr,0 = 0. Then we obtain the following iterated relation

P τr+1,j (k1, . . . , kj )

= P τr,j (k1, . . . , kj )

j∑i=1

ω(ki)τr+1 + P τr,j−1(k1, . . . , kj−1)ω(kj )

τr+1. (2.8)

We note thatP τr,r (k1, . . . , kr) = ∏ri=1ω(ki)

τi . It is easy to see, using inductionhypothesis (2.7) forr and (2.8), that

P τr+1,j (k1, . . . , kr+1) =∑

(is )1...r+1∈Sr+1,j

ω(ki1)τ1 . . . ω(kir+1)

τr+1. 2

COROLLARY 2.2. Let α, ν ,τ := (τi)1...r be a sequence of real numbers. For

ψ ∈ D(Nαν

∏ri=1N

12τi )∥∥∥∥∥Nα

ν

r∏i=1

N12τi ψ

∥∥∥∥∥2

=r∑j=1

∫P τr,j (k1, . . . , kj )

∥∥∥∥∥(Nν +

j∑i=1

ω(ki)ν

)α j∏i=1

a(ki)ψ

∥∥∥∥∥2

×

×dk1 . . .dkj , (2.9)

whereP τr,j is the function defined by(2.7).


LEMMA 2.3. Letp, q ∈ N and τ ′ := (τ ′i )1...p, τ := (τi)1...q be two sequences ofreal numbers. LetWr,s be a Wick monomial such thatr 6 p, s 6 q. Then∥∥∥∥∥

(p∏i=1

(Nτ ′i + 1)−12

)Wr,s

q∏i=1

(Nτi + 1)−12

∥∥∥∥∥6 ‖(P τ ′p,r )−

12 w (P τq,s)

− 12‖B(K⊗h⊗s ,K⊗h⊗r ).

Proof.Letψ,φ ∈K ⊗ 0fin(S(Rd)).∣∣(Wr,s ψ |φ)∣∣

=∣∣∣∣∣(w

s∏i=1

a(ki)ψ

∣∣∣∣∣r∏i=1

a(k′i )φ

)K⊗h⊗r⊗0(h)

∣∣∣∣∣6∣∣∣∣∣((P τ

′p,r)− 1

2 w (P τq,s)− 1

2 (P τs,q)12

s∏i=1

a(ki)ψ

∣∣∣∣∣(P τ ′p,r ) 12

r∏i=1

a(k′i )φ

)∣∣∣∣∣6 ‖(P τ ′p,r )−

12 w (P τq,s)

− 12‖B(K⊗h⊗s ,K⊗h⊗r )

(∫P τq,s

∥∥∥∥∥s∏i=1

a(ki)ψ

∥∥∥∥∥2

dk

) 12

×

×(∫

P τ′

p,r

∥∥∥∥∥r∏i=1

a(k′i )φ

∥∥∥∥∥2

dk′) 1

2

6 ‖(P τ ′p,r )−12 w (P τq,s)

− 12‖B(K⊗h⊗s ,K⊗h⊗r )

∥∥∥∥∥q∏i=1

N12τi ψ

∥∥∥∥∥×∥∥∥∥∥p∏i=1

N12τ ′iφ

∥∥∥∥∥.This inequality shows that the quadratic form

p∏i=1

(Nτ ′i + 1)−12Wr,s

q∏i=1

(Nτi + 1)−12

can be extended to a bounded operator with norm less than∥∥(P τ ′p,r)− 12w (P τq,s)

− 12∥∥

B(K⊗h⊗s ,K⊗h⊗r ). 2COROLLARY 2.4. Letvi ∈ B(K,K⊗h); i = 1, . . . , n. Then there existsc > 0such that

(i)

∥∥∥∥∥(N + 1)pn∏i=1

a](vi)(N + 1)−p−n2

∥∥∥∥∥ 6 c

n∏i=1

‖vi‖B(K,K⊗h).

(ii)

∥∥∥∥∥(N + 1)pn∏i=1

φ(vi)(N + 1)−p−n2

∥∥∥∥∥ 6 c

n∏i=1

‖vi‖B(K,K⊗h).

(iii ) Let r, s ∈ N, (τ ′i )1...r , (τi)1...s be sequences of real numbers. Then

226 ZIED AMMARI

∥∥∥∥∥(

r∏i=1

(Nτ ′i + 1)−12

)Wr,s

s∏i=1

(Nτi + 1)−12

∥∥∥∥∥6∥∥∥∥∥

r∏i=1

ω(k′i)− τ′i2 w

s∏i=1

ω(ki)− τi2∥∥∥∥∥

B(K⊗h⊗s ,K⊗h⊗r )

.

Proof. Clearly (i) gives (ii). Forp = 0 andn = 1, (i) follows from the lastlemma by takingω = 1 andr = p = 1, s = q = 0 or r = p = 0, s = q = 1. Forp, n ∈ N, commutation properties reduces the inequality to the casep = 0, n = 1.(iii) is a direct application of Lemma 2.3. 2

We recall now well known estimates, see [Ro].

LEMMA 2.5. Letb be a positive operator. Then

d0(bα1) 6 N1−α1d0(b)α1, where α1 6 1.

d0(bτ1)α1 6 d0(bτ2)α2d0(bτ3)α3,

whereα1 = α2+ α3, and α1τ1 = α2τ2+ α3τ3.

Combining Lemma 2.5 and Lemma 2.3 we obtain a slightly more general esti-mate.

LEMMA 2.6. Letr, s, p, q ∈ N. Let τ ′j := (τ ′ij )1...p, τj := (τ ij )1...q , j = 1 . . . 3

be sequences of real numbers such thatτ ′i1 = τ ′i2+ τ ′i3, τ i1 = τ i2 + τ i3. Then∥∥∥∥∥p∏i=1

(Nτ ′i2 + 1)−12 (Nτ ′i3 + 1)−

12Wr,s

q∏i=1

(Nτi2+ 1)−

12 (Nτi3

+ 1)−12

∥∥∥∥∥6 ‖(P τ ′1p,r )

− 12 w (P τ1q,s)

− 12‖.

The following estimate is an immediate application of Lemma 2.6.

COROLLARY 2.7. Letr, s, α, β be positive integers such thatα 6 r, β 6 s. Thenthe following assertion holds

‖(N + 1)−r−α

2 (d0(ω)+ 1)−α2 Wr,s (N + 1)−

s−β2 (d0(ω)+ 1)−

β2 ‖

6 inf{τ ′i,τi∈[0,1]|∑

τ ′i=α,∑τi=β}

∥∥∥∥∥r∏i=1

ω(k′i)− τ′i2 w

s∏i=1

ω(ki)− τi2∥∥∥∥∥

B(K⊗h⊗s ,K⊗h⊗r )

.


2.3. THE NELSON MODEL

The Nelson model [Ne] describes a system ofP nonrelativistic particles coupledto a scalar relativistic field of bosons by a local, translation invariant interaction. Itexhibits a relatively mild ultraviolet divergence, and was the first QFT model, onwhich a renormalization procedure was rigorously carried on.

We consider the atomic Hamiltonian of the system ofP nonrelativistic confinedparticles as follows

K := 1

2M

P∑j=1

D2j + V (x1, . . . , xP ).

It acts on the Hilbert spaceL2(R3P ,dx) which we denote in the sequel byK. Weassume thatV ∈ L2

loc(R3P ) andV > 0. Kato’s inequality gives thatK is essentially

self-adjoint onC∞0 (R3P ), see [RS, I–IV, Thm. X.28]. We set〈x〉 := (|x|2 + 1)12 .

We will also assume

V > c∑i

〈xi〉α, α > 2.

We notice that for 06 β 6 1, 〈D〉β(K + 1)−β2 , 〈x〉β(K + 1)−

β2 are bounded

operators.The boson one particle space is the Hilbert spaceh := L2(R3,dk), wherek

denotes the boson momentum observable. The boson position observable−(∇k/i)will be denotes by the italic letterx. This should not be confused with the nucleonposition observable denoted by the roman letter x. The free bosonic Hamiltonian isdefined by the second quantization of a single boson energy. It acts on the bosonicFock space0(h).

Hb := d0(ω),

ω(k) := (|k|2+m2)12 , m > 0.

It is essentially self-adjoint on0fin(D(ω)).The Hilbert space of the joint system isH := K ⊗ 0(h). The Hamiltonian

without interaction is given by

H0 := K ⊗ 1+ 1⊗Hb.This is a self-adjoint operator sinceK ⊗ 1 and 1 ⊗ Hb commute onH . Thelocal translation invariant interaction between nucleons and bosons is given by theformal expression

P∑j=1

ϕ(xj ), whereϕ(x)

:= [2(2π)3]− 12

∫e−i〈k,x〉(a∗(k)+ a(−k)) dk

ω(k)12

.

228 ZIED AMMARI

The interaction term cannot be defined as an operator onH with a dense domain.This comes from the fact thatω−

12 /∈ L2(R3,dk), because the integral diverges for

largek. This phenomenon is known as anultraviolet problem. In order to have awell defined operator, one introduces cutoff interactions:

ϕκ(x) := [2(2π)3]− 12

∫e−i〈k,x〉(a∗(k)+ a(−k)) χκ(k)

ω(k)12

dk,

Iκ :=P∑j=1

ϕκ(xj ), vκ := [(2π)3]− 12χκ(k)

ω(k)12

e−i〈k,x〉.

Hereχ is a positive function inC∞(R3) such that 06 χ(k) 6 1, χ(k) = 1 for|k| 6 1,χ(k) = 0 for |k| > 2 andχ(−k) = χ(k). We setχκ(k) := χ(k/κ).LEMMA 2.8. One has forα > 0

(i) d0(ω)α 6 Hα0 ,

(ii) Kα 6 Hα0 .

Proof.Forψ ∈ C∞0 (R3)⊗ 0fin(S(R3)), which is a core forH0, one has

(d0(ω)ψ,ψ) 6 (H0ψ,ψ),

(Kψ,ψ) 6 (H0ψ,ψ).

This means that d0(ω) 6 H0 andK 6 H0. SinceH0,K and d0(ω) commute, thespectral theorem gives the inequalities announced in the lemma. 2Iκ are well-defined operators onD((N + 1)

12 ) as long asκ < ∞ and they are

H0-bounded with infinitesimal bound. We setHκ := H0+ Iκ .THEOREM 2.9. For κ <∞, Hκ is a self-adjoint operator onD(H0).

Proof.Using Corollary 2.4 and Lemma 2.8, we prove forc independent fromλ

‖Iκ(H0+ λ)−1‖ 6 c λ−12 ‖vκ‖.

Then by the Kato–Rellich theorem, one sees thatHκ is a selfadjoint operator onD(H0). 2

3. Construction of the Nelson Hamiltonian

In this section we recall the construction in [Ne] of the Nelson Hamiltonian. Itconsists in applying to the cutoff HamiltoniansHκ a cutoff dependent unitarytransformationUκ , letting thenκ to∞.


3.1. DRESSING TRANSFORMATION

For a fixedκ0 andκ <∞, we define

gκ(k) := −i(2π)− 3

2

ω12 (k)

χκ(k)− χκ0(k)

ω(k)+ k2

2M

∈ C∞0 (R3), (3.1)

Gκ :=P∑j=1

e−i〈k,xj 〉 gκ(k) ∈ B(K,K ⊗ h), (3.2)

Cκ := 1

2

∑16j,`6P

∫|gκ(k)|2 sin〈k, (xj − x`)〉dk ∈ S(R3P ), (3.3)

Eκ := P

2(2π)3

∫1

ω(k)

(χκ(k)− χκ0(k))2

(ω(k)+ k2

2M )dk−

− P

(2π)3

∫χκ(k)

ω(k)

χκ(k)− χκ0(k)

(ω(k)+ k2

2M )dk, (3.4)

rκ(x) := −ik e−i〈k,x〉gκ(k), (3.5)

Uκ := eiφ(Gκ )+iCκ , (3.6)

Hκ := Uκ(Hκ − Eκ)U ∗κ . (3.7)

In order to simplify the writing of some formulas, we will replace oftenrκ(xj ),vκ(xj ) by rjκ , v

jκ .

LEMMA 3.1. For a fixedκ0 and forκ0 < κ <∞, Hκ is a selfadjoint operator onthe domainD(H0) and equal to

Hκ = H0+∑

16i<j6PVκ(xi − xj )⊗ 1+ Iκ , (3.8)

where

Iκ := Iκ0 +1

2M

P∑j=1

Rj(rκ(xj )),

Rj (v) := 1

2a2(v)+ 1

2a∗

2(v) + a∗(v) a(v)−√2Dj a(v)−

√2a∗(v)Dj ,

Vκ(x) := Re∫ω(k) |gκ(k)|2 e−i〈k,x〉 dk − 2Im

∫gκ(k) vκ(k)e

−i〈k,x〉 dk.

230 ZIED AMMARI

Proof.We computeUκHκU ∗κ using commutation relations. We notice that termscoming fromHb andK have been mixed. This creates a translation invariantpotential and second-order interacting terms:

Uκ HbU∗κ = Hb − φ(iωGκ) +

+∑

16i<j6PV (1)κ (xi − xj )+ P2

∫ω(k)|gκ(k)|2 dk, (3.9)

where

V (1)κ (x) := Re

∫ω(k)|gκ(k)|2 e−i〈k,x〉 dk.

UκKU∗κ =

1

2M

P∑j=1

(Dj + φ(ik gκ(k)e−i〈k,xj 〉))2 +

+V (x1, . . . , xP ). (3.10)

Uκ IκU∗κ = Iκ +

∑16i<j6P

V (2)κ (xi − xj ) −

− P

(2π)3

∫χκ(k)

ω(k)

χκ(k)− χκ0(k)

ω(k)+ k2

2M

dk,

where

V (2)κ (x) := −2Im

∫gκ (k) vκ(k)e−i〈k,x〉 dk.

Using the following computation

(Dj − φ(rjκ ))2

= D2j +

1

2a2(rjκ )+

1

2a∗

2(rjκ )+ a∗(rjκ ) a(rjκ )−

√2Dja(r

jκ )−

−√2a∗(rjκ )Dj − φ(ik2e−i〈k,xj 〉gκ)+ 1

2

∫k2|gκ(k)|2 dk,

in the second term (3.10) and collecting similar terms together, we obtain (3.8).It is easy to see, by (3.9)–(3.10), thatUκ preservesD(H0) for κ < ∞, hence

D(Hκ) = UκD(H0) = D(H0). 2


3.2. REMOVAL OF THE ULTRAVIOLET CUTOFF

We set

g∞(k) := −i(2π)− 3

2

ω12 (k)

1− χκ0(k)

(ω(k)+ k2

2M ),

G∞ :=P∑j=1

e−i〈k,xj 〉g∞.

V∞ := Re∫ω(k) |g∞(k)|2 e−i〈k,x〉 dk −

−2Im∫g∞(k)

1

ω(k)12

e−i〈k,x〉 dk,

C∞ := 1

2

∑16j,`6P

∫|g∞(k)|2 sin〈k, (xj − x`)〉dk,

U∞ := eiφ(G∞)+iC∞.

LEMMA 3.2. Lettingκ to∞, we obtain the following limits:

(i) Eκ →−∞,(ii) gκ → g∞ in h,

(iii) Cκ → C∞ in L∞(R3P ),(iv) Vκ → V∞ in L∞(R3)+ Ls(R3) for s ∈ ]2,+∞[,(v) Uκ → U∞ strongly inB(H).

Proof. (i) is obvious. (ii) follows from the monotone convergence theorem. (iii)follows from (ii), since |gκ |2 converges inL1(R3). V (1)

κ converges inL∞(R3),which follows from the fact that its integrand converges inL1(R3). V (2)

κ convergesin Ls(R3) for s ∈ ]2,+∞[, by using Hausdorff–Young inequality. This proves (iv).Using the fact that the maph 3 v → eiφ(v) is strongly continuous and the limits(ii), (iii), we see thats-limk→∞ Uκ exists and is equal to eiφ(G∞)+iC∞. 2

We give now a lemma which will be useful in this section and in Section 4.

LEMMA 3.3. For s ∈ [0,1], andvi ∈ B(K,K ⊗ h), i = 1,2 we have:

(i) ‖(N + 1)−s2 a(v1) (H0+ 1)−

1−s2 ‖ 6 ‖ω s−1

2 v1‖B(K,K⊗h).

(ii) ‖(H0+ 1)− s2 a∗(v1) (N + 1)− 1−s2 ‖ 6 ‖ω− s2 v1‖B(K,K⊗h).

(iii ) ‖(N + 1)−s a(v1) a(v2) (H0+ 1)−1+s‖6 ‖ω− 1−s

2 v1‖B(K,K⊗h) ‖ω− 1−s2 v2‖B(K,K⊗h).

(iv) ‖(H0+ 1)−s a∗(v1)a∗(v2)(N + 1)−1+s‖

6 ‖ω− s2 v1‖B(K,K⊗h) ‖ω− s2 v2‖B(K,K⊗h).

232 ZIED AMMARI

Proof.We see clearly that suitable choice ofr, s, α, andβ in Corollary 2.7 givessimilar inequalities with d0(ω) in the place ofH0. Now using Lemma 2.8 we obtain(i)–(iv). 2LEMMA 3.4. One has forvi ∈ B(K,K ⊗ h), such that[V, vi] = 0, whereV isthe potential of the atomic Hamiltonian ands, β ∈ [0,1]:(i) ‖(N + 1)−

12a(v1) (H0+ 1)−

12‖ 6 c ‖(V + 1)−

s2ω

s−12 v1‖B(K,K⊗h).

(ii) ‖(H0+ 1)− 12a∗(v1) (N + 1)− 1

2‖ 6 c ‖(V + 1)− s2ω s−12 v1‖B(K,K⊗h).

(iii ) ‖(N + 1)−1a(v1)a(v2)(H0+ 1)−12‖

6 c ‖(V + 1)−βs2 ω

s−14 v1‖ × ‖(V + 1)−

(1−β)s2 ω

s−14 v2‖.

(iv) ‖(H0+ 1)−12a∗(v1)a

∗(v2)(N + 1)−1‖6 c ‖(V + 1)−

βs2 ω

s−14 v1‖ × ‖(V + 1)−

(1−β)s2 ω

s−14 v2‖.

(v) ‖(N + 1)− 12 (H0+ 1)− 1

2a∗(v1)a(v2)(H0+ 1)− 12‖

6 c‖(V + 1)−1−s

2 ω−s2v1‖B(K,K⊗h) × ‖(V + 1)−

s2ω−

1−s2 v2‖B(K,K⊗h).

Proof.Using Lemma 3.3, we see that (i)–(v) are true with(H0+ 1)−12 replaced

by (H0+1)−1−s

2 (V+1)−s2 . We use then the fact that(V+1)

s2 (H0+1)−

s2 is bounded

for s ∈ [0,1]. 2We set forκ <∞:

Bκ(φ) :=( ∑

16i<j6PVκ(xi − xj )+ Iκφ|φ

), φ ∈ D(H

12

0 ).

LEMMA 3.5. There existsκ0,0 6 a < 1,0 6 b independent fromκ, such thatfor κ > 2κ0

|Bκ(φ)| 6 a‖H12

0 φ‖2+ b‖φ‖2, φ ∈ D(H12

0 ).

Proof.One has using Lemma 3.3

‖(H0+ λ)− 12 Iκ (H0+ λ)− 1

2‖6 c(λ−

12‖ω− 1

2vκ0‖ + ‖ω−14 rκ‖2+ ‖ω− 1

2 rκ‖2+ ‖ω− 12 rκ‖). (3.11)

We notice thatω−14 rκ contains the termχκ − χκ0, for κ > 2κ0, which is arbitrarily

small for κ0 large enough. Using this fact and (3.11) we see that there existsκ0

such thatIκ has a smallH0-form bound forκ > 2κ0. The integrand ofV (1)κ , V (2)

κ

contain the termχκ −χκ0, then there existκ0 such thatV (1)κ (resp. V (2)

κ ) has a smallL∞(R3) (resp. Ls(R3), s > 2) norm for κ > 2κ0. After a change of variablesto separate the motion of the center of mass, the termV (2)

κ becomesV (3)κ + V (4)

κ ∈


L∞+Lq where 26 q <∞. Using the Sobolev injectionH 1(R3)→ Lq′(R3), 26

q ′ 6 6, we obtain, by a convenient choice ofq andq ′, thatV (2)κ isH0-form bounded

[Ne]. 2In all the sequel a limit of a sequence of operators written without a prefix shall

be understand as a norm limit.

THEOREM 3.6. There is a unique selfadjoint operatorH∞ acting onH , satisfy-ing

(i) limκ→∞(Hκ − z)−1 = (H∞ − z)−1,

(ii) s-limκ→∞ e−it Hκ = e−it H∞ for t ∈ R.

The domain ofH∞ satisfiesD(H∞) ⊂ D(H12∞) = D(H

12

0 ).Proof. The proof is based on the Theorem A.1 in the appendix. Let us apply

now this theorem with

Bκ(φ) =( ∑

16i<j6PVκ(xi − xj )+ Iκφ|φ

).

Using Lemma 3.3, one has forφ ∈ D(H12

0 )

|Bκ(φ)− Bκ ′(φ)|

6 c(‖V (1)κ − V (1)κ ′ ‖L∞ + ‖V (2)

κ − V (2)κ ′ ‖Ls + ‖ω−

12 (rκ − rκ ′)‖+

+‖ω− 14 (rκ − rκ ′)‖ + ‖ω− 1

4 (rκ − rκ ′)‖2)× ‖(H0+ 1)12φ‖2. (3.12)

We see clearly, using Lemma 3.2(ii) and (iv) in the right-hand side of (3.12), thatBκ satisfies hypothesis of Theorem A.1. The KLMN theorem applied forB∞, gives

thatD(H12∞) = D(H

12

0 ). So this proves the theorem. 2DEFINITION 3.7. The HamiltonianH := U ∗∞H∞U∞ is called the Nelson Hamil-tonian andH∞ the modified Hamiltonian.

THEOREM 3.8. One has

(i) lim κ→∞(Hκ − Eκ − z)−1 = (H − z)−1,(ii) s-limκ→∞ e−it (Hκ−Eκ) = e−itH for t ∈ R.

(iii) D(H) ⊂ D(H12 ) = U ∗∞D(H

12

0 ).

234 ZIED AMMARI

Proof.One has

‖U ∗κ (Hκ − z)−1Uκ − U ∗∞(H∞ − z)−1U∞‖6 ‖(U ∗κ − U ∗∞)(Hκ − z)−1‖++‖(U ∗κ − U ∗∞)(H∞ − z)−1‖++‖(Hκ − z)−1− (H∞ − z)−1‖. (3.13)

Using (3.13), Theorem 3.6 and the fact that the maph→ W(h)(N + 1)−ε, ε > 0,is norm continuous, we obtain

‖U ∗κ (Hκ − z)−1Uκ − U ∗∞(H∞ − z)−1U∞‖6 c(‖(W(Gκ)−W(G∞))(N + 1)−ε‖++‖(Hκ − z)−1− (H∞ − z)−1‖++‖eiCκ − eiC∞‖).

The application of Lemma 3.2 completes the proof of (i). (ii) follows from theequivalence of the convergence in the strong resolvent sense and the strong conver-gence of unitary groups (Trotter theorem). (iii) is obvious. 2

Let χ ′κ be another cutoff function and defineH ′ to be the Nelson Hamiltonianconstructed using the later cutoff.

PROPOSITION 3.9.There exists a finite constantE, such that

H ′ = H + E.Proof. We defineH ′′ (resp.H ′′κ ) to be the Nelson Hamiltonian (resp. the cutoff

modified Hamiltonian) obtained using the dressing transformation given by

g′′κ := −i(2π)− 3

2

ω12 (k)

χ ′κ (k)− χκ0(k)

ω(k)+ k2

2M

.

It is easy to see thatH ′ = H ′′ + E, whereE is a finite constant. Using similarcalculus with (3.12), we obtain thatH∞ = H ′′∞. SinceU∞ = U ′′∞ we haveH =H ′′. ThenH ′ = H + E. 2

4. Higher Order Estimates

In this section we prove some estimates which allow to bound powers ofN andH0 by powers ofH . They play an important role. Fröhlich has proved a higherestimates in the massless case [Fr], but they are different from what we intend toprove. Their proofs are based in the following principle of cutoff independence[Ro].


LEMMA 4.1. Let {Nj } and{Hj } be sequences of operators such that, for c inde-pendent of j,

‖Njψ‖ 6 c ‖Hjψ‖, for ψ ∈ D(Hj).

Suppose thatNj is self-adjoint and thatNj → N in the strong resolvent sense,where N is self-adjoint, andHj → H in the strong graph limit. Then

‖Nψ‖ 6 c ‖Hψ‖, for ψ ∈ D(H).

Let 0 (resp.0j ) denotes the graph ofH (resp.Hj ). We recall thatHj → H inthe strong graph sense if for all(ψ, ϕ) ∈ 0, there exist a sequence(ψj , ϕj ) ∈ 0jwhich converges to(ψ, ϕ) in H ×H .

The following easy first order estimate follows from the proof of Theorem 3.6.

LEMMA 4.2. There existsc > 0 independent forκ such that forκ0 6 κ 6∞H0 6 (Hκ + c).

4.1. ROSEN ESTIMATES

In this subsection we prove higher order estimates using a technique due to Glimmand Jaffe to prove similar estimates for theY2 and(ϕ4)2 models. It has been takenup by Rosen in [Ro] for the general(ϕ2n)2 model and Fröhlich [Fr]. This techniqueis based in the so calledpull-through formulawhich is the identity that we obtain,in a formal way, when we move the resolvent through a product of annihilationoperatorsa(ki). But some care must be taken when we want to rigorously provethe pull through formula, since we need a dense subspaceH0 ⊂ D(H0) on which∏i a(ki) acts as an operator and satisfies

∏i a(ki)H0 ⊂ D(H0) and

∏i a(ki) can

be defined as operator onH0H0. We need also a resolvent control of the com-mutator of the modified cutoff interaction witha(ki), which allows to define itas locally integrable function with values in bounded operators onD(H0). ThisrequiresH0H0 to be dense.

Let {Ji}1...j+1 be a set of disjoint subsets of{1, . . . , n} so that within each subsetJi the elements are taken in their natural order. We introduce the notation:

HJκ := [a(ki1), . . . , [a(kij ), Iκ] . . .], where J = {i1, . . . , ij },

R(z) := (z − Hκ)−1,

R`(z) :=(z−

∑i

ω(ki)− Hκ)−1

,

where the sum runs overi ∈ J` ∪ J`+1 ∪ · · · ∪ Jj+1.

236 ZIED AMMARI

LEMMA 4.3. Letα ,τ := (τi)1...r be a sequence of real numbers. For

ψ ∈ D

((H0+ 1)α

r∏i=1

N12τi

)

∥∥∥∥∥(H0+ 1)αr∏i=1

N12τi ψ

∥∥∥∥∥2

=r∑j=1

∫P τr,j (k1, . . . , kj )

∥∥∥∥∥(H0+

j∑i=1

ω(ki)+ 1

)α×

×j∏i=1

a(ki)ψ

∥∥∥∥∥2

dk1 . . .dkj . (4.1)

Proof.This lemma is similar to Corollary 2.2. We prove (4.1) forψ ∈ D(K)⊗0fin(S(R3)) and then we extend it toψ ∈ D((H0 + 1)α

∏ri=1N

12τi ). This ex-

tends∏j

i=1 a(ki) to bounded operator fromD((H0 + 1)α∏ri=1N

12τi ) into (H0 +∑j

i=1ω(ki)+ 1)−αL2(R3j , (P τr,j )12 dk)⊗H . 2

Using the fact that(H0+∑j

i=1ω(ki)+ 1)−1H0 is bounded and (4.1), we havefor ψ ∈ (H0+ 1)−1D(N

r2 )∫ ∥∥∥∥∥N p−r

2 H0

r∏i=1

a(ki)ψ

∥∥∥∥∥2

dk

6 c

∫ ∥∥∥∥∥(H0+

r∑i=1

ω(ki)+ 1

)r∏i=1

a(ki)Np−r

2 ψ

∥∥∥∥∥2

dk

6 c

∫ ∥∥∥∥∥r∏i=1

a(ki)(H0+ 1)Np−r

2 ψ

∥∥∥∥∥2

dk

6 c ‖N p2 (H0+ 1)ψ‖2.

Hence∏ri=1 a(ki) can be defined as bounded operator from(H0+1)−1D(N

p2 ) into

L2(R3r ,dk)⊗ (H0+ 1)−1D(Np−r

2 ). It is easy to see, using commutation relationsand Lemma 3.1, that(N + 1)−

p2 IκN

p2 (H0 + 1)−1 is bounded forκ < ∞, hence

Iκ can be defined as bounded operator from(H0 + 1)−1D(Np2 ) into D(N

p2 ).

So the commutatorH {1}κ acts as bounded operator from(H0 + 1)−1D(Np2 ) into


L2(R3,dk1)⊗D(Np−1

2 ). Now for J := {1, . . . , r} we can prove by induction onrthat

H {J }κ : (H0+ 1)−1D(Np2 )→ L2

(R3r ,

r∏i=1

dki

)⊗D(N

p−r2 ). (4.2)

For #J = 1 (4.2) is already done. Assume that (4.2) holds for #J = r. Using thefact thata(kr+1)mapsL2(R3r ,

∏ri=1 dki)⊗D(N

p−r2 ) intoL2(R3(r+1),

∏r+1i=1 dki)⊗

D(Np−r−1

2 ) and maps(H0+1)−1D(Np2 ) intoL2(R3,dkr+1)⊗(H0+1)−1D(N

p−12 ),

we prove (4.2).A simple computation gives:

H {1}κ =1√2

P∑j=1

vjκ0(k1)+ 1

2M

P∑j=1

rjκ (k1)a∗(rjκ )+

+ rjκ (k1)a(rjκ )−√

2Djrjκ (k1), on (H0+ 1)−1D(N

12 ),

H {1,2}κ = 1

2M

P∑j=1

rjκ (k1)rjκ (k2), on (H0+ 1)−1D(N),

HJκ = 0, for all J such that #J > 3.

Before starting with the pull-through formula we will prove two lemma whichwill be useful in the sequel.

LEMMA 4.4. Let r be an integer andz /∈ σ (Hκ), then

(H0+ 1)(Hκ − z)−1 : D(N r2 )→ D(N

r2 ),

is a bijective map.Proof. Since(N + 1)− r2 IκN

r2 (H0 + 1)−1 is bounded forκ < ∞, we see that

(Hκ − z)(H0+1)−1 mapsD(Nr2 ) into D(N

r2 ). So it is enough to show that(H0+

1)Nr2 (Hκ − z)−1(1+ N)−r2 is bounded. We define[N, iHκ ] as quadratic form on

D(H0)

[N, iHκ ] =P∑j=1

φ(ivjκ0)+ 1

2M

P∑j=1

ia∗2(rjκ )− ia2(rjκ ) +

+ i√

2Dja(rjκ )− i

√2a∗(rjκ )Dj .

For κ < ∞, using Corollary 2.4 and the fact that〈D〉(K + 1)− 12 is bounded, we

see that[N, iHκ ] can be defined as a bounded operator onD(H0).We set ad`N . := [N, i ad`−1

N .] and ad1N. := [N, i.]. So ad`NHκ , which is similar toad1NHκ is defined by induction in as a bounded operator onD(H0) and equal to:

ad`NHκ =P∑j=1

φ(i` vjκ0)+ 1

2M

P∑j=1

i`2`−1a∗2(rjκ )− i`(−2)`−1 a2(rjκ )−

− (−i)`√

2Dj a(rjκ )− i`

√2a∗(rjκ )Dj .

238 ZIED AMMARI

SinceD(N) ⊃ D(Hκ) = D(H0), the resolvent(z − Hκ)−1 preserves the domainof N . This means that the following identity

N(z − Hκ)−1 = (z − Hκ)−1N + (z − Hκ)−1[N, Hκ ](z − Hκ)−1, (4.3)

holds in the sense of bounded operators onH . Using repeatedly (4.3) we noticethat(z− Hκ)−1 preservesD(Np) and we obtain onD(Np)

Np(z − Hκ)−1

= Np−1(z − Hκ)−1N − iNp−1(z− Hκ)−1ad1NHκ (z− Hκ)−1. (4.4)

We move now all factors ofN in each term to the right, we obtain the followingidentity between bounded operators

Np (z− Hκ)−1N−p = (z− Hκ)−1+k∑`=1

(z − Hκ)−1B`(z)N−`,

whereB`(z) is a polynomial in adjNHκ (z− Hκ)−1, j 6 `. Using Lemma 3.3 withs = 0, we see thatB`(z) is bounded forκ < ∞. Hence(z − Hκ)−1(H0 + 1) isa bijective map fromD(Np) into D(Np). Using Hadamard’s three lines lemma[RS, I–IV] for

f (ζ ) := ((z − Hκ)−1H0N−ζψ,Nζ φ),

in S := {ζ ∈ C, p 6 Re(ζ ) 6 p + 1},whereψ,φ ∈ K ⊗ 0fin(h). f (ζ ) is a bounded continuous analytic function onS,satisfying

|f (p + iλ)| 6 c ‖φ‖ ‖ψ‖, λ ∈ R,|f (p + 1+ iλ)| 6 c ‖φ‖ ‖ψ‖, λ ∈ R.

We obtain

|f (ζ )| 6 c ‖φ‖ ‖ψ‖, for ζ ∈ S.

Let 2p 6 r 6 2p+ 2. SinceK ⊗ 0fin(h) is a core forNr2 then(z− Hκ)−1H0(1+

N)−r2ψ ∈ D(N

r2 ) and we have also

‖N r2 (z− Hκ)−1H0(1+N)− r2ψ‖ 6 c ‖ψ‖, ψ ∈K ⊗ 0fin(h).

Then we have(H0+ 1)Nr2 (Hκ − z)−1(1+ N) r2 bounded. 2

LEMMA 4.5. There existε > 0, b < 0 andc independent forκ

(i) ‖R 12` (b)H

{1}κ R

12`+1(b)‖ 6 c

(P∑j=1

|vjκ0(k1)| + |rjκ (k1)ω(k1)

−ε|).

(ii) ‖R 12` (b)H

{1,2}κ R

12`+1(b)‖ 6 c

(P∑j=1

|rjκ (k1)ω(k1)−εrjκ (k2)ω(k2)

−ε|).


Proof. Let D be a dense set of analytic vectors forK. HJiκ can be defined on

D ⊗ 0fin(S(R3)), thenHJiκ R

12`+1(b) is well defined onDκ(b) := (Hκ − b) 1

2 D ⊗0fin(S(R3)). Furthermore sinceD ⊗ 0fin(S(R3)) is a core for(Hκ − b), Dκ(b) isdense inD((Hκ − b) 1

2 ), which is dense inH . Hence it is enough to show (i)–(ii) in Dκ(b). Lemma 3.3 and Lemma 2.8 give the bounds uniformly inκ. Then

R12` (b)H

Jκ R

12`+1(b) extends fromDκ (b) to a bounded operator onH . 2

The following lemma is thegeneralized pull through formula.

LEMMA 4.6. The following identity holds for allφ ∈ D(Nr2 ):

r∏i=1

a(ki)(z− Hκ)−1φ

= R1(z)

r∏i=1

a(ki)φ+

+∑part.

R1(z)HJ1κ R2(z) . . . R`−1(z)H

J`−1κ R`(z)

∏j∈J`

a(kj )φ+

+∑part.

R1(z)HJ1κ R2(z) . . . R`−1(z)H

J`κ R(z) φ.

The sum in right-hand side is taken over all the partitions of the set{1, . . . , r} intoordered subsets.

Proof. We prove this lemma by induction onr. By Lemma 4.4, we know thatthere existψ ∈ (H0 + 1)−1D(N

12 ) such thatφ = (z − Hκ)ψ . We considera(k1)

as bounded operator

a(k1) : (H0+ 1)−1D(N12 )→ (H0+ ω(k1)+ 1)−1L2(R3,dk1)⊗H .

Then we can write forκ <∞a(k1)ψ = (z− ω(k1)− Hκ)−1(z− ω(k1)− Hκ)a(k1)ψ.

By the justification in the beginning of this subsection, we see thatH {1}κ ψ ∈L2(R3,dk1)⊗H . We have the following identity onL2(R3,dk1)⊗H

(z − ω(k1)− Hκ)a(k1)ψ = a(k1)(z − Hκ)ψ +H {1}κ ψ. (4.5)

This proves the pull through formula forr = 1.The formula (4.5) can be generalized by induction inr.

240 ZIED AMMARI

We claim that forψ ∈ (H0+ 1)−1 D(Nr2 ), Ir := {1 . . . r}, we have in

L2

(R3r ,

r∏i=1

dki

)⊗H :(

z−∑i∈Ir

ω(ki)− Hκ)∏i∈Ir

a(ki)ψ

=∏i∈Ir

a(ki)(z − Hκ)ψ +HIrκ ψ +

∑Ir=(J1,J2)

H J1κ

∏i∈J2

a(ki)ψ, (4.6)

where the sum is over all partitions(J1, J2) of Ir . Let us prove (4.6), by inductionon r. We have forψ ∈ (H0+ 1)−1D(N

r+12 ):(

z−∑i∈Ir+1

ω(ki)− Hκ) ∏i∈Ir+1

a(ki)ψ

= a(kr+1)

(z−

∑i∈Ir

ω(ki)− Hκ)∏i∈Ir

a(ki)ψ +H {r+1}κ

∏i∈Ir

a(ki)ψ

=∏i∈Ir+1

a(ki)(z− Hκ)ψ +H {r+1}κ

∏i∈Ir

a(ki)ψ +

+ a(kr+1)∑

Ir=(J1,J2)

H J1κ

∏J2

a(ki)ψ + a(kr+1)HIrκ ψ.

Moving a(kr+1) throughHJ1κ andHIr

κ and using the identity

a(kr+1)HJ1κ = HJ1

κ a(kr+1)+HJ1∪{r+1}κ , on (H0+ 1)−1D(N

#J1+12 ),

we obtain (4.6).Now assume that the pull through formula holds forr, and let us prove it for

r + 1. We have∏i∈Ir+1


=(z−

∑i∈Ir+1

ω(ki)− Hκ)−1(

z−∑i∈Ir+1

ω(ki)− Hκ) ∏i∈Ir+1

a(ki)ψ.

Using now the iterated formula (4.6), we obtain∏i∈Ir+1


=(z−

∑i∈Ir+1

ω(ki)− Hκ)−1 ∏

i∈Ir+1

a(ki)φ+


+(z −

∑i∈Ir+1

ω(ki)− Hκ)−1

HIr+1κ (z− Hκ)−1φ+

+(z −

∑i∈Ir+1

ω(ki)− Hκ)−1 ∑

Ir+1=(J1,J2)

H J1κ

∏i∈J2

a(ki)(z− Hκ)−1φ.

Using the induction hypothesis we obtain:∑Ir+1=(J1,J2)

H J1κ

∏i∈J2

a(ki)(z− Hκ)−1

=∑part.

R1(z)HJ1κ R2(z) . . . R`−1(z)H

J`−1κ R`(z)

∏i∈J`

a(ki).

This completes the proof. 2THEOREM 4.7. Letν 6 1,06 τ andr ∈ N. Then

‖N 12ν N

r−12−τ ψ‖ 6 c ‖(Hκ + b) r2ψ‖, for ψ ∈ D(H

r2κ ), (4.7)

wherec, b are constants independent ofκ.Proof. We prove the theorem by induction onr. The caser = 1 follows from

Lemma 4.2. Assume that (4.7) holds for allj 6 r. Letφ ∈ (H0+1)− 12 D(N

r−12 ) ⊂

D(Nr2 ) andψ := R(−b)φ. Sinceφ ∈ D(N

r2 ), by Lemma 4.4, we see thatψ ∈

D(N12ν N

r2−τ ). Clearly, we have alsoφ ∈ D(N

12ν N

j−22−τ ), j 6 r.

We have

‖N 12ν N

r2−τψ‖2

=r∑j=1

∫P−τr,j

∥∥∥∥∥(Nν +

j∑i=1

ω(ki)ν

) 12 j∏i=1

a(ki)R(−b)φ∥∥∥∥∥

2

dk. (4.8)

(4.8) follows by Corollary 2.2. We recall thatHJκ is an operator-valued function in

variableski, i ∈ J . If we denote by dJ := dki1 . . .dkip whereJ = {i1, . . . , ip},then by Lemma 4.5 we have‖R 1

2` (−b)HJ

κ R12`+1(−b)‖ ∈ L2(R3p,dJ ). Using the

pull through formula in the right-hand side of (4.8) and the fact that(Nν +∑j

i=1ω(ki)ν)

12R

121 (−b) is bounded, we obtain:

‖N 12ν N

r2−τψ‖2

6 c

r∑j=1

∫P−τr,j

∥∥∥∥∥R 121 (−b)

j∏i=1

a(ki)φ

∥∥∥∥∥2

dk+

242 ZIED AMMARI

+ cr∑j=1

∑part.

∫P−τr,j

d∏`=1

‖R 12` (−b)HJ

j`

κ R12`+1(−b)‖2×

×∥∥∥∥R 1

2d+1(−b)

∏i∈J jd+1

a(ki)φ

∥∥∥∥2 d+1∏`=1

dJ j` +

+ cr∑j=1

∑part.

∫P−τr,j

d ′∏`=1

‖R 12` (−b)HJ

j`

κ R12`+1(−b)‖2×

×‖R 12 (−b)φ‖2

d ′∏`=1

dJ j`

=: I + II + III .

We recall thatP−τj,j =∏j

i=1ω(ki)−τ and by (2.7) we notice that

P−τr,j 6 c P−τj,j . (4.9)

In II using (4.9) we can separate the integral in variableski /∈ J jd+1 andki ∈ J jd+1.

Since‖R 12` (−b)HJ

j

`κ R

12`+1(−b)‖ ∈ L2(R3p,dJ ), then we obtain

II 6 cr∑j=1

∑part.

∫ ∏i∈J jd+1

ω(ki)−τ∥∥∥∥R 1

2d+1(−b)

∏i∈J jd+1

a(ki)φ

∥∥∥∥2

dJ jd+1. (4.10)

Now reordering the terms in the right-hand side of (4.10) and taking into accountthe fact thatJ jd+1 6= ∅, we have

II 6 cr−1∑j=1

∫P−τj,j

∥∥∥∥∥(Hκ +

j∑i=1

ω(ki)ν + b

)− 12 j∏i=1

a(ki)φ

∥∥∥∥∥2

dk. (4.11)

Doing the same thing for III, we obtain

III 6 c ‖R 12 (−b)φ‖2. (4.12)

Collecting (4.11)–(4.12) we obtain

‖N 12ν N

r2−τψ‖2

6 c

(r∑j=1

∫P−τr,j

∥∥∥∥∥(Hκ +

j∑i=1

ω(ki)ν + b

)− 12 j∏i=1

a(ki)φ

∥∥∥∥∥2

dk+

+r−1∑j=1

∫P−τj,j

∥∥∥∥∥(Hκ +

j∑i=1

ω(ki)ν + b

)− 12 j∏i=1

a(ki)φ

∥∥∥∥∥2

dk+


+‖R 12 (−b)φ‖2

)

6 c

(r∑j=1

∫P−τj,j

∥∥∥∥∥(Hκ +

j∑i=1

ω(ki)ν + b

)− 12 j∏i=1

a(ki)φ

∥∥∥∥∥2

dk+

+‖R 12 (−b)φ‖2

). (4.13)

(4.13) follows by (4.9). Now using the fact that(Hκ +

j∑i=1

ω(ki)ν + b

)− 12(H0+

j∑i=1

ω(ki)ν + b

) 12

is bounded uniformly inki and Corollary 2.2, we obtain

‖N 12ν N

r2−τR(b)φ‖2

6 c

(r∑j=1

‖(H0+ b)− 12N

j2−τφ‖2+ ‖R 1

2 (−b)φ‖2)

(4.14)

6 c

(r∑j=1

‖N 12ν N

j−22−τ φ‖2+ ‖R 1

2 (−b)φ‖2). (4.15)

In fact, using the fact thatN−τ (H0 + b)− 12 (1 + Nν)− 1

2 is bounded, we see thatthe right-hand side in (4.14) is less than (4.15), which holds for allφ ∈ (H0 +1)−

12 D(N

r−12 ). Since

D ⊗ 0fin(C∞0 (R

3)) ⊂ (H0+ 1)−12 D(N

r−12 )

is a core forN12ν N

r−22−τ , we see that(H0 + 1)− 1

2 D(Nr−1

2 ) is dense inD(N12ν N

r−22−τ )

and hence (4.15) holds for allφ ∈ D(N12ν N

r−22−τ ). Now let

φ := (Hκ + b)ψ,ψ ∈ D((Hκ + b) r+12 )

andb > 0. Then

φ ∈ D((Hκ + b) r−12 ) ⊂ D(N

12ν N

r−22 ).

The induction hypothesis and (4.15) give

‖N 12ν N

r2−τψ‖ 6 c ‖(Hκ + b) r+1

2 ψ‖, ψ ∈ D((Hκ + b) r+12 ). 2

244 ZIED AMMARI

COROLLARY 4.8. Letγ > 0 andε < 1/r, wherer ∈ N. Then

Nrε 6 c (Hκ + b)r , (4.16)

H1−γ0 Nr−1+γ 6 c (Hκ + b)r , (4.17)

where c and b are constants independent ofκ.Proof.The inequalities follow from Lemma 2.5 and Theorem 4.7. (4.16) follows

with ν = 1, τ = (1− εr)/(1− r) in Theorem 4.7 andb = ω, τ1 = ε, τ2 =1, τ3 = −(1− εr)/(1− r), α1 = r, α2 = 1, α3 = r − 1 in Lemma 2.5. (4.17)follows with ν = 1, τ = γ /(r − 1) in Theorem 4.7 andb = ω, τ1 = 0, τ2 =1, τ3 = −γ /(r − 1), α1 = r − 1+ γ, α2 = γ, α3 = r − 1 in Lemma 2.5 and thefact that d0(ω)γ 6 Hγ−1

0 d0(ω). 2Using the principle of the cutoff independence, formulated in Lemma 4.1, we

deduce similar estimates forH∞.

THEOREM 4.9. Letγ > 0 andε < 1/r, wherer ∈ N. c, b are positive constants.Then

Nrε 6 c (H∞ + b)r ,

H1−γ0 Nr−1+γ 6 c (H∞ + b)r .

Theorem 4.9 forε = 0 and Hadamard’s three lines lemma in [Ro] give the follow-ing corollary.

COROLLARY 4.10. For r ∈ R+ there are c,b positive constants such that

(N + 1)r 6 c (H∞ + b)r .COROLLARY 4.11. For r ∈ R+ there are c,b positive constants such that

(N + 1)r 6 c (H + b)r .Proof. If we prove thatNrU∞(N + 1)−r is bounded forr positive integer, the

corollary follows from Corollary 4.10.We have onD(N) the identity

U ∗∞NU∞ = N − iφ(iG∞)− 12‖G∞‖2.

SoU∞ preservesD(N). By iteration we have onD(Nr)

U ∗∞NrU∞ =

(N − iφ(iG∞)− 1

2‖G∞‖2)r.

Then we obtain the boundness of the operatorNrU∞(N + 1)−r , since(N − iφ(iG∞)− 1

2‖G∞‖2)r(N + 1)−r ,

is bounded. 2


4.2. NUMBER-ENERGY ESTIMATES

We say that a sequence of operatorsAκ(R),R ∈ R or C is of class O(Rγ ) (resp.o(Rγ )) uniformly in κ, if there existsc constant independent fromκ andR suchthat‖Aκ(R)‖ 6 cRγ (resp.‖Aκ(R)‖R−γ → 0, whenR→∞).LEMMA 4.12. We have uniformly inκ for z in a bounded set ofC \ R andm ∈N, γ > 0:

(i) (H0+ 1)1−γ

2 (N + 1)γ2+m (z− Hκ)−k (N + 1)−m+k−1 ∈ O(|Im(z)|cm,k ).

(ii) (N + 1)m (z− Hκ)−k (N + 1)−m+k+γ−2

2 (H0+ 1)1−γ

2 ∈ O(|Im(z)|cm,k ).(iii) Letχ ∈ C∞0 (R) and n, q ∈ N:

‖Nnχ(Hκ)Nq‖ <∞.

Proof. We recall the identity (4.4) onD(Nk), which had been proved in theproof of Lemma 4.4

Nk(z − Hκ)−1 = Nk−1(z − Hκ)−1N − iNk−1(z− Hκ)−1ad1NHκ(z − Hκ)−1.

We move now all factors ofN in each term to the right, we obtain the followingidentity between bounded operators

Nk (z− Hκ)−1N−k = (z− Hκ)−1 +k∑`=1

(z − Hκ)− 12B`(z)(z− Hκ)− 1

2N−`,

whereB`(z) is a polynomial in(z − Hκ)− 12 adjNHκ(z − Hκ)− 1

2 , j 6 `. UsingLemmas 3.3 and 4.2, we see that‖B`(z)‖ 6 c|Im(z)|−c` , uniformly in κ. Using

Corollary 4.8, (4.17), we see that‖H1−γ

20 N

γ2 (z − Hκ)− 1

2‖ 6 |Im(z)|− 12 , which

proves (i) fork = 1. Fork 6= 1, we write

H1−γ

20 Nm+ γ2 (z − Hκ)−kN−m+k−1

= H1−γ

20 Nm+ γ2 (z− Hκ)−1N−m

k∏`=0

Nm−`(z − Hκ)−1N−m+`−1.

This proves (i) fork 6= 1. The proof of (ii) is similar to (i). (iii) follows from thehigher-order estimates in Theorem 4.7 withτ = ν = 0. 2

We set:

Hext := H ⊗ 0(h), H extκ = Hκ ⊗ 1+ 1⊗ d0(ω),

N0 := N ⊗ 1, N∞ := 1⊗N, acting inHext.

LEMMA 4.13. We have uniformly inκ and z in a bounded set ofC\R andm ∈ N

246 ZIED AMMARI

(i) (N0+N∞)m (z− H extκ )−1 (N0+N∞)−m+1 ∈ O(|Im(z)|−cm).

(ii) (H ext0 + 1)

12 (N0+N∞)m (z − H ext

κ )−1 (N0 +N∞)−m ∈ O(|Im(z)|−cm).

Proof.The proof is analogous to the proof of Lemma 4.12. In fact we have

ad`N0+N∞Hextκ = ad`NHκ ⊗ 1,

(N0+N∞)m (z− H extκ )−1 (N0+N∞)−m

= (z − H extκ )−1+

m∑`=1

(z− H extκ )−

12B`(z) (z− H ext

κ )−12 (N0+ N∞)−`,

whereB`(z) ∈ O(|Imz|−c`). 2

4.3. COMMUTATOR ESTIMATES

Let q ∈ C∞0 (R3),0 6 q 6 1, q = 1 near 0. We setqR := q(x/R). We recall thatwe considerh in its momentum representationL2(R3,dk) andx = ∇k/i.

We use the following functional calculus formula, see [DG1], forχ ∈ C∞0 (R)andA a self-adjoint operator:

χ(A) = i

2π

∫C∂zχ(z)(z− A)−1 dz ∧ dz, (4.18)

whereχ is an almost analytic extension ofχ , such that

χ|R = χ, |∂zχ(z)| 6 cn|Imz|n, n ∈ N.

LEMMA 4.14. Letχ ∈ C∞0 (R), then one has uniformly forκ 6∞Nn[χ(Hκ), 0(qR)]Nm ∈ O(R−1).

Proof. Commutation relations allow to compute[Hκ, 0(qR)] as a sesquilinearform onD(H0), which byNτ -estimates is a bounded operator onD(H0), whenκ <∞.

We have

[H0, 0(qR)] = d0(qR, [ω, qR]),

[φ(vκ0), 0(qR)] = 1√

2a∗((1− qR)vκ0)0(q

R)− 1√20(qR)a((1− qR)vκ0),

[a2(rjκ ), 0(qR)] = −0(qR)a((1− qR)rjκ )a((1+ qR)rjκ ),

[a∗2(rjκ ), 0(qR)] = a∗((1− qR)rjκ )a∗((1+ qR)rjκ )0(qR), (4.19)


[a∗(rjκ )a(rjκ ), 0(qR)]= −a∗(rjκ )0(qR)a((1− qR)rjκ )+ a∗((1− qR)rjκ )0(qR)a(rjκ ),[a∗(rjκ )Dj , 0(q

R)] = a∗((1− qR)rjκ )0(qR)Dj ,

[Dja(rjκ ), 0(q

R)] = −Dj0(qR)a((1− qR)rjκ ).

Let χ1 ∈ C∞0 (R) such thatχ1χ = χ . Using (4.18), we have

Nn[χ(Hκ), 0(qR)]Nm

= Nnχ1(Hκ)[χ(Hκ), 0(qR)]Nm +Nn[χ1(Hκ), 0(qR)]χ(Hκ)Nm

= i

2π

∫C∂zχ(z)N

nχ1(Hκ)(z − Hκ)−1×(4.20)×[Hκ , 0(qR)](z − Hκ)−1Nm dz ∧ dz+

+ i

2π

∫C∂zχ1(z)N

n(z− Hκ)−1 [Hκ, 0(qR)]×

× (z − Hκ)−1χ(Hκ)Nm dz ∧ dz.

Moving the power ofN towardχ(Hκ), χ1(Hκ) and then using Lemma 4.12 andCorollary 4.8, we see that it is enough to show that forb > 0, (N + 1)−n(H0 +b)−

12 [Hκ, 0(qR)](H0+ b)− 1

2 ∈ O(R−1), uniformly in κ, to have the lemma. Usingnow Lemma 3.4, we obtain

‖(N + 1)−n(H0+ b)− 12 [Hκ, 0(qR)](H0+ b)− 1

2‖6 c(‖(1− qR) (V + 1)−

12vκ0‖+

+‖(V + 1)−s2ω

s−14 (1− qR)rκ‖‖ω s−1

4 (1+ qR)rκ‖++‖N−1d0(qR, [ω, qR])‖+ (4.21)

+‖(V + 1)−s2ω

s−12 (1− qR)rκ‖‖(V + 1)−

s2ω

s−12 rκ‖+

+‖(V + 1)−s2ω

s−12 (1− qR)rκ‖‖(K + i)−

12D‖).

Using the inequality (2.3) recalled in Subsection 2.1,

‖N−1 d0(qR, [ω, qR])‖ 6 ‖[ω, qR]‖,and the fact that[ω, qR] ∈ O(R−1), we see that‖N−1d0(qR, [ω, qR])‖ ∈ O(R−1).Now for the other kind of terms we will proceed as follows. SinceV > 〈x〉α, α >

248 ZIED AMMARI

2, we can pickµ > 0 ands < 1, such that(V + 1)−s2 〈x〉1+µ is bounded. Then

using Lemma A.2, we obtain

‖〈x〉1+µω s−12 (1− qR)rκ‖ ∈ O(R−1−µ),

‖〈x〉1+µω s−14 (1− qR)rκ‖ ∈ O(R−1−µ).

Hence we have:

‖(N + 1)−n(H0+ b)− 12 [Iκ , 0(qR)](H0+ b)− 1

2‖ ∈ O(R−1−µ). (4.22)

Then the integrand in (4.20) is|Im(z)|−2O(R−1). This ends the proof. 2Let j0 ∈ C∞0 (R3), j∞ ∈ C∞(R3), 06 j0, 06 j∞ , j2

0 + j2∞ 6 1, j0 = 1 near0. We set forR > 1, jR := (jR0 , jR∞), where

jR0 := j0

(x

R

), jR∞ := j∞

(x

R

).

We setj := j1.

LEMMA 4.15. One has uniformly forκ 6∞(i) χ(H ext

κ ) I ∗(jR)− I ∗(jR) χ(Hκ) ∈ O(R−1).(ii) Letχ ∈ C∞0 (R), then

(N0+ N∞)n (χ(H extκ ) I ∗(jR)− I ∗(jR) χ(Hκ))Nm ∈ O(R−1).

Proof. The proof is similar to the previous one. Instead of (4.20), we use theidentities:

H ext0 I ∗(jR)− I ∗(jR)H0

= dI ∗(jR, [ω, jR]), where[ω, jR] = ([ω, jR0 ], [ω, jR∞]),φ(vκ0)⊗ 1I ∗(jR)− I ∗(jR)φ(vκ0)

= φ((1− jR0 )vκ0)⊗ 1I ∗(jR)− 1⊗φ(jR∞vκ0)I∗(jR),

a∗(rjκ )Dj ⊗ 1I ∗(jR)− I ∗(jR)a∗(rjκ )Dj

= a∗((1− jR0 )rjκ )Dj ⊗ 1I ∗(jR)− 1⊗a∗(jR∞rjκ )DjI∗(jR),

Dj a(rjκ )⊗ 1I ∗(jR)− I ∗(jR)Dj a(r

jκ )

= Dj a((1− jR0 )rjκ )⊗ 1I ∗(jR)− 1⊗Dja(jR∞r

jκ )I∗(jR),

a2(rjκ )⊗ 1I ∗(jR)− I ∗(jR)a2(rjκ )

= −I ∗(jR)a((1− jR0 )rjκ )a((1+ jR0 )rjκ ),a∗

2(rjκ )⊗ 1I ∗(jR)− I ∗(jR)a∗2(rjκ )


= (−2a∗(jR0 rjκ )⊗ a∗(jR∞rjκ )− 1⊗a∗2(jR∞rjκ ))× I ∗(jR),

a∗(rjκ )a(rjκ )⊗ 1I ∗(jR)− I ∗(jR)a∗(rjκ )a(rjκ )

= a∗(rjκ )⊗ 1I ∗(jR)a((−1+ jR0 )rjκ )++ (a∗((1− jR0 )rjκ )⊗ 1+ 1⊗a∗(jR∞rjκ ))× I ∗(jR)a(rjκ ).

We notice that forv ∈ B(K,K ⊗ h), h1 = h2 = h, we define

1⊗a](v) :K ⊗ 0(h1)⊗ 0(h2)→K ⊗ 0(h1)⊗ 0(h2),

1⊗a](v) := T −1 10(h1) ⊗ a](v) T ,whereT is the natural identification:

T :K ⊗ 0(h1)⊗ 0(h2)→ 0(h1)⊗K ⊗ 0(h2).

Using (2.4) withj20 + j2∞ 6 1, we see that(H ext

0 I ∗(jR)− I ∗(jR)H0)(N + 1)−1 isbounded, which shows thatI ∗(jR) : D(H0)→ D(H ext

0 ). We obtain the followingidentity onH

C(z) := (z − H extκ )−1I ∗(jR)− I ∗(jR)(z− Hκ)−1

= (z − H extκ )−1(H ext

κ I ∗(jR)− I ∗(jR)Hκ)(z− Hκ)−1.

Using (4.18), we obtain:

χ(H extκ ) I ∗(jR)− I ∗(jR) χ(Hκ) = i

2π

∫C∂zχ (z) C(z)dz ∧ dz.

Using Corollary 4.8 and Lemma 3.4 we obtain forb < 0

‖C(b)‖6 c(‖(N∞ +N0)

−1dI ∗(jR, [jR, ω])‖ + ‖(V + 1)−s2ω

s−12 (1− jR0 )vκ0‖+

+‖(V + 1)−s2ω

s−12 jR∞vκ0‖+

+‖(V + 1)−s2ω

s−14 (1− jR0 )rκ‖ ‖ω

s−14 (1+ jR0 )rκ‖+

+‖(V + 1)−s2ω

s−14 jR∞rκ‖ ‖ω

s−14 jR0 rκ‖ + ‖(V + 1)−

s4ω

s−14 jR∞rκ‖2+

+‖(V + 1)−s2ω

s−12 rκ‖ × ‖(V + 1)−

s2ω

s−12 (1− jR0 )rκ‖+

+ (‖(V + 1)−s2ω

s−12 (1− jR0 )rκ‖ + ‖(V + 1)−

s2ω

s−12 jR∞rκ‖)×

×‖(K + i)−12D‖+

+‖(V + 1)−s2ω

s−12 rκ‖ × ‖(V + 1)−

s2ω

s−12 jR∞rκ‖). (4.23)

250 ZIED AMMARI

Applying (2.4) with[jR0 , ω]2+ [jR∞, ω]2 ∈ O(R−2), we obtain

‖(b − H extκ )−1(H ext

0 I ∗(jR)− I ∗(jR)H0)(b − Hκ)−1‖ ∈ O(R−1).

For the other terms ofC(b), we use (4.23), the fact that we can pickµ > 0 ands < 1 such that(V + 1)−

s2 〈x〉1+µ is bounded and Lemma A.2. We obtain:

‖(V + 1)−s2ω

s−12 f Rε rκ‖ ∈ O(R−1−µ),

‖(V + 1)−s2ω

s−14 f Rε rκ‖ ∈ O(R−1−µ), (4.24)

‖(V + 1)−s4ω

s−14 jR∞rκ‖2 ∈ O(R−1−µ),

wheref Rε denotesjR∞ or 1− jR0 . Then we have

‖(b − H extκ )−1× (Iκ ⊗ 1 I ∗(jR)− I ∗(jR)Iκ)(b − Hκ)−1‖

∈ O(R−1−µ). (4.25)

Hence we have‖C(z)‖ ∈ |Im(z)|−2O(R−1). This proves (i).Let χ1 ∈ C∞0 (R) such thatχ1χ = χ . As in the previous lemma, we have using

(4.18):

(N0+N∞)nχ(H extκ )I ∗(jR)− I ∗(jR)χ(Hκ)Nm

= (N0+ N∞)nχ1(Hextκ )(χ(H ext

κ )I ∗(jR)− I ∗(jR)χ(Hκ))Nm++ (N0+N∞)n(χ1(H

extκ )I

∗(jR)− I ∗(jR)χ1(Hκ))χ(Hκ)Nm

= i

2

∫C∂zχ (z)(N0+N∞)nχ1(H

extκ )C(z)N

m dz ∧ dz+

+ i

2

∫C∂zχ1(z)(N0+N∞)nC(z)χ(Hκ)Nm dz ∧ dz.

Moving (N0 +N∞)n (resp. Nm) towardχ(Hκ) (resp.χ1(Hextκ )) in the last expres-

sion and then using (i) and Lemma 4.13 we prove (ii). 2

5. Spectral Theory for the Nelson Hamiltonian

We study in this section the spectral properties of both Nelson and modified Hamil-tonians. In Subsection 5.1 we prove the existence of ground state for the NelsonHamiltonian. We use essentially the fact thatHκ −Eκ are Pauli–Fierz Hamiltonianwhich converge in the norm resolvent sense toH . This is the subject of Theo-rem 5.1. In Subsection 5.2 we prove a Mourre estimate for the modified Hamil-tonian, which gives that pure point spectrum is locally finite outside its thresholds.


5.1. HVZ THEOREM

THEOREM 5.1. One has

σess(H) = [inf σ (H)+m,+∞[,and inf σ (H) is a discrete eigenvalue ofH .

Proof.Hκ is an example of a Pauli–Fierz Hamiltonian, see [DG2, Section 3].The HVZ theorem proved in [DG2] for the Pauli–Fierz Hamiltonians gives forκ <∞

σess(Hκ) = σess(Hκ − Eκ) = [inf σ (Hκ)+m,+∞[.Using the fact that(z − Hκ)−1 → (z − H∞)−1 and [RS, I–IV, Thm. VIII.23], wesee that

limκ→∞ (inf σ (Hκ)) = inf σ (H∞). (5.1)

Let χ ∈ C∞0 (] −∞, inf σ (H∞)+m[), thenχ ∈ C∞0 (] −∞, inf σ (Hκn)+m[) fora sequenceκn → +∞. By the HVZ theorem forHκn , χ(Hκn) is compact and byTheorem 3.6χ(H∞) is compact. So we obtain

σess(H∞) ⊂ [inf σ (H∞)+m,+∞[.Let us now show that[inf σ (H∞) + m,+∞[⊂ σess(H∞). Let λ such thatλ >

inf σ (H∞) + m. By (5.1) and the HVZ theorem forHκ , there exists a sequenceκn→ +∞ such thatλ ∈ σess(Hκn), or equivalently

µ := (λ+ c)−1 ∈ σess((Hκn + c)−1), c� 1.

Let us show thatµ ∈ σess((H∞ + c)−1), i.e. :λ ∈ σess(H∞):Assume the contrary and letχ ∈ C∞0 (R), such thatχ(µ) = 1, χ((H∞ + c)−1)

compact. Letϕκn,j be Weyl sequences for(Hκn + c)−1 atµ such that

‖ϕκn,j‖ = 1, limj→∞((Hκn + c)

−1− µ)ϕκn,j = 0,

and

w- limj→∞ ϕκn,j = 0.

One has

‖χ((H∞ + c)−1)ϕκn,j − ϕκn,j‖6 ‖χ((Hκn + c)−1)− χ((H∞ + c)−1)‖ + ‖χ((Hκn + c)−1)ϕκn,j − ϕκn,j‖.

Sinceχ((H∞ + c)−1) compact, there exists forε > 0, aκ1 andj1 such that

‖χ((H∞ + c)−1)ϕκ1,j1 − ϕκ1,j1‖ 6 2ε,

‖χ((H∞ + c)−1)ϕκ1,j1‖ 6 ε.

We obtain‖ϕκ1,j1‖ 6 3ε, this gives a contradiction, if we chooseε < 1/3. 2

252 ZIED AMMARI

5.2. MOURRE ESTIMATE

We denote byb the operator acting onh defined by

b := 1

2(∇ω .Dk +Dk .∇ω), onC∞0 (R

3).

[ABG, Prop. 4.2.3] yields that the closure ofb is the infinitesimal generator ofthe strongly continuous unitary groupUt associated to the vector field∇ω in thefollowing sense

UtF := [det∇φ−t (k)] 12 F(φ−t (k)), F ∈ S ′(R3), (5.2)

whereφt is the flow of the vector field∇ω. MoreoverC∞0 (R3) is a core forb. Wedenote in the sequel byb its closure. We setB := d0(b). ClearlyB is essentiallyselfadjoint on0fin(C

∞0 (R3)).

We denote byτ the set ofthresholds,

τ := σpp(H∞)+mN∗.Let S be a selfadjoint operator onH , we say thatS is of classC1(B), see [ABG],if the map

t 7→ eitB(S − z)−1 e−itB,

is stronglyC1 for somez ∈ C \ σ (S). By [ABG, Lemma 6.2.9]S ∈ C1(B) if andonly if the sesquilinear form[B, (z−S)−1] onD(B) is continuous for the topologyof H , i.e:

|((S − z)−1ϕ,Bϕ)− (Bϕ, (S − z)−1ϕ)| 6 c ‖ϕ‖2 for ϕ ∈ D(B). (5.3)

We recall here a well known theorem, see [ABG, Thm. 6.2.10].

THEOREM 5.2. Let S,B two selfadjoint operators acting on Hilbert space. IfS ∈ C1(B) then

(i) D(S) ∩D(B) is dense inD(S),(ii) ([B, S]u, u) 6 c ‖Su‖2, u ∈ D(S) ∩D(B),(iii) [B, (z − S)−1] = (z − S)−1[B, S](z − S)−1,

where(iii) is understood as identity between bounded operators in the followingsense:

H(z−S)−1−→ D(S)

[B,S]−→ D(S)∗ (z−S)−1−→ H .


LEMMA 5.3. Hκ is of classC1(B) if κ <∞.Proof.We will prove

(i) eitB preservesD(H0).(ii) |(Hκu, Bu)− (Bu, Hκu)| 6 c ‖H0u‖2+ ‖u‖2, u ∈ D(H0) ∩D(B).

[ABG, Thm. 6.3.4] and [ABG, Prop. 6.3.5] give that (i), (ii) impliesHκ ∈ C1(B).Let us prove (i). It is enough to show thatH0eitB(H0+ 1)−1e−itB is bounded to

have (i). We have using (5.2)

eitBd0(ω)e−itB = d0(eitbωe−itb)

= d0(ω(φ− t2(k))).

Since∇ω is a bounded completeC∞ vector field, we have|φt(k) − k| 6 c |t|uniformly in k. So this implies|∑N

1 ω(φt(ki)) − ω(ki)| 6 c |t|N and, hence,H0eitB(H0+ 1)−1e−itB is bounded. This prove (i).

Let us prove (ii). We compute[Hκ, iB]:(iBu, Hκu)− (iHκu, Bu)

= (d0(|∇ω|2)u, u)−P∑j=1

(φ(ibvjκ0)u, u)+ 1

2M

P∑j=1

(ia(rjκ )a(brjκ )u, u)−

− (ia∗(rjκ )a∗(brjκ )u, u)+ (ia∗(rjκ )a(brjκ )− ia∗(brjκ )a(rjκ )u, u)−

−√2(iDja(brjκ )u, u)+

√2(ia∗(brjκ )Dju, u).

A simple computation yields

(i) 〈x〉−1bvκ0 ∈ B(K,K ⊗ h),

(ii) 〈x〉−1brκ ∈ B(K,K ⊗ h) for κ <∞,(iii ) 〈k〉−ε〈x〉−1brκ ∈ B(K,K ⊗ h) for ε > 0, uniformly in κ.

(5.4)

Now using Lemma 3.4 withβ = 1 and (5.4)(i)–(ii) we obtain foru ∈ D(H0)∩D(B)

|(iBu, Hκu)− (iHκu, Bu)|6 c(‖rκ‖ ‖(V + 1)−

12brκ‖ + ‖(V + 1)−

12bvκ0‖+

+‖(K + i)−12D‖ ‖(V + 1)−

12brκ‖)× (‖H0u‖2+ ‖u‖2).

SinceV >∑

i〈xi〉α, α > 2, we prove (ii). This completes the proof. 2LEMMA 5.4. H∞ is of classC1(B).

254 ZIED AMMARI

Proof. SinceHκ is of classC1(B) we know by Theorem 5.2 that(z − Hκ)−1 :D(B)→ D(B) and

(z − Hκ)−1[Hκ , iB](z − Hκ)−1 = [(z − Hκ)−1, iB], on H .

Forφ ∈ D(B), one has

(i(Hκ + i)−1φ,Bφ)− (iBφ, (Hκ − i)−1φ)

= ((Hκ + i)−1φ, [Hκ , iB](Hκ − i)−1φ).

Using Lemma 3.3 and (4.17) we obtain

|((Hκ + i)−1φ,Bφ)− (Bφ, (Hκ − i)−1φ)|6 c (‖d0(|∇ω|2)(N + 1)−1‖ + ‖〈k〉−εrκ‖ ‖〈k〉−ε〈x〉−1brκ‖++‖〈x〉−1bvκ0‖ + ‖(K + i)−

12D‖ ‖〈k〉−ε〈x〉−1brκ‖)‖φ‖2, ε > 0,

where c is independent fromκ. Then lettingκ → ∞ and using Theorem 3.6 weobtain

|((H∞ + i)−1φ, iBφ)− (iBφ, (H∞ − i)−1φ)| 6 c ‖φ‖2 for φ ∈ D(B).This impliesH∞ ∈ C1(B). 2LEMMA 5.5. We have forχ ∈ C∞0 (R)(i) w-limκ→∞[(Hκ − i)−1, iB] = [(H∞ − i)−1, iB],(ii) limκ→∞ χ(Hκ)[Hκ, iB]χ(Hκ) = χ(H∞)[H∞, iB]χ(H∞).

Proof. (i) follows from the proof of Lemma 5.4. Let us prove (ii):

χ(Hκ)[Hκ, iB]χ(Hκ)− χ(Hκ ′)[Hκ ′, iB]χ(Hκ ′)= (χ(Hκ)− χ(Hκ ′))[Hκ, iB]χ(Hκ)++χ(Hκ ′)([Hκ, iB] − [Hκ ′, iB])χ(Hκ)+ (5.5)

+χ(Hκ)[Hκ ′, iB](χ(Hκ)− χ(Hκ ′)).We first claim that

limκ→∞χ(Hκ)H

12

0 = χ(H∞)H12

0 . (5.6)

To have (5.6), we see using the functional calculus formula (4.18), that it is enoughto show that

limκ→∞(Hκ − z)

−1H12

0 = (H∞ − z)−1H12

0 .


Forκ, κ ′ <∞, we have the following operator identity onH

(Hκ − z)−1− (Hκ ′ − z)−1 = (Hκ − z)−1(Hκ ′ − Hκ)(Hκ ′ − z)−1.

It follows from the proof of Theorem 3.6 that

limκ,κ ′→∞

(H0+ 1)−12 (Hκ ′ − Hκ)(H0+ 1)−

12 = 0.

Then we obtain (5.6). We also claim that

limκ→∞ (H0+ 1)−

12 [Hκ, iB](H0 + 1)−

12 (N + 1)−1 (5.7)

= (H0+ 1)−12 [H∞, iB](H0+ 1)−

12 (N + 1)−1.

In fact, using Lemma 3.4 and the fact thatV > c∑

i〈xi〉2, we get

‖(H0+ i)−12a∗(brκ ′ − brκ)a∗(rκ ′)(H0+ i)−

12 N−1‖

6 ‖〈k〉−ε〈x〉−1(brκ ′ − brκ)‖ ‖〈k〉−εrκ ′‖,‖(H0+ i)−

12a(rκ ′ − rκ)a(brκ ′)(H0+ i)−

12 N−1‖

6 ‖〈k〉−ε(rκ ′ − rκ)‖ ‖〈k〉−ε〈x〉−1brκ ′ ‖,‖(H0+ i)−

12a∗(brκ ′ − brκ)a(rκ ′)(H0+ i)−

12 N−1‖

6 ‖〈k〉−ε〈x〉−1(brκ ′ − brκ)‖ ‖〈k〉−εrκ ′‖,‖(H0+ i)−

12a∗(brκ ′ − brκ)Dj(H0+ i)−

12 N−1‖

6 ‖〈k〉−ε〈x〉−1(brκ ′ − brκ)‖ ‖(K + 1)−12D‖,

‖(H0+ i)−12Dja(brκ ′ − brκ)(H0+ i)−

12 N−1‖

6 ‖〈k〉−ε〈x〉−1(brκ ′ − brκ)‖ ‖(K + 1)−12D‖.

Using these estimates and (5.4), we obtain (5.7). Now using (5.5), (5.6) and (5.7)we obtain (ii). 2

We have to prove a localization estimate for[Hκ, iB] similar to the one in theLemma 4.15.

LEMMA 5.6. We have uniformly inκ:

χ(H extκ )

([H extκ , iBext]I ∗(jR)− I ∗(jR)[Hκ, iB]

)χ(Hκ) ∈ o(R0).

256 ZIED AMMARI

Proof.We set

C(z) := (z− H extκ )−1([H ext

κ , iBext]I ∗(jR)− I ∗(jR)[Hκ, iB])(z − Hκ)−1,

whereBext := B ⊗ 1+ 1⊗ B. A simple computation gives

[H ext0 , iBext]I ∗(jR)− I ∗(jR)[H0, iB] = dI ∗(jR, [|∇ω|2, jR]),

a(rjκ )a(brjκ )⊗ 1I ∗(jR)− I ∗(jR)a(rjκ )a(brjκ )

= I ∗(jR)(a((jR0 − 1)rjκ )a(jR0 br

jκ )+ a(rjκ )a((j0− 1)brjκ )),

a∗(rjκ )a∗(brjκ )⊗ 1I ∗(jR)− I ∗(jR)a∗(rjκ )a∗(brjκ )

= −(a∗(jR0 rjκ )⊗ a∗(jR∞brjκ )+ a∗(jR0 brjκ )⊗ a∗(jR∞rjκ )++1⊗ a∗(jR∞rjκ )a∗(j∞brjκ )).

We have also similar identities fora∗a, aDj ,Dja∗, replacingrjκ by brjκ , in the

proof of Lemma 4.15. As in the proof of Lemma 4.15 we use Corollary 4.8 andLemma 3.3.

We have forβ < 0:

‖C(β)‖6 c(‖(N0 +N∞)−1dI ∗(jR, [|∇ω|2, jR])‖++‖〈k〉−ε〈x〉−1(1− jR0 )rκ‖ ‖〈k〉−ε〈x〉−1jR0 brκ‖++‖〈k〉−ε〈x〉−1(1− jR0 )brκ‖ ‖〈k〉−ε〈x〉−1jR0 rκ‖++‖〈k〉−ε〈x〉−1jR0 rκ‖ ‖〈k〉−ε〈x〉−1jR∞brκ‖++‖〈k〉−ε〈x〉−1jR0 brκ‖ ‖〈k〉−ε〈x〉−1jR∞rκ‖++‖〈k〉−ε〈x〉−1jR∞brκ‖ ‖〈k〉−ε〈x〉−1jR∞rκ‖++‖〈k〉−ε〈x〉−1brκ‖ ‖〈k〉−ε〈x〉−1(1− jR0 )rκ‖++‖〈k〉−ε〈x〉−1rκ‖ ‖〈k〉−ε〈x〉−1(1− jR0 )brκ‖++‖〈k〉−ε〈x〉−1rκ‖ ‖〈k〉−ε〈x〉−1jR∞brκ‖++‖〈k〉−ε〈x〉−1brκ‖ ‖〈k〉−ε〈x〉−1jR∞rκ‖++‖〈k〉−ε〈x〉−1(1− jR0 )brκ‖ + ‖〈k〉−ε〈x〉−1jR∞brκ‖).

Using Lemma A.2, Lemma 3.3 and 5.4(iii) we obtain

C(z) ∈ |Im(z)|−2o(R0), uniformly in κ. 2THEOREM 5.7. The following three assertions hold:


(i) Letλ ∈ R \ τ. Then there existε > 0, C0 > 0 and compact operatorK0 suchthat

1[λ−ε,λ+ε](H∞)[H∞, iB]1[λ−ε,λ+ε](H∞) > C0 1[λ−ε,λ+ε](H∞)+K0.

(ii) For all [λ1, λ2] such that[λ1, λ2] ∩ τ = ∅, one has

dim1pp[λ1,λ2](H∞)H <∞.

Consequentlyσpp(H∞) can accumulate only atτ , which is a closed countableset.

(iii) Letλ ∈ R \ (τ ∪ σpp(H∞)). Then there existsε > 0, C0 > 0 such that

1[λ−ε,λ+ε](H∞)[H∞, iB]1[λ−ε,λ+ε](H∞) > C01[λ−ε,λ+ε](H∞).

Proof.We set

d(λ) := inf

{n∑i=1

|∇ω(ki)|2; τ +n∑i=1

ω(ki) = λ, n = 1,2 . . . , τ ∈ σpp(H∞)

},

d(λ) := inf

{n∑i=1

|∇ω(ki)|2; τ +n∑i=1

ω(ki) = λ, n = 0,1, . . . , τ ∈ σpp(H∞)

}.

1µλ := [λ− µ, λ + µ], µ > 0,

dµ(λ) := infν∈1µλ

d(ν),

dµ(λ) := infν∈1µλ

d(ν),

E0 := inf σ (H∞).

We will follow the logic of the proof of Mourre estimate in the case of a Pauli–FierzHamiltonian [DG2]. Let us recall the statements that we will prove by inductionin n:H1(n): Let ε > 0 andλ ∈ [E0, E0 + nm[. There exists a compact operatorK0,

an interval1 3 λ such that

11(H∞)[H∞, iB]11(H∞) > (d(λ)− ε)11(H∞)+K0.

H2(n): Let ε > 0 andλ ∈ [E0, E0 + nm[. There exists an interval1 3 λ suchthat

11(H∞)[H∞, iB]11(H∞) > (d(λ)− ε)11(H∞).

258 ZIED AMMARI

H3(n): Let µ > 0, ε0 > 0. There existsδ > 0 such that for allλ ∈ [E0, E0 +nm− ε0], one has

11δλ(H∞)[H∞, iB]11δλ(H∞) > (dµ(λ)− ε)11δλ(H∞).S1(n): τ is closed countable set in[E0, E0 + nm].S2(n): For allλ1 6 λ2 6 E0 + nm with [λ1, λ2] ∩ τ = ∅, one has

dim 1pp[λ1,λ2](H∞)H <∞.

The sketch of the proof is given by

S2(n− 1)⇒ S1(n),

(S1(n),H3(n− 1))⇒ H1(n),

H1(n)⇒ H2(n),

H2(n)⇒ H3(n),

H1(n)⇒ S2(n).

H(1) andS(1) are immediate because the spectrum ofH∞ is discrete in[E0, E0+m[. S2(n− 1)⇒ S1(n) is obvious.H1(n)⇒ H2(n),H2(n)⇒ H3(n) follow usingarguments in [CFKS], [Mr]. The implicationH1(n)⇒ S2(n) is based in the Virialtheorem which holds sinceH∞ ∈ C1(B), see [ABG, Prop. 7.2.10]. So we haveonly to prove the implication(S1(n),H3(n− 1))⇒ H1(n).

Let χ ∈ C∞0 (R)

χ(Hκ)

= I (jR)1{0}(N∞)I ∗(jR)χ(Hκ)+ I (jR)1[1,∞[(N∞)I ∗(jR)χ(Hκ) (5.8)

= 0(qR)χ(Hκ)+ I (jR)1[1,∞[(N∞)χ(H extκ )I ∗(jR)+ o(R0). (5.9)

(5.8) follows from the fact thatI (jR)I ∗(jR) = 1 and I (jR) is bounded. (5.9)follows from the fact that

0(qR) = I (jR)1{0}(N∞)I ∗(jR), qR = (jR)2; I (jR) = 0(jR)U,and Lemma 4.15. We notice that the term0(qR)χ(Hκ) is compact since0(qR)(H0

+1)−12 is compact which is proved in [DG2, Lemma 4.2].

Letλ ∈ [E0, E0+nm[. SinceS2(n−1)⇒ S1(n), the setτ is closed in[E0, E0+nm], which givesd(λ) = supµ>0 d

µ(λ). So we can chooseµ such thatdµ(λ) >d(λ)− ε

3.H3(n− 1) gives forλ1 < E0+ (n− 1)m

11δλ1(H∞)[H∞, iB]11δλ1

(H∞) >(dµ(λ1)− ε3

)11δλ(H∞).


Replacingλ1 with λ− d0(ω(k)), we obtain

11δλ(H∞ + 1⊗ d0(ω(k)))([H∞, iB] + 1⊗ d0(|∇ω|2))××11δλ(H∞ + 1⊗ d0(ω(k)))1[1,∞[(N∞)

> 11δλ(H∞ + 1⊗ d0(ω(k)))×

×(dµ(λ− 1⊗ d0(ω(k)))+ 1⊗ d0(|∇ω|2)− ε

3

)1[1,∞[(N∞)

>(dµ(λ)− ε

3

)11δλ(H∞ + 1⊗ d0(ω(k)))(H∞ + 1⊗ d0(ω))1[1,∞[(N∞)

>(dµ(λ)− 2ε

3

)11δλ(H∞ + 1⊗ d0(ω(k)))(H∞ + 1⊗ d0(ω))1[1,∞[(N∞).

Let χ ∈ C∞0 (R), χ1 ∈ C∞0 (R) such thatχ1χ = χ . One has uniformly inκ:

χ(Hκ)[Hκ, iB]χ(Hκ)= 0(qR)χ(Hκ)[Hκ, iB]χ(Hκ ) . . .++ I ∗(jR)1[1,+∞[(N∞)χ(H ext

κ )I ∗(jR)χ1(Hκ)[Hκ , iB]χ(Hκ)+ (5.10)

+o(R0)

= 0(qR)χ(Hκ)[Hκ, iB]χ(Hκ )++ I ∗(jR)1[1,+∞[(N∞)χ(H ext

κ )I ∗(jR)[Hκ, iB]χ(Hκ)+ o(R0) (5.11)

= 0(qR)χ(Hκ)[Hκ, iB]χ(Hκ )++ I ∗(jR)1[1,+∞[(N∞)χ(H ext

κ )[H extκ , iBext]χ(H ext

κ )I ∗(jR)+ (5.12)

+o(R0).

(5.10) follows by (5.9). Lemma 4.15(i) gives (5.11) and (5.12) follows by Lemma5.6.

Lemma 5.5(ii) proves that

0(qR)χ(Hκ)[Hκ, iB]χ(Hκ)→ 0(qR)χ(H∞)[H∞, iB]χ(H∞)norm limit. Now, lettingκ → ∞ in the expression (5.12) which holds uniformlyin κ and using the fact that0(qR)χ(Hκ)[Hκ, iB]χ(Hκ) is compact, we obtain:

χ(H∞)[H∞, iB]χ(H∞)= K1(R)+ I ∗(jR)1[1,+∞[(N∞)χ(H ext

∞ )[H ext∞ , iB

ext]χ(H ext∞ )I

∗(jR)++o(R0),

260 ZIED AMMARI

whereK1(R) is a compact operator. This gives forχ such that suppχ ⊂ [λ − δ,λ+ δ[

χ(H∞)[H∞, iB]χ(H∞) >(d(λ)− 2ε

3

)χ2(H∞)+K1(R)+ o(R0).

ChoosingR large enough, we obtainH1(n). Properties (ii), (iii) are standard con-sequences of (i). 2

6. Construction of the Wave Operators

6.1. ASYMPTOTIC FIELDS

In this subsection we prove the existence of asymptotic fields using the Cookmethod, see, e.g., [H-K]. We setht := e−itω(k)h, for h ∈ h and we denote byh0 the spaceC∞0 (R3 \ {0}). We introduce Heisenberg derivatives:

IDκ := ∂t + i[Hκ, ·],ID := ∂t + i[H, ·].

Since the existence of asymptotic fields in time±∞ is similar, we will restraintproofs of this subsection to the case+∞.

THEOREM 6.1. (i)For h ∈ h the strong limits

W±(h) := s − limt→±∞ eitHW(ht )e

−itH (6.1)

exist and are called asymptotic Weyl operators.(ii) Furthermore

W±(h)(H + i)−1 = limt→±∞eitHW(ht)(H + i)−1e−itH . (6.2)

(iii) The map

h 3 h 7→ W±(h) is strongly continuous,

h 3 h 7→ W±(h)(H + 1)−1 is norm continuous.

(iv) The Weyl commutation relations hold:

W±(h)W±(g) = ei2 Im(h|g)W±(h+ g),

W±(h)∗ = W±(−h).(v) The Hamiltonian preserves the asymptotic Weyl operators:

eitHW±(h)e−itH = W±(h−t ).


Proof.We have the relation onH

W(ht ) = e−itH0W(h)eitH0.

Hence we can define∂tW(ht ) as quadratic form onD(H0)

∂tW(ht ) = −i[H0,W(ht )]. (6.3)

Using (6.3) and Theorem 2.9 we have, since Im(ht |vκ) ∈ B(K), the followingidentity onH

∂t(eitHκW(ht )e

−itHκ ) = ieitHκ Im(ht |vκ)W(ht)e−itHκ . (6.4)

We will first prove (6.1) and (6.2) forh ∈ h0 then we extend toh ∈ h. Let h ∈h0, we notice in this case that(hs|vκ) = (hs|vκ1), for κ > κ1. Since eitHκ is astrongly continuous unitary group and using the inequality (2.1) we see thatt 7→eitHκ Im(ht |vκ1)W(ht )e

−itHκ is strongly measurable. Hence, by integrating (6.4) weobtain onH the identity:

eitHκW(ht )e−itHκ (6.5)

= W(h)+ i∫ t

0eisHκ Im(hs|vκ1)W(hs)e

−isHκ ds, h ∈ h0.

Using Theorem 3.8 and the convergence dominated theorem, lettingκ → ∞ in(6.5), we obtain

eitHW(ht)e−itH (6.6)

= W(h)+ i∫ t

0eisH Im(hs|vκ1)W(hs)e

−isH ds, h ∈ h0.

Moreover

eitHW(ht)(H + i)−1e−itH (6.7)

= W(h)(H + i)−1+ i∫ t

0eisH Im(hs|vκ1)W(hs)(H + i)−1e−isH ds.

Clearly there existsε > 0 such that∑

i〈xi〉1+ε(H + i)−1 is bounded since∑i〈xi〉1+ε(H∞ + i)−1 is bounded. Lemma A.2 gives that

∑i〈xi〉−1−ε Im(hs|vκ1) ∈

O(s−1−ε). Then the existence of the limits (6.2) and consequently (6.1) forh ∈ h0

follows.Let h ∈ h andhn ∈ h0 a sequence such that limn→∞ hn = h in h. Corollary

4.11 and inequality (2.1) gives

‖(eisHW(hs)e−isH − eitHW(ht )e

−itH ) (H + i)−1‖6 c (‖(eisHW(hn,s)e

−isH − eitHW(hn,t )e−itH )(H + i)−1‖ + ‖hn − h‖ε).

262 ZIED AMMARI

This inequality gives the existence of (6.2) and (6.1) forh ∈ h. This shows (i),(ii). Using (2.1) and Corollary 4.11 we have

‖eisH (W(hs)−W(gs))e−isH (H + i)−1‖ 6 c ‖h− g‖−ε. (6.8)

Taking the limits →∞ in (6.8) we obtain

‖(W+(h)−W+(g))(H + 1)−1‖ 6 c ‖h− g‖−ε.This proves (iii). The rest follows from simple computations. 2THEOREM 6.2. The five following assertions hold:

(i) There exist self-adjoint operatorsφ±(h), called asymptotic fields, such that

W±(h) = eiφ±(h) for h ∈ h.

(ii) For hi ∈ h, i = 1 . . . n. We haveD((H + i)n2 ) ⊂ D(

∏n1 φ±(hi)) and

n∏i=1

φ±(hi)(H + i)−n2 = lim

t→±∞eitHn∏i=1

φ(hi,t )e−itH (H + i)−

n2 .

(iii) The map

(h1, . . . , hn) 7→n∏i=1

φ±(hi)(H + i)−n2 is norm continuous.

(iv) The commutation relations hold as quadratic forms onD(φ±(h))∩D(φ±(g))[φ±(h), φ±(g)] = iIm(h|g).

(v) We have

eitHφ±(h)e−itH = φ±(h−t ).Proof. Sinces → W+(sh) is strongly continuous using Theorem 6.1(iii), (i)

follows from Stone’s theorem.We intend to show the existence of the following limit forhi ∈ h, i = 1 . . . n

limt→+∞ eitH

n∏i=1

φ(hi,t )e−itH (H + 1)−

n2 . (6.9)

Lethi ∈ h0, i = 1 . . . n. As in the previous proof we have the following identityas quadratic form onD(H0) which extends as an operator identity onH :

∂t

(n∏i=1

φ(hi,t )(N + 1)−n2

)= −i

[H0,

n∏i=1

φ(hi,t )

](N + 1)−

n2 . (6.10)


Now we compute the derivative

∂t

(eitHκ

n∏i=1

φ(hi,t )(Hκ − Eκ + i)−n2 e−itHκu, v

)

=(∂t

n∏i=1

φ(hi,t )(Hκ − Eκ + i)−n2 e−itHκu,e−itHκ v

)+

+(

n∏i=1

φ(hi,t )(Hκ − Eκ + i)−n2 e−itHκ u,Hκe

−itHκ v

)−

−(Hκ(Hκ − Eκ + i)−

n2 e−itHκu,

n∏i=1

φ(hi,t )e−itHκ v

).

We have(hi,t |vκ) = (hi,t |vκ1) for κ > κ1. Hence, we have onH

∂t

(eitHκ

n∏i=1

φ(hi,t )(Hκ − Eκ + i)−n2 e−itHκ

)

= ieitHκn∑j=1

n∏i 6=j

Im(hj,t |vκ1)φ(hi,t )(Hκ − Eκ + i)−n2 e−itHκ . (6.11)

Lettingκ →∞ in (6.11) we obtain

∂t

(eitH

n∏i=1

φ(hi,t )(H + i)−n2 e−itH

)(6.12)

= ieitHn∑j=1

n∏i 6=j

Im(hj,t |vκ1)φ(hi,t )(H + i)−n2 e−itH .

Since∑

i〈xi〉−1−εIm(hi,t |vκ1) ∈ O(t−1−ε), the dominated convergence theoremgives the existence of (6.9) forh ∈ h0.

Let hi ∈ h, i = 1 . . . n andhi,` ∈ h0 sequences such that lim` hi,` = hi, in h.Using Corollary 2.4 and Corollary 4.8, we obtain the inequality∥∥∥∥∥

(eisH

n∏i=1

φ(hi,s)e−isH − eitH

n∏i=1

φ(hi,t )e−itH

)(H + 1)−

n2

∥∥∥∥∥6 c

(∥∥∥∥∥(

eisHn∏i=1

φ(hi,`,s)e−isH − eitH

n∏i=1

φ(hi,`,t )e−itH

)(H + 1)−

n2

∥∥∥∥∥++

n∑i=1

‖hi − hi,`‖).

264 ZIED AMMARI

Cauchy criterion for the convergence, proves the existence of the limit (6.9).To complete (ii) it suffices to show by induction inn and for u ∈ H , the

existence of following limit

lims→0

limt→+∞eitH

(1

s

(W(sh1,t )− 1

) n∏i=2

φ(hi,t )−

− in∏i=1

φ(hi,t )

)(H + 1)−

n2 e−itHu = 0. (6.13)

We first prove (6.13) foru ∈ D((H +1)−ε), ε > 0, then by an argument of densitywe obtain (6.13) foru ∈ H . We recall (2.2):

lims→0

sup‖h‖6c

s−1‖(W(sh)− 1− isφ(h))(N + 1)−12−ε‖ = 0.

We see that

lims→0

supt∈R

∥∥∥∥∥(

1

s(W(sh1,t )− 1)

n∏i=2

φ(hi,t )− in∏i=1

φ(hi,t )

)(H + 1)−

n2−ε∥∥∥∥∥ = 0.

This completes the proof of (ii). (iii) follows from (ii) and Corollary 2.4 (ii). (iv)follows from the properties of CCR representations. 2DEFINITION 6.3. We define the asymptotic creation and annihilation operatorsonD(φ±(h)) ∩D(φ±(ih))

a±(h) := 1√2(φ±(h)− iφ±(ih)),

a±∗(h) := 1√2(φ±(h)+ iφ±(ih)).

We denote bya±](h) the operatora±∗(h) or a±(h).

We formulate now a theorem which follows from the previous one.

THEOREM 6.4. (i)a±∗(h) anda±(h) are closed operators.(ii) For hi ∈ h, i = 1 . . . n. We haveD((H + i)

n2 ) ⊂ D(

∏ni=1 a

±](hi)) and

n∏i=1

a±](hi)(H + i)−n2 = lim

t→±∞ eitHn∏i=1

a](hi,t )e−itH (H + i)−

n2 .

(iii) The map

(h1, . . . , hn) 7→n∏i=1

a±](hi)(H + i)−n2 is norm continuous.


(iv) The commutation relations hold as quadratic form onD(a±(h))∩D(a±(g))[a±(h), a±∗(g)] = (h|g)1,[a±(h), a±(g)] = [a±∗(h), a±∗(g)] = 0.

(v) We have

eitH a±](h)e−itH = a±](h−t ).

Similar results hold for the modified HamiltonianH∞. We formulate this in thefollowing theorem.

THEOREM 6.5. (i)For h ∈ h the following limit exists

W±(h) : = U ∗∞W±(h)U∞

= s- limt→±∞eit H∞W(ht )e

−it H∞.

(ii) h → W±(h) is a CCR representation. We denote bya±∗(h), a±(h) thecreation and annihilation operators associated to this representation.

(iii) For hi ∈ h, i = 1 . . . n. We haveD((H∞ + i)n2 ) ⊂ D(

∏ni=1 a

±](hi)) and

n∏i=1

a±](hi)(H∞ + i)−n2 = lim

t→±∞eit H∞n∏i=1

a](hi,t )e−it H∞(H∞ + i)−

n2 ,

wherea±](h) denote eithera±∗(h) or a±(h).(iv) The map

(h1, . . . , hn) 7→n∏i=1

a±](hi)(H∞ + i)−n2 is norm continuous.

(v) We have

eit H∞ a±](h)e−it H∞ = a±](h−t ).Proof.The existence of the strong limit follows from (6.1) and the fact that

U ∗∞W(ht )U∞ = e−iIm(G∞|ht )W(ht ),

w- limt→+∞ht = 0.

This prove (i). Theorem 6.1(iv) and (i) give

W±(h) W±(g) = ei2 Im(h|g)W±(h+ g),

W±(h)∗ = W±(−h).

266 ZIED AMMARI

This proves CCR representation. (iii) is a consequence of Theorem 6.2(ii) and thefact that

U ∗∞φ(ht )U∞ = φ(ht )+ Im(G∞|ht).The rest follows from Theorem 6.4. 2

6.2. WAVE OPERATORS

We recall the construction of the Fock subrepresentation of a CCR representation.Details can be found in [BR], [DG3]. Letg be pre-Hilbert space, and denote byg

its completion. We define the space ofvacuaassociated to a CCR representationπover g:

Kπ := {u ∈ H | aπ(h)u = 0, h ∈ g}.

PROPOSITION 6.6. (i)Kπ is a closed space.(ii) Kπ is contained in the set of analytic vectors ofφπ(h), h ∈ g.

Let Hπ :=Kπ ⊗ 0(g). We define

�π : Kπ ⊗ 0fin(g)→ H ,

�πψ ⊗ φ(h)p� := φπ(h)pψ, h ∈ g, ψ ∈Kπ .

PROPOSITION 6.7.The map�π extends to an isometric map

�π : Hπ → H ,

satisfies�π1⊗ a](h) = a]π (h), h ∈ g.

Theorem 6.1 shows that asymptotic Weyl operators define a CCR representa-tion. Then we define the space ofvacuain our case

K± := {u ∈ H | a±(h)u = 0, h ∈ h}.We denote byH± the spaceK± ⊗ 0(h).

PROPOSITION 6.8.The following three assertions hold:

(i) K± is closedH -invariant space.(ii) For hi ∈ h, i = 1 . . . n. One hasK± ⊂ D(

∏ni=1 a

±∗(hi)).(iii) Ran1pp(H) ⊂K±.


Proof.The fact thatK± isH -invariant follows from Theorem 6.4(v). (i) and (ii)follow by Proposition 6.7. Let us prove now (iii). Letu ∈ H such thatHu = Eu,one has

limt→±∞ eitHa(ht)e

−itHu = 0,

since

s- limt→±∞ a(ht) = 0

and

eitH a(ht)e−itHu = (E + i)eit (H−E)a(ht )(H + i)−1u.

This meansa±(h)u = 0. 2We define

H± := H |K± ⊗ 1+ 1⊗ d0(ω),

and thewave operator

�± : H± → H ,

�±ψ ⊗n∏i=1

a∗(hi)� :=n∏i=1

a±∗(hi)ψ, for ψ ∈ K±, hi ∈ h, i = 1 . . . n.

THEOREM 6.9. �± is a unitary map satisfying

a±](h)�± = �± 1⊗ a](h), f or h ∈ h,

H�± = �±H±.Proof. Proposition 6.8 gives that�± is isometric and satisfies properties an-

nounced in the theorem. Let prove that�± is unitary. Using [DG3, Thm. 3.3], itsuffices to show that the CCR representationh→ W±(h) admits a densely definednumber operator. For each finite-dimensional spacef ⊂ h, we define as quadraticform the following expression

n±f(u) :=

dimf∑i=1

‖a±(hi)u‖2, {hi} is an orthonormal basis off, u ∈ H .

Now we show thatn±(u) := supf n±f(u) is densely defined:

‖n±f(u)‖2 6 lim

t→±∞

dimf∑i=1

‖a(hi,t )e−itHu‖2

6 limt→±∞(e

−itHu,Ne−itHu).

268 ZIED AMMARI

We conclude using Corollary 4.11 thatn±(u) 6 c ‖(H + b) 12u‖. ThusD(H

12 ) ⊂

D(n±) and Ran�± = H . 2We define anextended wave operator

�ext,± :∞⊕n=0

D((H∞ + 1)n2 )⊗⊗ns h→ H ,

�ext,±ψ ⊗n∏i=1

a∗(hi)� :=n∏i=1

a±∗(hi)ψ, ψ ∈ D((H∞ + 1)n2 ).

We set

K± := U ∗∞K±, H± := K± ⊗ 0(h).Then we have a wave operator of the modified Hamiltonian:

�± : H± → H ,

�±ψ ⊗n∏i=1

a∗(hi)� :=n∏i=1

a±∗(hi)ψ, for ψ ∈ K±, hi ∈ h, i = 1 . . . n.

We notice that�ext,±|H± = �±. This suggests to treat sometimes�ext,± as a par-

tial isometry. Another construction of the extended wave operator is given by thefollowing theorem, see [DG2, Thm. 5.7]:

THEOREM 6.10. (i)Letu ∈ D(�ext,±). Then one has

limt→±∞ eit H∞Ie−it Hext∞ u = �ext,±u,

whereI is the scattering identification operator defined in the Subsection2.1.(ii) Let χ ∈ C∞0 (R). Then Ranχ(H ext∞ ) ⊂ D(�ext,±) and the operators

Iχ(H ext∞ ), �ext,±χ(H ext∞ ) are bounded. Moreover

limt→±∞ eit H∞Ie−it Hext∞ χ(H ext

∞ ) = �ext,±χ(H ext∞ ).

7. Propagation Estimates

We make the following notations for the Heisenberg derivatives

dl0 := ∂t + i[ω(k), ·],ID0 := ∂t + i[d0(ω), ·],ÎDκ := ∂t + i[Hκ, ·].


PROPOSITION 7.1.Letχ ∈ C∞0 (R). ForR′ > R > 1, there existsc independentfromκ such that we have forκ 6∞∫ ∞

1

∥∥∥∥d0

(1[R,R′]

( |x|t

)) 12

χ(Hκ)e−it Hκ u

∥∥∥∥2 dt

t6 c‖u‖2.

Proof. We use a standard method in scattering theory of the N-body problem[Gr], [SS]. It is based on a technical lemma, see, e.g., [DG1, Lemma B.4.1].

Let F ∈ C∞(R) be a cutoff function equal to 1 near∞, to 0 near the origin,with F ′(s) > 1[R,R′](s). We consider the observable

8(t) := χ(Hκ)d0(F

( |x|t

))χ(Hκ).

By Lemma A.3, it is enough to show that

ÎDκ8(t) > t−1C0 χ(Hκ)d0

(F ′( |x|t

))χ(Hκ)+O(t−1−µ) (7.1)

uniformly in κ to have the inequality.One has

ÎDκ8(t) = χ(Hκ) d0

(dl0F

( |x|t

))χ(Hκ) +

+χ(Hκ)[Iκ , i d0

(F

( |x|t

))]χ(Hκ).

Using the fact that

dl0F

( |x|t

)> c0

tF ′( |x|t

)+O(t−2),

it is sufficient to show that the second term in the previous identity is O(t−1−µ), µ >0 uniformly inκ, to have (7.1).

By simple commutation relations we obtain:[φ(vκ0),d0

(F

( |x|t

))]= iφ

(iF

( |x|t

)vκ0

),

[a∗

2(rjκ ),d0

(F

( |x|t

))]= 2a∗(rjκ )a

∗(F

( |x|t

)rjκ

),

[a2(rjκ ),d0

(F

( |x|t

))]= −2a(rjκ )a

(F

( |x|t

)rjκ

), (7.2)

[a∗(rjκ )a(r

jκ ),d0

(F

( |x|t

))]

270 ZIED AMMARI

= a∗(rjκ )a(F

( |x|t

)rjκ

)+ a∗

(F

( |x|t

)rjκ

)a(rjκ ),[

Dj a(rjκ ),d0

(F

( |x|t

))]= Dj a

(F

( |x|t

)rjκ

),[

a∗(rjκ )Dj ,d0

(F

( |x|t

))]= a∗

(F

( |x|t

)rjκ

)Dj.

Using the functional calculus formula (4.18) and Corollary 4.8, it is enough toestimate

(N + 1)−n(H0+ c)− 12

[Iκ , i d0

(F

( |x|t

))](H0+ c)− 1

2 , c > 0.

Using (7.2) and Lemma 3.4, we obtain:

‖(N + 1)−n(H0+ c)− 12

[Iκ , i d0

(F

( |x|t

))](H0+ c)− 1

2‖

6 c

(∥∥∥∥(V + 1)−12F

( |x|t

)vκ0

∥∥∥∥++‖ω s−1

4 rjκ ‖∥∥∥∥(V + 1)−

s2ω

s−14 F

( |x|t

)rjκ

∥∥∥∥+ (7.3)

+‖(V + 1)s−1

2 ω−s2 rjκ ‖

∥∥∥∥(V + 1)−s2ω

s−12 F

( |x|t

)rjκ

∥∥∥∥++‖D (K + c)− 1

2‖∥∥∥∥(V + 1)−

s2ω

s−12 F

( |x|t

)rjκ

∥∥∥∥).It remains to see that the terms∥∥∥∥(V + 1)−

12F

( |x|t

)vκ0

∥∥∥∥, ∥∥∥∥(V + 1)−s2 ω−

s−12 F

( |x|t

)rκ

∥∥∥∥ and∥∥∥∥(V + 1)−s2 ω−

s−14 F

( |x|t

)rκ

∥∥∥∥are integrable for(1− s) small enough. They are O(t−1−µ) by (4.24). Then usingLemma A.3 we finish the proof of the estimate announced in the proposition.2PROPOSITION 7.2.Letχ ∈ C∞0 (R), 0< c0 < c1, and

2[c0,c1](t) := d0

(⟨x

t−∇ω(k),1[c0,c1]

( |x|t

)(x

t−∇ω(k)

)⟩).

One has uniformly inκ 6∞∫ ∞1‖2[c0,c1](t)

12χ(Hκ)e

−it Hκ u‖2 dt

t6 c‖u‖2.


Proof.LetR0(x) ∈ C∞ be a function such that:

R0(x) = 0, for |x| 6 c0

2,

R0(x) = 1

2x2 + c, for |x| > 2c1,

∇2xR0 > 1[c0,c1](|x|).

We choosec1 > 2, c2 > c1+ 1 and we define the function

R(x) := F(|x|)R0(x),

whereF(s) = 1, if s 6 c1, F (s) = 0, if s > c2.We set

b(t) := R(x

t

)− 1

2

(⟨∇R

(x

t

),x

t− ∇ω(k)

⟩+ hc

).

We consider the observable

8(t) := χ(Hκ)d0(b(t))χ(Hκ).Pseudodifferential calculus gives

χ(Hκ)ID0d0(b(t))χ(Hκ)

> χ(Hκ)

(1

t2[c0,c2](t)−

1

td0

(1[2,c2]

( |x|t

)))χ(Hκ)+O(t−2).

The first term will serves in the application of Lemma A.4 and the second is in-tegrable along the evolution using Proposition 7.1. To complete the proof of theproposition, it suffices to show uniformly inκ that:

χ(Hκ)[Iκ , i d0(b(t))]χ(Hκ ) ∈ O(t−1−µ), µ > 0. (7.4)

As in the Proposition 7.1, using (7.2) and (7.3), we see that (7.4) is bounded by asum of terms

‖(V + 1)−12b(t)vκ0‖, ‖(V + 1)−

s2 ω−

1−s2 b(t)rκ‖

and

‖(V + 1)−s2 ω−

1−s4 b(t)rκ‖.

By (4.24) these terms are O(t−1−µ), µ > 0, for (1− s) small enough. We end theproof by using Lemma A.3. 2PROPOSITION 7.3.Let 0 < c0 < c1, J ∈ C∞0 ({c0 < |x| < c1}), χ ∈ C∞0 (R).For 16 i 6 3, one has uniformly inκ 6∞∫ ∞

1

∥∥∥∥d0

(∣∣∣∣J(xt)(

xi

t− ∂iω(k)

)+ hc

∣∣∣∣) 12

χ(Hκ)e−it Hκ u

∥∥∥∥2dt

t6 c‖u‖2.

272 ZIED AMMARI

Proof.We set

A :=(x

t−∇ω(k)

)2

+ t−δ,

b(t) := J(x

t

)A

12J

(x

t

).

Let J1 ∈ C∞0 ({c0 < |x| < c1}), 0 6 J 6 1, J = 1 near the support ofJ1. Weconsider the observable

8(t) := −χ(Hκ)d0(b(t))χ(Hκ).

One has

χ(Hκ)ID0 d0(b(t))χ(Hκ) = −χ(Hκ)d0(dl0b(t))χ(Hκ),

and we have using [DG2, Lemma 6.4]

χ(Hκ)ID0 d0(b(t))χ(Hκ)

> c0

tχ(Hκ)d0

(∣∣∣∣J1

(x

t

)(xi

t− ∂iω(k)

)+ hc

∣∣∣∣)χ(Hκ)−− ctχ(Hκ)d0

(⟨x

t− ∇ω, J2

(x

t

)(x

t−∇ω

)⟩)χ(Hκ)+O(t−1−µ).

The second term is integrable along the evolution by Proposition 7.2. It’s enoughto show that

χ(Hκ)[Iκ , i d0(b(t))]χ(Hκ ) ∈ O(t−1−µ), µ > 0, uniformly in κ.

This follows by using (7.3), the fact thatJ (xt)A

12 ∈ O(1) and Lemma A.2. Using

Lemma A.3 we end the proof. 2PROPOSITION 7.4.Letχ ∈ C∞0 (R), supported inR\ (τ ∪σpp(H∞)). There existε > 0, C independent inκ and a sequence ofHκ such that forκ 6∞, we have∫ ∞

1

∥∥∥∥0(1[0,ε]( |x|t))χ(Hκ)e−it Hκ u

∥∥∥∥2dt

t6 C‖u‖2.

Proof. We notice that Proposition 7.4 is a minimal velocity estimate for a se-quence ofHκ which is uniform inκ. Let χ supported nearλ such thatλ ∈ R \(τ ∪ σpp(H∞)). Then there exists a sequenceHκ such thatλ ∈ R \ (τ ∪ σpp(Hκ)).Lemma 5.5 in Mourre estimate section gives

χ(Hκ)[Hκ, iB]χ(Hκ) > cκχ2(Hκ).


Let ε > 0. Letq ∈ C∞0 (|x| 6 2ε) such that 06 q 6 1, q(x) = 1, if |x| 6 ε.We set

8κ(t) := χ(Hκ)0(qt )Bt0(qt )χ(Hκ).

The Heisenberg derivative of8κ(t) is

ÎDκ8κ(t) = χ(Hκ)d0(qt ,dl0qt )B

t0(qt )χ(Hκ)+ hc+

+χ(Hκ)[Iκ , i0(qt )] Bt0(qt )χ(Hκ)+ hc+

+ t−1χ(Hκ)0(qt )[Hκ, iB]0(qt )χ(Hκ)−

− t−1χ(Hκ)0(qt )B

t0(qt )0(qt )χ(Hκ)

=: R1κ + R2

κ + R3κ + R4

κ .

We claim that

R2κ ∈ O(t−1−µ), µ > 0. (7.5)

To prove this, we use the estimates (4.22) in the proof of Lemma 4.14 to obtain

χ(Hκ)[Iκ , i0(qt )](d0(ω)+ 1)−12 ∈ O(t−1−µ), µ > 0. (7.6)

To prove (7.5) it suffices by Corollary 4.8 to show that

(d0(ω)+ 1)12B

t0(qt )(d0(ω)+ 1)−

12 (N + 1)−2 ∈ O(1). (7.7)

A simple explicit calculus gives

(d0(ω)+ 1)12B

t0(qt ) (7.8)

= B

t0(qt )(d0(ω)+ 1)

12 +

[(d0(ω)+ 1)

12 ,B

t

]0(qt )+

+ Bt[(d0(ω)+ 1)

12 , 0(qt )].

Clearly Bt0(qt )(N + 1)−2 ∈ O(1) and[

(d0(ω)+ 1)12 ,B

t

]0(qt ) = (d0(ω)+ 1)−

12 d0(|∇ω|2)0(qt ) ∈ O(1).

274 ZIED AMMARI

To estimate the last term in (7.8), we write on then-particle sector:

[(d0(ω)+ 1)12 , 0(qt )]|K⊗snh

=n∑j=1

j−1∏i=1

q

(xi

t

)[( n∑i=1

ω(ki)+ 1

) 12

, q

(xj

t

)] n∏i=j+1

q

(xi

t

)

=:n∑j=1

Rj(t).

Pseudodifferential calculus gives that

xkRj(t) ∈ O(1), uniformly ink, j.

This proves (7.7). Now (7.6) and (7.7) imply thatR2κ ∈ O(t−1−ν), for κ <∞.

We consider nowR1κ . We have:

dl0qt = − 1

2t

⟨x

t−∇ω(k),∇q

(x

t

)⟩+ hc + rt =: 1

tgt + rt ,

wherert ∈ O(t−2). We have using (2.3) and Corollary 4.8:∥∥∥∥χ(Hκ)d0(qt , rt )B

t0(qt )χ(Hκ)

∥∥∥∥ ∈ O(t−2).

We set

B1 := χ(Hκ)d0(qt , gt )(N + 1)−12 , B2 := (N + 1)

12B

t0(qt )χ(Hκ).

So we obtain the inequality

R1κ > −ε−1

0 t−1B1B∗1 − ε0t

−1B2B∗2 .

Using arguments in [DG2, Prop. 6.5], we obtain

−B2B∗2 > −C1χ(Hκ)0(q

t )2χ(Hκ)− Ct−1,∫ ∞1‖B1e

−it Hκ u‖ 6 C‖u‖2.

Using Lemma 4.14 and Theorem 5.7, we have

R3κ > C0t

−1χ(Hκ)0(qt )2χ(Hκ)− Ct−2.

We have

−R4κ 6 C2

ε

tχ(Hκ)0(q

t )2χ(Hκ)+ Ct−2.


Collecting the four terms, we obtain

ÎDκφκ(t) > −ε0t−1B2B

∗2 + R2

κ + R3κ + R4

κ

> (C0− ε0C1− εC2)t−1χ(Hκ)0(q

t )2χ(Hκ)+ Ct−2

> C0χ(Hκ)t−10(qt )2χ(Hκ)− R(t),

whereR(t) is integrable. By Lemma A.3 we obtain the inequality announced in theproposition forχ supported near a one energy levelλ. Then we complete the prooffor an arbitraryχ using a standard argument, see, e.g., [DG1, Proposition 4.4.7].2

8. Asymptotic Completeness

In this section we prove the main result of this paper, which is the asymptoticcompleteness of the Nelson Hamiltonian. This is the subject of Theorem 8.5, wherewe prove Ran1pp(H) =K±.

THEOREM 8.1. Let q, q ∈ C∞0 (R3) such that0 6 q, q 6 1, q, q = 1 on aneighborhood of zero andqt := q(x

t).

(i) The following limits exist

0±(q) := s- limt→±∞ eit H∞ 0(qt )e−it H∞

= limκ→+∞ s- lim

t→±∞ eit Hκ 0(qt )e−it Hκ .

(ii) We have

0±(qq) = 0±(q)0±(q),06 0±(q) 6 0±(q) 6 1, if 0 6 q 6 q,

[H∞, 0±(q)] = 0.

(iii) We haveRan0±(q) ⊂ K±.

Proof. It is sufficient using a density argument and Lemma 4.14 to show forχ ∈ C∞0 (R) the existence of the limit

s- limt→±∞ eit H∞χ(H∞) 0(qt )χ(H∞)e−it H∞ . (8.1)

Using Lemma A.4, we see that as for all asymptotic limits these amounts to boundHeisenberg derivatives uniformly inκ. We have onH :

∂t(eit Hκ χ(Hκ) 0(q

t )χ(Hκ)e−it Hκ )

= eit Hκ χ(Hκ)d0(qt ,dl0qt )χ(Hκ)e

−it Hκ ++eit Hκ χ(Hκ)[Iκ , i0(qt )]χ(Hκ)e−it Hκ .

276 ZIED AMMARI

By (4.22) we have uniformly inκ:

χ(Hκ)[Iκ , i0(qt )]χ(Hκ) ∈ O(t−1−ε). (8.2)

We use now an argument introduced in [DG2]:

dl0qt = 1

tgt + rt ,

where

gt = −1

2

((x

t− ∂ω(k)

)∂q

(x

t

)+ hc

)and rt ∈ O(t−2).

The estimate (2.3) and Corollary 4.8 give uniformly inκ

‖χ(Hκ)d0(qt ,dl0rt )χ(Hκ)‖ ∈ O(t−2). (8.3)

The other term will be estimated:

|(e−it Hκ u, χ(Hκ)d0(qt , gt )χ(Hκ)e

−it Hκ u)| (8.4)

6 ‖d0(|gt |) 12χ(Hκ)e

−it Hκ u‖2, u ∈ H .

Using (8.2)–(8.4), Proposition 7.3 and Lemma A.4, we obtain the existence of thelimit (8.1).

The first statement in (ii) follows from the fact that

0(qt qt ) = 0(qt )0(qt ).The second statement follows by

06 0(qt ) 6 0(qt ) 6 1 if 0 6 q 6 q.

The last statement is a consequence of (i) and Lemma 4.14.Let us prove (iii). SinceH∞, 0±(q) commute,0±(q) preservesD(H∞). Con-

sequentlyD(H∞) ∩ Ran0±(q) is dense in Ran0±(q). SinceK± is closed, itis enough to show thatD(H∞) ∩ Ran0±(q) ⊂ K± to prove (iii). Let v ∈D(H∞) ∩ Ran0±(q), v = 0±(q)u.

(H∞ + b)−1a±(h)0±(q)u

= limt→±∞ eit H∞(H∞ + b)−1a(ht)e

−it H∞0±(q)u

= limt→±∞ eit H∞(H∞ + b)−1a(ht)0(q

t )e−it H∞u

= limt→±∞ eit H∞(H∞ + b)−10(qt )a(qtht )e

−it H∞u.

If h ∈ h0, by a stationary phase argument we haveqtht ∈ o(1), t → ±∞. Usingthe fact thath → (H∞ + i)−1a(ht) is continuous, we obtaina±(h)v = 0 for allh ∈ h. This ends the proof. 2


COROLLARY 8.2. Let {qn} ∈ C∞0 (R3) be a decreasing sequence of functionssuch that0 6 qn 6 1, qn = 1 on a neighborhood of0 and

⋂∞n=1 suppqn = {0}.

Then the following limit exist and it does not depend in the choose of the sequence

(i) P±0 := limn→∞0

±(qn),

(ii) RanP±0 ⊂ K±.

Moreover,P±0 is an orthogonal projection.Proof. The existence of the limit (i) follows from Theorem 8.1(ii) and Lemma

A.5. The independence from the choose of the sequence follows from the fact thatthere exists an indexmn such thatqn > qmn, qn > qmn ; limn→∞mn = +∞ andTheorem 8.1(ii). (ii) is a consequence of Theorem 8.1(iii) and (i). 2

Let j0 ∈ C∞0 (R3),0 6 j0,0 6 j∞, j20 + j2∞ 6 1, j0 = 1 near 0. Setj :=

(j0, j∞) andj t := (j t0, j t∞), wherej t0 := j0(xt), j t∞ := j∞( xt ). We recall thatI (j t )

is the operator introduced in Subsection 2.1.

THEOREM 8.3. (i)The following limits exist

s- limt→±∞ eit Hext∞ I ∗(j t )e−it H∞ =: W±(j),

s- limt→±∞ eit H∞I (j t )e−it Hext∞ = W±(j)∗.

(ii) For a bounded Borel function F, we have

W±(j) F (H∞) = F(H ext∞ )W

±(j).

(iii) Let q0, q∞ ∈ C∞(R3), ∇q0, ∇q∞ ∈ C∞0 (R3), 0 6 q0, q∞ 6 1, q0 = 1near0. Setj := (q0j0, q∞j∞). Then

0±(q0)⊗ 0(q∞(∇ω(k)))W±(j) = W±(j ).(iv) Letq ∈ C∞(R3), ∇q ∈ C∞0 (R3), 06 q 6 1, q = 1 near0. Then

W±(j)0±(q) = W±(qj), whereqj = (qj0, qj∞).

(v) Let j = (j0, j∞) be another pair satisfying the conditions stated before thetheorem. Then

W±(j )∗W±(j) = 0(j0j0+ j∞j∞),in particular if j2

0 + j2∞ = 1, thenW±(j) is isometric.(vi) Let j0+ j∞ = 1. If χ ∈ C∞0 (R), then

�ext,±χ(H ext∞ )W

±(j) = χ(H∞).

278 ZIED AMMARI

Proof.To prove (i) we use the same arguments as in Theorem 8.1. Using LemmaA.4, it is enough to prove the existence of the limit

s- limt→±∞ eit Hext∞ χ(H ext

∞ )I∗(j t )e−it H∞χ(H∞), (8.5)

for someχ ∈ C∞0 (R). We compute

∂t(eit Hext

κ χ(H extκ )I ∗(j t )χ(Hκ)e−it Hκ )

= eit Hextκ (χ(H ext

κ )D0I∗(j t )χ(Hκ)+

+ iχ(H extκ )(Iκ ⊗ 1I ∗(j t )− I ∗(j t )Iκ)χ(Hκ))e−it Hκ ,

whereD0 is the asymmetric Heisenberg derivative∂t + iH ext0 . − .iH0. We have

D0I∗(j t ) = dI ∗(j t , d0j

t ).Pseudodifferential calculus gives

d0jt = 1

tgt + rt ,

gt = (gt0, gt∞), gtε = −1

2

((x

t− ∂ω(k)

)∂jε

(x

t

)+ hc

), ε = 0,∞

with rt ∈ O(t−2). Using Corollary 4.8 and (2.4) we obtain

‖χ(H extκ )dI ∗(j t , rt )χ(Hκ)‖ ∈ O(t−2). (8.6)

Using now (2.5) withuti := eit Hκ ui , one obtain

|(ut1|χ(H extκ )dI ∗(j t , gt )χ(Hκ)ut2)|

6 ‖d0(|gt0|)12 ⊗ 1χ(H ext

κ )ut2‖ ‖d0(|gt0|)12χ(Hκ)u

t1‖+

+‖(1⊗ d0(|gt∞|)12 )χ(H ext

κ )ut2‖ ‖d0(|gt∞|)12χ(Hκ)u

t1‖.

Then theκ-uniform integrability of the termχ(H extκ )D0I

∗(j t )χ(Hκ) follows usingProposition 7.1.

Using (4.25) we obtain uniformly inκ

χ(H extκ )(Iκ ⊗ 1I ∗(j t )− I ∗(j t )Iκ)χ(Hκ) ∈ O(t−1−µ).

Then the existence of the limit in (i) follows.(ii) follows by Lemma 4.15. (iii) follows using the fact that

limt→±∞ eitd0(ω)0(qt )e−itd0(ω) = 0(q(∇ω)),

0(qt0)⊗ 0(qt∞)I ∗(j t ) = I ∗(j t ).


(iv) is true since

I ∗(j t )0(qt ) = I ∗((jq)t ).(v) is a consequence of the fact

I (j t )I ∗(j t )0(j t0jt0 + j t∞j t∞).

(vi) One has

H ext∞ 1[k,∞[(N∞) > mk + E0.

Let χ ∈ C∞0 (R). There existsn0 ∈ N such that forn > n0

χ(H ext∞ )1]n,∞[(N∞) = 0. (8.7)

We have

�ext,±χ(H ext∞ )W

±(j)

= �ext,±1[0,n](N∞)χ(H ext∞ )W

±(j) (8.8)

= s- limt→±∞ eit H∞I1[0,n](N∞)χ(H ext

∞ )I∗(j t )e−it H∞ (8.9)

= s- limt→±∞ eit H∞I1[0,n](N∞)I ∗(j t )e−it H∞χ(H∞). (8.10)

(8.8) follows from (8.7). (8.9) follows from the limit (i) and Theorem 6.10. Lemma4.15 and the boundeness of the operatorI1[0,n](N∞)(N0 + 1)−

n2 gives (8.10). We

use now an estimate proved in [DG3]:

‖I1]n,∞[(N∞)I ∗(j t )(N + 1)−1‖ 6 (n+ 1)−1. (8.11)

SinceII ∗(j t ) = 1, lettingn→∞ we obtain�ext,±χ(H ext∞ )W±(j) = χ(H∞). Thiscompletes the proof. 2THEOREM 8.4. Let jn = (j0,n, j∞,n) be a sequence satisfying the hypothesisstated in the beginning of Theorem8.3 such thatj0 + j∞ = 1 and for anyε > 0there existsm,∀n > m, suppj0,n ⊂ [−ε, ε]. Then

�±∗ = w- limκ→+∞ W

±(jn),

K± = RanP±0 .

Proof. Let q ∈ C∞0 (R),0 6 q 6 1 andq = 1 in a neighborhood of zero suchthatqj0,n = j0,n for n large enough. Using Theorem 8.3(iii) and Corollary 8.2 weobtain

0±(q)⊗ 1 W±(jn) = W±(jn),

280 ZIED AMMARI

w- limn→l+∞

P±0 ⊗ 1 W±(jn)− W±(jn) = 0. (8.12)

Let χ ∈ C∞0 (R). We have

�±∗χ(H∞) = �±∗�ext,±χ(H ext∞ )W

±(jn) (8.13)

= w- limn→∞ �

±∗�ext,±χ(H ext∞ )P

±0 ⊗ 1 W±(jn) (8.14)

= w- limn→∞P

±0 ⊗ 1χ(H ext

∞ )W±(jn) (8.15)

= w- limn→∞χ(H

ext∞ )W

±(jn) (8.16)

= w- limn→∞ W

±(jn)χ(H∞). (8.17)

Formula (8.13) follows from Theorem 8.3(iv). (8.14) follows by (8.12). (8.15) istrue sinceP±0 commutes withH ext∞ and that RanP±0 ⊂ K±, �ext,±1K± ⊗ 1 = �±and�±∗�± = 1K± ⊗ 1. (8.16) follows from the fact thatP±0 commutes withH ext∞and (8.12). (8.17) is Theorem 8.3(ii). So we conclude by a density argument that

�±∗ = w- limn→+∞ W

±(jn),

P±0 ⊗ 1 �±∗ = �±∗.So we obtain

Ran�±∗ = K± ⊗ 0(h) ⊂ RanP±0 ⊗ 0(h) ⊂ K± ⊗ 0(h).Hence we prove thatK± = RanP±0 . 2THEOREM 8.5. We have

Ran1pp(H) =K±.

Proof.By Proposition 6.8(iii) we have

Ran1pp(H∞) ⊂ K±.

Then it suffices to show thatK± ⊂ Ran1pp(H∞). Proposition 7.4 gives the exis-tence ofε > 0 and a sequenceHκ such that∫ +∞

1‖0(qt )χ(Hκ)e−it Hκ u‖2 dt

t6 C‖u‖2,

whereχ ∈ C∞0 (R \ (τ ∪ σpp(H∞))) andq ∈ C∞0 ([−ε, ε]), q = 1 for |x| < ε/2.Theorem 8.1 gives that

‖0(qt )χ(Hκ)e−it Hκ u‖ → ‖0±(q)χ(H∞)u‖ = 0, t →±∞


then0±(q)χ(H∞) = 0. So we have RanP±0 ⊂ Ran1τ∪σpp(H∞)(H∞). Theorem 5.7

gives thatτ is a closed countable set andσpp(H∞) can accumulate only atτ , so1pp(H∞) = 1τ∪σpp(H∞)(H∞). This proves Ran1pp(H∞) = K±. Then we prove thetheorem. 2

Appendix

The following theorem follows from the KLMN theorem and [RS, I–IV, Thm.VIII.25].

THEOREM A.1. LetH0 be a positive self-adjoint operator onH . Let forκ 6∞,

Bκ be quadratic forms onD(H12

0 ) such that

|Bκ(ψ,ψ)| 6 a ‖H12

0 ψ‖2 + b ‖ψ‖2,

wherea < 1 uniformly inκ andBκ → B∞ onD(H12

0 ). Then

(i) There exist forκ 6 ∞ self-adjoint operatorsHκ with D(Hκ) ⊂ D(H12

0 )

and

(Hκψ,ψ) = Bκ(ψ,ψ)+ (H12

0 ψ,H12

0 ψ), ψ ∈ D(Hκ),(ii) lim

κ→∞(z −Hκ)−1 = (z−H∞)−1,

(iii) s- limκ→∞ e−itHκ = e−itH∞.

LEMMA A.2. LetF ∈ C∞(R), equal to0 near the origin and bounded near∞.we denote byFR the derivative operatorF(|x|/R). We recall thatx denote thenucleon position observable. One has uniformly inκ

(i) ‖〈x〉−sFRvκ0‖ ∈ O(R−s),

(ii) ‖〈x〉−sω(k)−εFRrκ‖ ∈ O(R−s), ε > 0.

Proof.We have〈x〉−svκ0 ∈ Hs(R3,B(K)). Since we have∥∥∥∥RsF( |Dk|R

)〈x〉−svκ0

∥∥∥∥ 6 c‖|Dk|s〈x〉−svκ0‖

we obtain (i). Let us prove (ii). Pseudodifferential calculus gives∥∥∥∥ω(k)−εF( |Dk|R> 1

)rκ

∥∥∥∥6 c

∥∥∥∥F( |Dk|R> 1

2

)ω(k)−εrκ

∥∥∥∥+ c

R2‖ω(k)−εrκ‖.

282 ZIED AMMARI

Then we obtain (ii) using (i) forω−εrκ , which isL2(R3,dk) uniformly in κ. 2Let H a Hilbert space. Let{Hκ} be a sequence of self-adjoint operators on a

common domainD ⊂ H . We suppose thats-limκ→∞ e−itHκ = e−itH∞ , whereH∞is a self-adjoint operator. We have forχ ∈ C∞0 (R), using [RS, I–IV, Thm. VIII.20;I–IV, Thm. VIII.21]

s- limκ→∞χ(Hκ) = χ(H∞).

LEMMA A.3. Let t → Bt ∈ B(H) andχ ∈ C∞0 (R) such that forκ 6∞‖B∗t Bt χ(Hκ)‖ 6 ct ,

wherect is κ-independent locally integrable function int . If there exist a constantc independent ofκ such that forκ <∞∫ ∞

1‖Bt χ(Hκ)e−itHκ u‖2 dt 6 c ‖u‖2, u ∈ H ,

then ∫ ∞1‖Bt χ(H∞)e−itH∞u‖2 dt 6 c ‖u‖2.

Proof.We have to prove, uniformly inT , that:∫ T

1‖Bt χ(H∞)e−itH∞u‖2 dt 6 c ‖u‖2. (A.1)

We apply the dominated convergence theorem. It is enough to show that

limκ→∞‖Bt χ(Hκ)e

−itHκu‖2 = ‖Bt χ(H∞)e−itH∞u‖2,

since we have‖Bt χ(Hκ)e−itHκu‖2 6 c′ ‖B∗t Bt χ(Hκ)‖‖u‖2. It is easy to see thatto prove (A.1) it suffices to show

w- limκ→∞χ(Hκ)B

∗t Btχ(Hκ) = χ(H∞)B∗t Btχ(H∞).

This follows from the hypothesis‖B∗t Btχ(Hκ)‖ 6 ct for κ 6∞. 2Let Hi , i = 1,2 be two Hilbert spaces. LetHi,κ , i = 1,2 be two sequences of

self-adjoint operators onHi, such thatHi,κ converge in the strong resolvent sensetoHi,∞.

The Lemma A.4 follows from the proof of Lemma A.3 and [DG1, LemmaB.4.2].


LEMMA A.4. Let t → C(t) ∈ B(H1,H2) andχ ∈ C∞0 (R). We suppose that theasymmetric Heisenberg derivatives:

DκC(t) := ∂tC(t)+ i(H1,κC(t)− C(t)H2,κ),

satisfies forκ <∞, the following hypothesis:

(i) DκC(t) = B(t)+ Rκ(t).(ii) ‖χ(H2,κ ) Rκ(t) χ(H1,κ)‖ 6 c t−1−ε, ε > 0, uniformly forκ <∞.(iii ) ‖χ(H2,κ )B(t)‖ 6 ct , ‖B(t) χ(H1,κ)‖ 6 ct , uniformly forκ 6∞,

wherect is κ-independent locally integrable function int.

(iv) |(u2|B(t)u1)| 6 cn∑j=1

‖B2,j (t)u2‖‖B1,j (t)u1‖, where

∫ ∞1‖Bi,j (t) χ(Hi,∞)e−itHi,∞u‖2 dt 6 c ‖u‖2, i = 1,2 and j = 1 . . . n.

Then the following limit exists

s- limt→+∞ eitH2,∞χ(H2,∞) C(t) χ(H1,∞)e−itH1,∞ .

Proof.Let u ∈ H1, v ∈ H2. We have:

(u|eit2H2,∞χ(H2,∞)C(t2)χ(H1,∞)eit2H1,∞)−− (u|eit1H2,∞χ(H2,∞)C(t1)χ(H1,∞)e−it1H1,∞v)

= limκ→∞(u|e

it2H2,κ χ(H2,κ)C(t2)χ(H1,κ)e−it2H1,κ v)−

− (u|eit1H2,κ χ(H2,κ)C(t1)χ(H1,κ)e−it1H1,κ v)

= limκ→∞

∫ t2

t1

(e−itH2,κ u|χ(H2,κ)IDκC(t)χ(H1,κ )e−itH1,κ v)dt

= limκ→∞

∫ t2

t1

(e−itH2,κ u|χ(H2,κ)(B(t)+ R(t))χ(H1,κ)e−itH1,κ v)dt.

Using Lebesgue dominated convergence theorem and, as in the proof of LemmaA.3, the fact that

(iii ) ⇒ w- limκ→∞χ(H2,κ)B(t)χ(H1,κ) = χ(H2,∞)B(t)χ(H1,∞),

284 ZIED AMMARI

we obtain

limκ→∞

∫ t2

t1

(e−itH2,κ u|χ(H2,κ)B(t)χ(H1,κ)e−itH1,κ v)dt (A.2)

=∫ t2

t1

(e−itH2,∞u|χ(H2,∞)B(t)χ(H1,∞)e−itH1,∞v)dt.

We have by (ii):

limκ→∞

∫ t2

t1

(e−itH2,κ u|χ(H2,κ)R(t)χ(H1,κ)e−itH1,κ v)dt

6 c t−ε1 ‖v‖‖u‖, if t1 < t2. (A.3)

Using (iv) we obtain:

limκ→∞

∫ t2

t1

(e−itH2,∞u|χ(H2,κ)B(t)χ(H1,κ)e−itH1,∞v)dt (A.4)

6 c

n∑j=1

∫ t2

t1

‖B2,j (t)e−itH2,∞u‖2 dt ×

∫ t2

t1

‖B1,j (t)e−itH1,∞u‖2 dt

6 c ‖u‖ ‖v‖.(A.2), (A.3) and (A.4) give the existence of the claimed limit. 2We recall here a convergence lemma of positive operators, see, e.g., [DG2,

Lemma A.3]

LEMMA A.5. LetQn be a sequence of commuting self-adjoint operators. If

06 Qn 6 1, Qn+1 6 Qn, Qn+1Qn = Qn+1.

Then there existQ a projection

Q = s-limnQn.

Acknowledgements

The author would like to thank Prof. C. Gérard for helpful discussions and forsuggestions during the writing of this paper.

References

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287

Generating Relations of the HypergeometricFunctions by the Lie Group-Theoretic Method

I. K. KHANNA, V. SRINIVASA BHAGAVAN and M. N. SINGHDepartment of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi-221005, India

(Received: 31 August 2000)

Abstract. In this paper, the generating relations for a set of hypergeometric functionsψα,β,γ,m(x)

are obtained by using the representation of the Lie group SL(2,C) giving a suitable interpretation tothe indexm in order to derive the elements of Lie algebra. The principle interest in our results liesin the fact that a number of special cases would inevitably yield too many new and known results ofthe theory of special functions, namely the Laguerre, even and odd generalized Hermite, Meixner,Gottlieb, and Krawtchouk polynomials.

Mathematics Subject Classifications (2000):Primary 33C10, Secondary 33C45, 33C80.

Key words: special functions, hypergeometric functions, Lie algebra, generating functions.

1. Introduction

Recently, we have defined a set of hypergeometric functionsψα,β,γ,m(x). It isinteresting to note that the functionψα,β,γ,m(x) is a product of binomial and hyper-geometric functions. Independently, these two do not satisfy the three-term recur-rence relation, whereas their product satisfies it. Thus, it is possible to derive manyproperties, including the ascending and descending recurrence relations, which areessential for obtaining the generating functions by the Lie group-theoretic method.In recent years, the development of advanced computers have made it necessary tostudy the functions with series representations from the numerical point of view.The most important functions of these type are hypergeometric in character. Onaccount of many properties, the multiple hypergeometric functions are used in anincreasing number of problems and are capable of being elegantly represented bytheir uses. Because of the important role which hypergeometric functions playin problems of physics and applied mathematics, the theory of generating func-tions has been developed in various directions and has found wide applicationsin various branches, e.g., Laguerre polynomials are very useful in the quantum-mechanical study of the hydrogen atom, Hermite polynomials have applicationsin the quantum-mechanical discussion of the harmonic oscillator and probabalisticdistribution. Also, orthogonal polynomials are of great importance in approxima-tion theory, physics, and the mathematical theory of mechanical quadratures, etc.

288 I. K. KHANNA ET AL.

The hypergeometric functions have also demonstrated their significance in scienceand technology. In this paper, we obtain generating functions forψα,β,γ,m(x) by us-ing the representation of the Lie group SL(2,C) [8] giving a suitable interpretationto the indexm, in order to derive the elements of Lie algebra.

The principle interest in our results lies in the fact that a number of special caseswould inevitably yield too many new and known results of the theory of specialfunctions. It is worth recalling that several of the classical polynomials, namely theLaguerre, even and odd generalized Hermite, Meixner, Gottlieb and Krawtchoukpolynomials [6], are derived as special cases of our results in Sections 2, 3 and 4.

2. Definition

We have defined a set of hypergeometric functions

ψα,β,γ,m(x) = βm(γ )m

m! (1− x)−m/22F1

[−m,α; γ ; x

β

], (2.1)

which is valid under the conditions

(i) α is a nonzero real number,(ii) γ is neither zero nor a negative integer,(iii) m is a nonnegative integer,(iv) in general, we shall insist thatα, β, γ are independent ofm because many

properties which are valid forα, β, γ independent ofm fail to be valid whenα, β, γ are dependent,

(v) x is any finite complex variable such that|x| < 1.

The following special cases ofψα,β,γ,m(x) have been obtained.

limα→∞ψα,1,1+γ,m

(xα

)= L(γ )m (x), (2.2)

whereL(γ )m (x) is the Laguerre polynomial [1].

limα→∞ψα,1,1/2,m

(x2

α

)= (−1)mH2m(x)/2

2mm! (2.3)

and

limα→∞ψα,1,3/2,m

(x2

α

)= (−1)mH2m+1(x)/2

2mm!2x, (2.4)

whereH2m(x) andH2m+1(x) are the even and odd Hermite polynomials respec-tively [2].

limα→∞ψα,1,(µ+1)/2,m

(x2

α

)= (−1)mHµ

2m(x)/22mm! (2.5)

GENERATING RELATIONS OF HYPERGEOMETRIC FUNCTIONS 289

and

limα→∞ψα,1,(µ+3)/2,m

(x2

α

)= (−1)mHµ

2m+1(x)/22mm!, (2.6)

whereHµ

2m(x) andHµ

2m+1(x) are the generalized even and odd Hermite polynomi-als respectively [4].

ψ−y,1,γ ,m(1− ρ−1) = (γ )m

m! ρm/2Mm(y; γ, ρ), (2.7)

provided 0< ρ < 1, y = 0,1,2, . . . , whereMm(y; γ, ρ) is the Meixner polyno-mial [7],

ψ−y,1,1,m(1− eλ) = emλ/2φm(y, λ), (2.8)

whereφm(y, λ) is the Gottlieb polynomial [1],

ψ−y,1,−N,m(P−1) = (−N)mm! (1− P−1)−m/2Km(y;P,N), (2.9)

provided 0< P < 1, y = 0,1,2, . . . , N , whereKm(y;P,N) is the Krawtchoukpolynomial [3].

The following recurrence relations forψα,β,γ,m(x) have been obtained:

d

dxψα,β,γ,m(x)

= 1

2x(1− x)(2β(1− γ −m)√1− xψα,β,γ,m−1(x)+

+m(2− x)ψα,β,γ,m(x)), (2.10)

d

dxψα,β,γ,m(x)

= 1

x(β − x)({β(−γ −m)+ αx + mx(β − x)

2(1− x)}ψα,β,γ,m(x)+

+ (m+ 1)√

1− x)ψα,β,γ,m+1(x). (2.11)

These two independent differential recurrence relations determine the linear ordi-nary differential equation:(

x(β − x) d2

dx2+(γβ + (m− α − 1)x − mx(β − x)

(1− x))

d

dx+

+mα + m(m− 2)x(β − x)4(1− x)2 −

− m

2(1− x) (γβ + (m− α − 1)x)

)ψα,β,γ,m(x) = 0. (2.12)


3. Representation of SL(2,C) and Generating Functions

Let sl(2,C) be the Lie algebra of a three-dimensional complex local Lie groupSL(2,C), a multiplicative 2× 2 matrix group with elements [5]

SL(2,C) ≡{g =

(a b

c d

): a, b, c, d ∈ C

}, (3.1)

which is the determinant of the matrixg, i.e. |g| = 1. A basis for sl(2,C) isprovided by the matrices

j+ =(

0 −10 0

), j− =

(0 0−1 0

), j3 =

( 12 00 −1

2

)(3.2)

with the commutation relations

[j3, j+] = j+, [j3, j−] = −j−, [j+, j−] = 2j3. (3.3)

Let us write the differential equation (2.12) in operator functional notation as

L

(x,

d

dx,m

)= x(β − x)D2+

(γβ + (m− α − 1)x − mx(β − x)

(1− x))D +

+mα + m(m− 2)x(β − x)4(1− x)2 − m

2(1− x) (γβ + (m− α − 1)x). (3.4)

In order to use the Lie group-theoretic method, we now construct the followingpartial differential equation by replacing d/dx by∂/∂x,m byy∂/∂y andψα,β,γ,m(x)by f (x, y):(

x(β − x) ∂2

∂x2+ xy

2(β − 2+ x)4(1− x)2

∂2

∂y2− (β − 1)xy

(1− x)∂2

∂x∂y+

+ (γβ − (α + 1)x)∂

∂x−

−(γβ − α(2− x)

2(1− x) + x(β − x)4(1− x)2

)y∂

∂y

)f (x, y) = 0. (3.5)

Thus

L = L

(x,

∂

∂x, y

∂

∂y

)≡ x(β − x) ∂

2

∂x2+ xy

2(β − 2+ x)4(1− x)2

∂2

∂y2− (β − 1)xy

(1− x)∂2

∂x∂y+

+ (γβ − (α + 1)x)∂

∂x−(γβ − α(2− x)

2(1− x) + x(β − x)4(1− x)2

)y∂

∂y. (3.6)

We need the following observations [3]:


OBSERVATION I. LetL(x,d/dx, n) be a linear differential operator containinga parametern. Assuming thatL is a polynomial inn, we construct a partial differ-ential operatorL(x, ∂/∂x, y∂/∂y) by substitutingy∂/∂y for n. Thenz = ynνn(x)is a solution ofL(x, ∂/∂x, y∂/∂y) z = 0 if and only if

u = νn(x) is a solution ofL(x,d/dx, n)u = 0. (3.7)

OBSERVATION II. LetG(x, y) have a convergent expansion of the form

G(x, y) =∑n

gn(x)yn, (3.8)

wheren is not necessarily a nonnegative integer.

If L(x, ∂/∂x, y∂/∂y)G(x, y) = 0, then within the region of convergence of theseries (3.8),u = gn(x) is a solution of (3.7). In particular, ifG(x, y) is regular atx = 0, u = gn(x) is also regular atx = 0.

In lieu of Observation I, we conclude thatf (x, y) = ymψα,β,γ,m(x) is a solutionof (3.5).

Let us now introduce the first-order linearly independent differential operatorsJ 3, J− andJ+ each of the form

A1(x, y)∂

∂x+ A2(x, y)

∂

∂y+ A3(x, y)

such that

J 3[ymψα,β,γ,m(x)] = amymψα,β,γ,m(x),

J−[ymψα,β,γ,m(x)] = bmym−1ψα,β,γ,m−1(x), (3.9)

J+[ymψα,β,γ,m(x)] = cmym+1ψα,β,γ,m+1(x),

wheream, bm andcm are expressions inm which are independent ofx andy, butnot necessarily ofα, β andγ . EachAi(x, y), i = 1,2,3, on the other hand, is anexpression inx andy which is independent ofm but not necessarily ofα, β andγ .

By using (3.9) and recurrence relations (2.10) and (2.11), we get the followingoperators:

J 3 = y∂

∂y+ γ

2,

J− = xy−1√

1− xβ

∂

∂x− (2− x)

2β√

1− x∂

∂y, (3.10)

J+ = xy(β − x)√1− x

∂

∂x+ y2

2(1− x)3/2 (2β − 3βx + x2)∂

∂y+ (γβ − αx)y√

1− x .

Clearly, the operatorsJ 3, J− andJ+ satisfy the commutation relations

[J 3, J±] = ±J±, [J+, J−] = 2J 3. (3.11)


It concludes that theJ -operators generate a three-dimensional Lie algebra isomor-phic to sl(2,C), which is the algebra of generalized Lie derivatives of a multiplierrepresentation SL(2,C).

In order to determine the multiplier representation of SL(2,C), we first computethe actions of exp(a′J+), exp(b′J−) and exp(c′J 3) on f ∈ F , whereF is thecomplex vector space of all functions ofx andy analytic in some neighbourhoodof the point(x0, y0) = (0,0).

Thus, iff ∈ F is analytic in a neighbourhood of(x0, y0), then the values of themultiplier representations of exp(a′J+)f , exp(b′J−)f and exp(c′J 3)f are givenby

[T (exp(a′j+))f ](x0, y0)

= [exp(a′J+)f ](x0, y0)

= (1− x0)γ /2[√1− x0− a′y0(β − x0)]−α[√1− x0− a′βy0]γ−α ×

× f(

x0√

1− x0√1− x0− a′y(β − x0)

,y0[(1− x0)

3/2− a′y0(β − x0)]1/2[√1− x0− a′βy0][√1− x0− a′y0(β − x0)]1/2

),

|x0| < 1,

∣∣∣∣a′y0(β − x0)√1− x0

∣∣∣∣ < 1,

∣∣∣∣ a′βy0√1− x0

∣∣∣∣ < 1;

[T (exp(b′j−))f ](x0, y0)

= f(

βx0y0

βy0− b′√1− x0,

[(βy0− b′√1− x0)(βy0

√1− x0− b′)

β2√

1− x0

]1/2),

|x0| < 1,

∣∣∣∣b′√1− x0

βy0

∣∣∣∣ < 1,

∣∣∣∣ b′

βy0√

1− x0

∣∣∣∣ < 1;

[T (exp(c′j3))f ](x0, y0) = exp

(γ c′

2

)f (x0, y0ec

′). (3.12)

Now, in the neighbourhood of the identity, everyg ∈ SL(2,C) can be expressedas

g = exp(a′j+))exp(b′j−))exp(c′j3)),

from which we get the operatorT (g) acting onf ∈ F given by

[T (g)f ](x, y)= (√

1− x)γ [√1− x − a′y(β − x)]−α[√1− x − a′βy]γ−α exp

(γ c′2

)f (ξ, ηec

′), (3.13)

where

ξ = βxy√

1− x[√1− x − a′y(β − x)][βy(1+ a′b′)− b′√1− x] ,


η = [βy(1+ a

′b′)− b′√1− x]{βy[(1− x)3/2− a′y(β − x)]++ [a′b′y(β − x)− b′√1− x][√1− x − a′βy]}

β2[√1− x − a′βy]2[√1− x − a′y(β − x)]

1/2

and

g =(a b

c d

)∈ SL(2,C).

Further, by setting

a′ = −bd, b′ = −cd, exp

(c′2

)= d−1 and ad − bc = 1,

we have

[T (g)f ](x, y) = (√1− x)γ [d√

1− x + by(β − x)]−α[d√1− x + bβy]γ−α f (ξ, ηd−2), (3.14)

where

ξ = βxy√

1− x[d√1− x + by(β − x)][aβy + c√1− x] ,

η = [aβy + c

√1− x]{βy[d(1− x)3/2+ by(β − x)]+

+ c[by(β − x)+ d√1− x][d√1− x + bβy]}β2d−3[d√1− x + bβy][d√1− x + by(β − x)]

1/2

,

provided

|x| < 1,

∣∣∣∣c√1− xaβy

∣∣∣∣ < 1,

∣∣∣∣ bβy

d√

1− x∣∣∣∣ < 1,∣∣∣∣by(β − x)

d√

1− x∣∣∣∣ < 1, and |arg(a),arg(d)| < π.

Hereg lies in a sufficiently small neighbourhood of the identity element[ 1 0

0 1

]∈ SL2.

Equation (3.14) defines a local multiplier representation of SL(2,C) and thedifferential operatorsJ+, J−, J 3 are generalized Lie derivatives ofT . In terms oftheJ -operators, we introduce the Casimir operator [5].

C = C1,0 = J+J− + J 3J 3− J 3

= x

β

(x(β − x) ∂

2

∂x2+ xy

2(β − 2+ x)4(1− x)2

∂2

∂y2− (β − 1)xy

(1− x)∂2

∂x∂y+

+[γβ − (α + 1)] ∂∂x−(γβ − α(2− x)

2(1− x) + x(β − x)4(1− x)2

)y∂

∂y

)+

+ γ4(γ − 2). (3.15)


It can be verified thatC commutes withJ+, J− andJ 3.Equation (3.15) enables us to write (3.5) as

Cf (x, y) = γ

4(γ − 2)f (x, y). (3.16)

To derive the generating functions, we search for the functionsf (x, y) whichsatisfy (3.16).

Now, we consider the following different cases:

Case 1.Whenf (x, y) is a common elgenfunction ofC andJ 3.The simultaneous equations

Cf (x, y) = γ

4(γ − 2)f (x, y) (3.17)

and

J 3f (x, y) =(ν + γ

2

)f (x, y) (3.18)

admit a solutionf (x, y) = yνψα,β,γ,ν(x), so that

[T (g)f ](x, y)= (1+ bc)− 1

2 (2γ+3ν)aγ+2ν

(1+ c

√1− xaβy

)ν/2×

×(

1+ by(β − x)d√

1− x)− 1

2 (ν+2α)(1+ bβy

d√

1− x)α−γ−ν

×

×{(

1+ by(β − x)d√

1− x)+ cd

(1+ by(β − x)

d√

1− x)(

1+ bβy

d√

1− x)}ν/2

×

×ψα,β,γ,ν[ x(

1+ by(β−x)d√

1−x)(1+ bc)(1+ c

√1−xaβy

)], (3.19)

which satisfies the relationC[T (g)f ](x, y) = 14γ (γ − 2)[T (g)f ](x, y).

If ν is not an integer, Equation (3.19) has a Laurent series expansion [3]:

[T (g)f ](x, y) =∞∑

p=−∞hp(g, x)y

ν+p. (3.20)

Considering that[T (g)f ](x, y) is regular atx = 0, then from the result of (3.8),we havehp(g, x) = jp(g)ψα,β,γ,ν+p(x), which implies that

[T (g)f ](x, y) =∞∑

p=−∞jp(g)ψα,β,γ,ν+p(x)yν+p. (3.21)


To determinejp(g), putx = 0 and equate the coefficient ofyp, and we get

jp(g) = aγ+2ν(1+ bc)−γ−ν0(1+ ν + p)0(1+ p)0(1+ ν)

(−bd

)p×

× 2F1

[ −ν, ν + γ + p; bcad1+ p;

]. (3.22)

Thus, the generating function (3.20) becomes

(1+ bc)−ν/2(

1+ c√

1− xaβy

)ν/2(1+ by(β − x)

d√

1− x)− 1

2 (ν+2α)

×

×(

1+ bβy

d√

1− x)α−γ−ν{(

1+ by(β − x)d(1− x)3/2

)+

+ cd

βy√

1− x(

1+ by(β − x)d√

1− x)(

1+ bβy

d√

1− x)}ν/2

×

×ψα,β,γ,ν[ x(

1+ by(β−x)d√

1−x)(1+ bc)(1+ c

√1−xaβy

)]

=∞∑

p=−∞

0(1+ ν + p)0(1+ ν)0(1+ p)

(−byd

)p ×× 2F1

[ −ν, ν + γ + p; bcad1+ p;

]ψα,β,γ,ν+p(x),

|x| < 1,∣∣∣ ca

∣∣∣ < ∣∣∣ βy√1− x

∣∣∣ < ∣∣∣bd

∣∣∣−1,

∣∣∣∣by(β − x)d√

1− x∣∣∣∣ < 1,

−π < arg(a), arg(d) < π and ad − bc = 1. (3.23)

Deductions.(I) By putting a = d = β = y = 1 andc = 0, Equation (3.23)yields

(1− b√1− x)− 12 (ν+2α)

(1− b√

1− x)α−γ−ν/2

ψα,1,γ ,ν

(x

1− b√1− x)

=∞∑p=0

(1+ ν)pp! ψα,1,γ ,ν+p(x)bp,

|b√1− x| < 1,

∣∣∣∣ b√1− x

∣∣∣∣ < 1. (3.24)

(II) Taking a = d = β = y = 1 andb = 0, we have{(1− c√1− x)

(1− c√

1− x)}ν/2

ψα,1,γ ,ν

(x

1− c√1− x)


=∞∑p=0

(1− γ − ν)pp! ψα,1,γ ,ν−p(x)cp,

|c√1− x| < 1,

∣∣∣∣ c√1− x

∣∣∣∣ < 1. (3.25)

Applications.Further, using Section 2.2, we get the following generating func-tions:

(i)∞∑k=0

(ν + 1)kL(α)ν+k(x)bk

k!

= (1− b)−1−α−ν exp( −xb

1− b)L(α)ν

( x

1− b), |b| < 1;

(ii)∞∑k=0

(−α − ν)kL(α)ν−k(x)ckk! = (1− c)νL(α)ν

( x

1− c), |c| < 1;

(iii)∞∑k=0

(−1)kHµ

2ν+2k(x)bk

k!22k

= (1− b)− 12 (2ν+2µ+1) exp

(−x2b

1− b)Hµ

2ν

[x√

1− b], |b| < 1;

(iv)∞∑k=0

(−2ν)2kHµ

2ν−2k(x)ck

k! = (1− c)νHµ

2ν

[x√

1− c], |c| < 1;

(v)∞∑k=0

(−1)kHµ

2ν+1+2k(x)bk

k!22k

= (1− b)− 12 (2ν+2µ+3) exp

(−x2b

1− b)Hµ

2ν

[x√

1− b], |b| < 1;

(vi)∞∑k=0

(−2ν − 1)2kHµ

2ν+1−2k(x)ck

k! = (1− c)νHµ

2ν+1

[x√

1− c], |c| < 1;

(vii)∞∑k=0

(ν + γ )kρk/2Mν+k(y; γ, ρ)bkk!

= (1− bρ1/2)−y−γ−ν(1− bρ−1/2)yMν

(y; γ, ρ

1/2− bρ−1/2− b

),

|bρ1/2| < 1, |bρ−1/2| < 1,

provided thatγ > 0, 0< ρ < 1, y = 0,1,2, . . . ;


(viii)∞∑k=0

(−ν)kρ−k/2Mν−k(y; γ, ρ)ckk!

= (1− cρ−1/2)νMν

(y; γ, ρ

1/2− cρ−1/2− c

), |cρ−1/2| < 1,


(ix)∞∑k=0

(ν + 1)kekλ/2φν+k(y, λ)bk

k!

= (1− beλ/2)y−ν(1− be−λ/2)−y−1φν

(y, log

(eλ/2− be−λ/2− b

)),

|beλ/2| < 1, |be−λ/2| < 1;

(x)∞∑k=0

(−ν)ke−kλ/2φν−k(y, λ)ckk!

= (1− ce−λ/2)νφν(y, log

(eλ/2− ce−λ/2− c

)), |ce−λ/2| < 1;

(xi)∞∑k=0

(−N + ν)kk! Kν+k(y;P,N)

(b√

1− P−1

)k= (1−b√1− P−1

)y(1− b√

1−P−1

)N−y−νKν(y;p−bp

√1− P−1, N),

|b√1− P−1| < 1,

∣∣∣∣ b√1−P−1

∣∣∣∣ < 1;

(xii)∞∑k=0

(−ν)kk! Kν−k(y;P,N)

(c√

1− P−1)k

= (1− c√1− P−1)νKν(y;p− cp√

1− P−1, N), |c√1− P−1| < 1,

provided that 0< P < 1, y = 0,1,2, . . . , N ,

which are the generating relations for the Laguerre, even and odd generalizedHermite, Meixner, Gottlieb, Krawtchouk and polynomials, respectively.

The generating function (3.23) is valid under the condition thatν is not aninteger. However, ifν is a nonnegative integer, sayν = k, the generating functioncan be further written as

(1+ bc)−k/2(

1+ c√

1− xaβy

)k/2(1+ by(β − x)

d√

1− x)− 1

2 (k+2α)

×


×(

1+ bβy

d√

1− x)α−γ−k{(

1+ by(β − x)d(1− x)3/2

)+

+ cd

βy√

1− x(

1+ by(β − x)d√

1− x)(

1+ bβy

d√

1− x)}k/2

×

×ψα,β,γ,k( x(

1+ by(β−x)√1−x

)(1+ bc)(1+ c

√1−xaβy

))

=∞∑p=0

p!k!(−byd

)p−k{0(p − k + 1)}−1×

× 2F1

[ −k, γ + p; bcadp − k + 1;

]ψα,β,γ,p(x),∣∣∣ c

a

∣∣∣ < ∣∣∣∣ βy√1− x

∣∣∣∣ < ∣∣∣bd

∣∣∣−1, ad − bc = 1. (3.26)

Case 2.Whenf (x, y) is a common elgenfunction of the operatorsC andJ−.Let f (x, y) be a solution of the simultaneous equations

Cf (x, y) = γ

4(γ − 2)f (x, y) (3.27)

and

J−f (x, y) = −f (x, y),which may be written as(

x(β − x) ∂2

∂x2+ xy

2(β − 2+ x)4(1− x)2

∂2

∂y2− (β − 1)xy

(1− x)∂2

∂x∂y+

+[γβ − (α + 1)x] ∂∂x−

−(γβ − α(2− x)

2(1− x) + x(β − x)4(1− x)2

)y∂

∂y

)f (x, y) = 0 (3.28)

and (xy−1√

1− xβ

∂

∂x+ (x − 2)

2β√

1− x∂

∂y+ 1

)f (x, y) = 0. (3.29)

Assuming the general solution of (3.28) and (3.29) in the form

f (x, y) = exp

(βy√1− x

)K

(xy√1− x

), (3.30)

and substituting this in (3.29), we get(u

d2

du2+ (γ + u) d

du+ α

)K(u) = 0, (3.31)


whereu = xy/√1− x. This is Kummer’s differential equation [3] and has for itssolution

K(u) = 1F1[α; γ ;−u], (3.32)

whereu = xy/√1− x. Thus, one solution of this system is

f (x, y) = exp

(βy√1− x

)1F1

[α; γ ;− xy√

1− x]. (3.33)

If this function is expanded in powers ofy, we get

exp

(βy√1− x

)1F1

[α; γ ;− xy√

1− x]=∞∑m=0

ψα,β,γ,m(x)ym

(γ )m, (3.34)

which is the generating function forψα,β,γ,m(x).From (3.33), we have

[T (g)f ](x, y)= (√1− x)γ [d√1− x + by(β − x)]−α[d√1− x + bβy]α−γ ×× exp(ξ)1F1[α; γ ;−φ], (3.35)

where

ξ =

(aβy + c√1− x)2{βy[d(1− x)3/2+ by(β − x)]++ [bcy(β − x)+ cd√1− x][d√1− x + bβy]}d[d√1− x + bβy]2{[d√1− x + by(β − x)]×× [aβy + c√1− x] − βxy√1− x}

1/2

and

φ =

x2y2(1− x){βy[d(1− x)3/2+ by(β − x)]++ [bcy(β − x)+ cd√1− x][d√1− x + bβy]}

d[d√1− x + by(β − x)]2[d√1− x + bβy]2××{[d√1− x + by(β − x)][aβy + c√1− x] − βxy√1− x}

1/2

satisfying the relation

C[T (g)f ](x, y) = 1

4γ (γ − 2)[T (g)f ](x, y). (3.36)

Since[T (g)f ](x, y) is analytic aty = 0, it can be expanded in the form

[T (g)f ](x, y) =∞∑m=0

Pm(g)ψα,β,γ,m(x)ym. (3.37)


To compute the coefficientsPm(g), we putx = 0 in (3.37) and we have

(d + byβ)−γ exp(aβy + cd + bβy

)=∞∑m=0

(γ )m

m! Pm(g)(βy)m, (3.38)

which may be written as

d−γ exp

(c

d

)(1+ bβy

d

)−γexp

(βy

d2(1+ bβy

d

))

=∞∑m=0

(γ )m

m! Pm(g)(βy)m. (3.39)

From the well-known generating function for the Laguerre polynomials [1], wehave

(1− t)−1−α exp( −xt

1− t)=∞∑n=0

L(α)n (x)tn. (3.40)

On comparing (3.39) with the well-known generating function (3.40), we find

Pm(g) = m!(γ )m

d−γ exp(c/d)(−bd

)mL(γ−1)m

( 1

bd

). (3.41)

Thus we have(1+ by(β − x)

d√

1− x)−α(

1+ bβy

d√

1− x)α−γ

exp(ξ ′)1F1[α; γ ;−φ′]

=∞∑m=0

m!(γ )m

exp(c/d)Lγ−1m

( 1

bd

)ψα,β,γ,m(x)

(−byd

)m,

∣∣∣byd

∣∣∣ < 1, (3.42)

where

ξ ′ =a2β2y2

(1+ c

√1−xaβy

)2{(1+ by(β−x)d(1−x)3/2

)+ cd

βy√

1−x(1+ by(β−x)

d√

1−x)(

1+ bβy

d√

1−x)}

d2(1+ bβy

d√

1−x)2{(1+ bc)(1+ by(β−x)

d√

1−x)(

1+ c√

1−xaβy

)− x}1/2

and

φ′ = x2y2

{(1+ by(β−x)

d(1−x)3/2)+ cd

βy√

1−x(1+ by(β−x)

d√

1−x)(

1+ bβy

d√

1−x)}

d4(1+ by(β−x)

d√

1−x)2(1+ bβy

d√

1−x)2{(1+ bc)(1+ by(β−x)

d√

1−x)(

1+ c√

1−xaβy

)− x}1/2

.

Deductions.If

a = i√w, b = i/√w, c = 0, d = −i/√w,i = √−1 and β = 1,


then we have:

(1− y√1− x)−α(

1− y√1− x

)α−γexp

(− wy√

1− x − y)×

× 1F1

[α; γ ; wxy

(√

1− x − y)(1− y√1− x)]

=∞∑m=0

m!(γ )m

L(γ−1)m (w)ψα,1,γ ,m(x)y

m, (3.43)

or

(1− y√1− x)−α(

1− y√1− x

)α−γexp

(− wy√

1− x − y)×

× 1F1

[α; γ ; wxy

(√

1− x − y)(1− y√1− x)]

=∞∑m=0

L(γ−1)m (w)2F1[−m,α; γ ; x]

(y√

1− x)m, (3.44)

which is the bilateral generating function involving the Laguerre polynomial and acertain terminating2F1.

4. Applications

As usual, we get the following applications from (3.34):

(i)∞∑m=0

L(λ)m (x)tm

(1+ λ)m = exp(t) 0F1[−;1+ λ;−xt];

(ii)∞∑m=0

(−1)mH2m(x)tm

22mm!(12

)m

= exp(t) 0F1

[−; 1

2;−x

2t

2

]or∞∑m=0

(−1)mH2m(x)z2m

(2m)! = exp(z2) cos(√

2xz);

(iii)∞∑m=0

(−1)mH2m+1(x)tm

2x22mm!(3/2)m = exp(t) 0F1

[−; 3

2;−x

2t

2

]or∞∑m=0

(−1)mH2m+1(x)z2m+1

(2m+ 1)! = exp(z2) sin(√

2xz);


(iv)∞∑m=0

(−1)mHµ

2m(x)tm

m!(µ+12

)m

= exp(t) 0F1

[−; µ+ 1

2;−x2t

];

(v)∞∑m=0

(−1)mHµ

2m+1(x)tm

m!(µ+32

)m

= x exp(−t) 0F1

[−; µ+ 3

2;−x2t

];

(vi)∞∑m=0

Mm(y; γ, ρ)(tρ1/2)m

m! = exp(tρ1/2) 1F1[−y; γ ; (ρ−1/2 − ρ1/2)t];

or, equivalently,

∞∑m=0

Mm(y; γ, ρ)zmm! = exp(z) 1F1[−y; γ ;−z(1− ρ−1)],


(vii)∞∑m=0

emλ/2φm(y, λ)tm

m! = exp(te−λ/2) 1F1[−y;1; (eλ/2 − e−λ/2)t],

which can be reduced to∞∑m=0

φm(y, λ)zm

m! = exp(z) 1F1[1+ y;1;−z(1− e−λ)t].

(viii)∞∑m=0

Km(y;P,N)m!

(t√

1− P−1

)m

= exp

(t√

1− P−1

)1F1

[−y;−N;− P−1t√

1− P−1

],

which can be rewritten as∞∑m=0

Km(y;P,N)zmm! = exp(z) 1F1[−y;−N;−zP−1],

0< ρ < 1, y = 0,1,2, . . . , N .

These are all well-known generating functions in one form or another for the La-guerre, even and odd (generalized) Hermite, Meixner, Gottlieb, and Krawtchoukpolynomials, respectively.

Remark.The corresponding bilateral (or bilinear) generating relations for theLaguerre, even and odd Hermite, Meixner, Gottlieb, and Krawtchouk polynomialscan be deduced from (3.43) by using the conditions of Section 2.


Acknowledgement

The authors wish to express their sincere thanks to the referee for the kind sugges-tions given to improve this paper.

References

1. Rainville, E. D.:Special Functions, Macmillan, New York, 1960.2. Szegö, G.:Orthogonal Polynomials, 4th edn, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math.

Soc., Providence, Rhode Island, 1975.3. Srivastava, H. M. and Manocha, H. L.:A Treatise on Generating Functions, Ellis Horwood,

England, 1984.4. Chihara, T. S.:An Introduction to Orthogonal Polynomials, Gordon and Breach, New York,

1978.5. Miller, W.: Lie Theory and Special Functions, Academic Press, New York, 1968.6. Khanna, I. K. and Srinivasa Bhagavan, V.: Weisner’s method to obtain generating relations for

the generalized polynomial set,J. Phys. A Math. Gen.32 (1999), 1–10.7. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.:Higher Transcendental

Functions, Vol. 2, McGraw-Hill, New York, 1953.8. Cohn, P. M.:Lie Groups, Cambridge Univ. Press, Cambridge, 1961.9. Vilenkin, N. Ja. and Klimyk, A. U.:Representation of Lie Groups and Special Functions,

Vols. 1 and 2, Kluwer Acad. Publ., Dordrecht, 1991, 1993.

Mathematical Physics, Analysis and Geometry 3: 305–321, 2000.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

305

Gauge Fields with Generic Singularities

SPYROS PNEVMATIKOS1 and DIMITRIS PLIAKIS2

1 Department of Mathematics, University of Patras, 26500 Patras, Greece.e-mail: [email protected] Department of Physics, University of Crete, 71409 Heraklion, Greece.

(Received: 10 March 2000; in final form: 16 January 2001)

Abstract. Let M be an n-dimensional manifold equipped with an Abelian Yang–Mills field withconnection form α. We consider an external potential function V and examine the existence andregularity of the vortex lines of the form α + V dt which define the motion of a particle weaklycoupled to the Yang–Mills field on M . These curves are smooth unless the curvature form dα issingular and in this paper we treat this singular case from a generic aspect. The problem reducesto the division properties for smooth functions and differential forms, the development of whichconstitutes the main part of the work presented here.

Mathematics Subject Classifications (2000): 70S15, 58A10, 58A35.

Key words: gauge theory, singularities, stratifications.

0. Introduction

The motion of a light particle on a smooth manifold M in the presence of a strongAbelian gauge field with connection form α and an external potential function V ,reduces to the classical variational problem on the space of paths γ on M, whichasks for the stationary points of the functional I (γ ) = ∫

γ(α + V dt). The cor-

responding Euler–Lagrange equations could be expressed by the vector field XV

which is tangent to the paths γ and is given by the equation iXVdα + dV = 0

under the assumption that the curvature form dα is nondegenerate on M. The cur-vature form coincides with the restriction of the canonical symplectic form of thecotangent bundle on the image of the section α: M → T ∗M and, evidently, couldappear with a degeneracy locus consisting of the points where its kernel is nottrivial. Several studies have appeared in the classical context with regular fibering,originating from Dirac’s works on the canonical quantization of electromagnetictheory and, in the geometrical intrinsic aspect, by Lichnerowicz, cf. [5, 12].

Generically, this locus is stratified in smooth submanifolds and a singular fiber-ing with vector spaces of variable dimension arises. Here, the problems causedby the successive degeneracies are tackled by essentially applying the ideas ofthe theory of stratified symplectic structures in a generic context, cf. [20]. Wefocus on the external potentials that could define dynamics everywhere on M

despite the apparent degeneracies; these functions constitute an algebra, the Dirac

306 SPYROS PNEVMATIKOS AND DIMITRIS PLIAKIS

algebra of the structure (M,α). More precisely, we prove that the Dirac algebraconsists of those smooth functions whose differential annihilates the kernel of thecurvature form on the points of the first singular stratum of the structure (M,α);also, we deduce the behavior of the motion in the neighborhood of the degeneracylocus. The problem reduces to the division properties in the context of smoothfunctions and smooth differential forms and the development of these propertiesis the main purpose of this paper. These division questions were studied in theholomorphic category in the 70’s ([14, 17, 24]). Here, we treat a special caseof the divisor form, relying on a smooth version of the Hironaka division by anideal generated by analytic functions that define the degeneracy locus. An essen-tial role is played by the ideal of the smooth functions that identically vanish onevery stratum of the degeneracy locus and its equality with the ideal generatedby the functions that define this stratum; in other words, we are dealing herewith a problem of the differentiable Nullstellensatz on the stratified degeneracylocus.

Finally, from the generic aspect, we describe the behavior of singular Lagrangiansystems on a manifold M; the singular locus of the motion is exactly the pro-jection on M of the critical locus of the corresponding Legendre transformationbetween TM and T ∗M. The Lagrangian functions L that possess a Lagrangianvector field XL defined everywhere on TM are characterized by pointwise con-ditions on the first stratum of the critical locus and the projection of its integralcurves gives the motion on M; in particular, the trajectories with initial conditionsin a stratum always remain in this stratum. In the case when the Lagrangian isdefined by a generic quadratic form on M, a singular connection, generalizingthe Levi-Civita connection, can be used for the determination of the geodesicsin the stratified singular locus. This description leads to the study of the singularvariational problem developed in [22].

1. The Motion in the Presence of an Abelian Gauge Field

1.1. We begin by illustrating our methods with the planar motion of a particle ofmass m and charge q in a strong magnetic field B = curl A with vector potential A.The motion is governed by the Euler-Lagrange equations that are implied by theusual Lagrangian with the minimal coupling substitution

L(x,υ) = 12(mυ + qA(x))2 − U(x),

where x,υ denote the position and velocity vectors of the particle on the plane andU(x) is an external potential. In the limit of a light particle strongly coupled withthe magnetic field, the Lagrangian is of first order with respect to the velocities

L(x,υ) = mqυ.A(x) − V (x) with V (x) = U(x) − 12 (qA(x))2

GAUGE FIELDS WITH GENERIC SINGULARITIES 307

and the equations of motion are put into the appealing form of a system of linearequations

B(x) × υ = −∇V (x).

Evidently, the zeroes of the magnetic field raise an obstruction to the existence ofmotion everywhere on the plane for an arbitrary external potential. Let us supposethat the magnetic field is written as

B(x, y) = φ(x, y)n

where n is the unit normal vector to the (x, y)-plane in R3 and the equations of

motion are written as

φ(x, y)n × υ = −∇V (x, y).

Clearly, the existence of dynamics everywhere on the plane is equivalent to thedivisibility of the right-hand side by φ(x, y) in the ring of smooth functions on R

2.A first elementary case, which illustrates the generic situation, appears when

(∂ix∂jy φ)(0, 0) = 0, i + j < k, (∂kxφ)(0, 0) �= 0.

Then, under a local change of coordinates (x, y) → (ξ, ζ ), we get B(ξ, ζ ) =ξ kn and, after the classical smooth division, the admissible external potentials arewritten as

V (ξ, ζ ) = ξ k+1V ′(ξ, ζ ),

where V ′ is a smooth function and, thus, the solution of the equation of motion is

υ = ξn × ∇V ′(ξ, ζ ) + 2V ′(ξ, ζ )e2,

where (e1, e2) denotes the base of the (x, y)-plane. In conclusion, the above exter-nal potentials are admissible for the magnetic field, i.e. the corresponding equationsof motion are everywhere defined, overpassing the obstruction raised by the zeroesof the magnetic field. Clearly, these potentials constitute an algebra, called Diracalgebra, with the bracket

{V1, V2} = ξ k(∇V1 × ∇V2) · n.If we also assume that the magnetic field defined by the vector potential

A(x, y) = (yp, xq),

where p, q ∈ N, g.c.d. (p, q) = 1; if p, q > 1, we observe that the equation ofmotion

(pyp−1 − qxq−1)n × υ = −∇V (x, y)


possesses a critical locus that is an algebraic curve with a singularity at (0, 0),unlike the simple case described above. Evidently, we have to study a more com-plicated division problem on this locus and this issue will be treated in the sequelin a rather general context.

1.2. Let M be a smooth n-dimensional manifold equipped with the connection formof an Abelian Yang–Mills field α: M → T ∗M, cf. [10]. The motion of a lightparticle on M, strongly coupled with the Yang–Mills field, with external potentialV : M → R, is defined by the Lagrangian

L := iCπ∗α + π∗V

with π : TM → M the canonical projection and C the tautological vector fieldon TM. The connection form is expressed locally as

α(x) =∑

1�i�n

ai(x) dxi

and the Lagrangian leads to Euler–Lagrange equations which are of first order:∑1�j�n

(∂xj ai(x) − ∂xiaj (x))xj = ∂xiV (x), i = 1, . . . , n

with the critical locus being the set of points x ∈ M where the velocities are not,a-priori, well defined:

det(∂xj ai − ∂xiaj )(x) = 0.

These equations are expressed intrinsically by the vector field XV where this couldbe defined by the equation of motion

iXV!α + dV = 0

outside of the degeneracy locus of the curvature form !α = dα. The curvatureform is the restriction of the canonical symplectic form on the image of the sectionα: M → T ∗M and the critical locus of the equations is exactly the degeneracylocus

"(!α) =⋃k>0

"k(!α)

with

"k(!α) = {x ∈ M/ dim kerx !α = k}, k ∈ N.

In the sequel, the manifold M is assumed to be orientable with a volume form w

and a Riemannian structure with inner product 〈, 〉 on the fibers of ∧rT ∗M, r =1, . . . , n, thus giving rise to the Hodge ∗-operator between the fibers of ∧rT ∗Mand ∧n−rT ∗M. Clearly

!sα(x) ∧ ∗!s

α(x) = ρα,s(x)w


with

ρα.s(x) = ∥∥!sα

∥∥2(x)

and, if the dimension of M is 2m or 2m + 1, then

ρ−1α,m(0) =

⋃k>1

"k(!α).

Let us note in passing the effects in the known situation when ρ−1α,m(0) = ∅. In the

even 2m-dimensional case, the equation of motion has a smooth solution XV on M

given by

iXVw = 1

ρα,mdV ∧ !m−1

α .

In the odd (2m+ 1)-dimensional case, we introduce ∗θV = iXVw and the equation

of motion is equivalent to a division identity of a 2-form by a 1-form

θV ∧ ∗!mα = ∗(dV ∧ !m−1

α ).

The kernel of ∗!mα is everywhere one-dimensional and, hence, defines a regular

foliation; as is known, the annihilation of dV on the kernel of !α is a necessaryand sufficient condition for the existence of XV on M (cf. [12]). This in turn yieldsthe condition which is necessary for the division question that will appear in thesequel. Furthermore, if the curvature form has constant but nonmaximal rank onM,i.e. ρα,s(x) �= 0 and ρα,s+1(x) = 0, for s < n, then since

s(iXV!α) ∧ !s−1

α = iXV!s

α,

the external potentials that lead to a smooth solution XV on M are given by

iXVw = ρα,s−1

ρα,s∗ (dV ∧ ∗!α).

The admissible external potentials here also constitute the Dirac algebra endowedwith the Poisson bracket

{V1, V2} = !α(XV1,XV2).

1.3. The connection forms on an n-dimensional smooth manifold M, with thecurvature form transverse to the natural stratification by the rank of the bundle∧2T ∗M, constitute an open and dense set in the space of connection forms on M

equipped with the Whitney C∞-topology (cf. [16, 20]). These are called genericconnection forms. The manifold M equipped with a generic connection form α isstratified into the smooth submanifolds

"k(!α) = {x ∈ M | dim kerx !α = k}, 0 � k � n,


of codimension k(k − 1)/2, if nonempty, whose closure satisfies the frontier con-dition

"k(!α) =⋃k′�k

"k′(!α).

The versal unfolding lemma, established for differential forms in [20], immedi-ately leads to a natural local classification of the generic connection forms throughthe following normal form on R

n given in local coordinates (x1, . . . , xn) with τ

arbitrary injection in {1, . . . , n} by

ωk =∑

1�i<j�k

xτ(i,j) dxi ∧ dxj + dxk+1 ∧ dxk+2 + · · · + dxn−1 ∧ dxn, k � n.

THEOREM 1. If α is a generic connection form on an n-dimensional smoothmanifold M and xo ∈ "k(!α), there exist both a germ of diffeomorphism and agerm of smooth map, respectively,

φ: (M, xo) → (Rn, 0), ψ : (M, xo) → GL(n,R)

such that, for every x in a neighborhood of xo, we have

(dα)(ψ(x)., ψ(x).) = φ∗ωk(., .).

The diffeomorphism φ respects the germs of the corresponding stratification, fork′ � k, in the following sense:

"k′(dα) = φ−1("k′(ωk)).

Remark 1. In the case k = 0 or 1, we find the formal expression of the classicalDarboux local model which always holds for the germs of closed 2-forms of con-stant rank. In the case k = 2, for even dimension, we find the formal expression ofthe singular Martinet’s local model:

dα = x1 dx1 ∧ dx2 + dx3 ∧ dx4 + · · · + dxn−1 ∧ dxn.

Remark 2. After a direct calculation on the normal form, the preceding theoremsuggests that the local equations of the algebraic set "k(!α) are given through thepolynomials

P(i,k) =∑σ

(−1)sgn σ xσ(τ(i1,j1) . . . xσ(τ(ik,jk)

where the sum is taken over all permutations of {(i1, j1), . . . , (ik, jk)} for all dis-joint couples (i1, j1), . . . , (ik, jk), 1 � il � jl � k ([20]). Such polynomials arecalled affine polynomials and provide the required division property in the ringof smooth functions essential for handling our problem. Also, Lemma 4 suggeststhat every nonempty stratum "k(!α), k > 1, is defined locally by k(k − 1)/2homogeneous affine polynomials that generate, in the ring of germs of smoothfunctions on M, the ideal

�k(!α) = {f ∈ C∞

xo(M) | f |"k(!α ) = 0

}, 1 < k � n.


2. The Dirac Algebra of External Potentials

2.1. Let α be the connection form of an Abelian gauge field on a smooth man-ifold M. We call Dirac algebra (denoted by D(M,α)) the algebra of smoothfunctions V : M → R that defines dynamics everywhere on M through the equationof motion iXV

!α +dV = 0 admitting a unique smooth solution XV everywhere onM, where !α is the curvature form of α. In this case, the Poisson bracket is welldefined, for V1, V2 ∈ D(M,α), by

{V1, V2} = !α(XV1,XV2).

THEOREM 2. The Dirac algebra D(M,α) relative to a generic connection formα on a smooth manifold M consists of those smooth functions V on M whosedifferential annihilates the kernel of the curvature form !α at the points of the firstsingular stratum of the structure (M, α). In other words, when dimM = 2n, resp.dimM = 2n + 1, then at every point of "2(!α), resp. "1(!α), this condition isexpressed as

!n−1α ∧ dV = 0, resp. !n

α ∧ dV = 0.

Proof. If dimM = 2n, the equation of motion admits a unique smooth solutionXV on the open dense stratum "o(!α). We stand at a point of "2(!α) and, in aneighborhood of it, we trivialize the vector bundles TM and T ∗M. Performing thelocal change of coordinates and the change of frames suggested by Theorem 1, thegerm of !α is expressed as

ω = x1dx1 ∧ dx2 + ω0 with ω0 = dx3 ∧ dx4 + · · · + dx2n−1 ∧ dx2n.

The equation of motion is consequently written as iXω = η, where η denotes thecorresponding smooth 1-form to −dV in the local context at the origin of R

2n.Setting ∗θX = iXw, where w is a volume form, the equation of motion is writtenas

∗ωnθX = ∗(η ∧ ωn−1).

Clearly

∗ωn = x1, ωn−1 = iK1iK2w + x1ωn−20 ∧ dx1 ∧ dx2

with KI = ∂/∂xi , i = 1, 2. In view of the assumption, since it implies that

∗(η ∧ ωn−1) = x1η′,

where η′ is a germ of a smooth 1-form, Hadamard’s lemma gives the existenceof a local smooth solution on "2(ω) and, consequently, the existence of a smoothsolution of the equation of motion on "o(!α) ∪ "2(!α). This further suggeststhe annihilation of the second member of the equation of motion on "2(!α) and,


hence, by continuity, on "(!α). Standing on a local chart centered at an arbitrarypoint of "(!α), the equation of motion, in matrix form, is written as

[∗!nα(x)]2XV (x) = B(x),

where B(x) is the matrix product of the second member by the comatrix of !α(x).Lemma 4, which we prove below, suggests that, in suitable coordinates, ∗!α(x)

is an affine polynomial that generates the ideal of germs of smooth functions thatvanish on "(!α). This property, used in two steps, assures the smooth division ofB(x) by the square of ∗!α(x), cf. [20], and this implies the local smooth solutionof the equation of motion that we extend to a global smooth solution by means ofa partition of unity on M.

If dimM = 2n + 1, we stand at a point a ∈ "3(!α) and, performing the localchange of coordinates and the change of frames suggested by Theorem 1, the germof !α is expressed as

ω = iR(dx1 ∧ dx2 ∧ dx3) + ω0,

where

ω0 = dx4 ∧ dx5 + · · · + dx2n ∧ dx2n+1

and

R(x) =∑

1�i,j�3

Rijxi∂/∂xj with det(Rij ) �= 0.

The equation of motion is consequently written as iXω = η, where η denotes thecorresponding smooth 1-form to −dV in the local context at the origin of R

2n+1.Setting

θX = ∗iXw, θR = ∗iRw, w3 = dx1 ∧ dx2 ∧ dx3,

this equation is written as

θX ∧ θR = ∗(η ∧ ωn−1o ) + ∗(η ∧ w3 ∧ ωn−2

0 )θR.

The compatibility condition is expressed as iRη = 0 and, furthermore, we have

θR ∧ ∗(η ∧ ωn−10 ) = 0.

Lemma 5, which we prove below (see also de Rham’s division theorem [17]),suggests that this equation has a local smooth solution on "3(!α) and we concludethe same for the equation of motion. By continuity, we conclude that

∗!nα ∧ ∗(dV ∧ !n−1

α ) = 0

holds on M and this is exactly the condition of Lemma 5. Standing at a pointof an arbitrary stratum "k(!α), we deduce that, in suitable coordinates, ∗!α isequivalent to a 1-form whose components are affine polynomials. Therefore, wedirectly apply Lemma 5 and conclude that there exists a local smooth solution ofthe equation of motion which can be extended to a global smooth solution by meansof a partition of unity on M. ✷


Remark 1. The above theorem asserts that if the differential of the externalpotential annihilates the kernel of the curvature form !α at the points of the firstsingular stratum, "2(!α) or "1(!α), then it does so consequently in the remainingstrata of degeneracy as well. Also, in both cases the exactness of the curvature formimmediately leads to the property that if V ∈ D(M,α), then the vector field XV

is tangent to every stratum "k(!α) of M. Clearly, these vector fields constitute asubalgebra in the Lie algebra of smooth vector fields on M.

Remark 2. In the case of a 2n-dimensional manifold M, under the additionalgeneric assumption that the restriction of !n−1

α on "2(!α) doesn’t vanish at thepoint xo ∈ "2(!α), ([20]), through the singular local Martinet model ([9, 16]),the above theorem suggests that V ∈ D(M,α) if and only if there exist localcoordinates (ξ, ζ ) around xo such that, for (ξ ′, ζ ′) = (ξ2, . . . , ξn, ζ2, . . . , ζn) andsmooth functions a(ξ, ζ ), b(ξ ′, ζ ′),

V (ξ, ζ ) = ξ 21a(ξ, ζ ) + b(ξ ′, ζ ′).

2.2. In this section we present the tools necessary for the establishment of the abovetheorem. In the ring R{{X}} of convergent power series of n variables, we consider

G =∑k∈Nn

ckXk

with Xk = Xk11 . . . Xkn

n , k = (k1, . . . , kn) ∈ Nn, and we say that

suppG = {k ∈ Nn: ck �= 0}.

The lexicographic ordering of (n + 1)-uples (|k|, k1, . . . , kn) where |k| = k1 +· · · + kn, induces a total ordering on N

n. The initial exponent expG is defined asthe smallest element of suppG and the corresponding monomial is called the initialmonomial of G. If G1, . . . ,Gs ∈ R{{X}}, we introduce

9i = (expGi + Nn) −

i−1⋃j=1

9j, i = 1, . . . , s,

and we note

�0 = Nn −

s⋃i=1

9i.

THEOREM 3. Let f be the germ of a smooth function and G1, . . . ,Gs the germsof analytic functions at the origin of R

n. Then there exist germs of smooth functionsQ1, . . . ,Qs, R at the origin of R

n such that

f =∑

1�i�s

GiQi + R

and, in addition, the Taylor series of R satisfies that expR ∈ �0.


This division theorem of a smooth function by an ideal generated by analyticfunctions, established in [23], constitutes the basic tool of this analysis. It gener-alizes the classical division theorem of an analytic function by an ideal generatedby analytic functions, established by Hironaka in [8]. Now, using this result, weestablish the division property of a smooth function by an ideal generated by affinepolynomials in the ring of smooth functions, employed in the proof of Theorem 2.

LEMMA 4. If α is a generic connection form on an n-dimensional smooth mani-fold M then, every nonempty stratum "k(!α), k > 1, is defined locally by k(k −1)/2 homogeneous affine polynomials that generate the ideal �k(!α) in the ring ofgerms of smooth functions on M.

Proof. Theorem 1 suggests that "k(!α) is locally diffeomorphic to the smoothstratum of the algebraic variety

Vk :=⋂

i=1,...,s

V (P(k,i)),

where each variety

V (P (k,i)) = {x ∈ Nn | P(k,i)(x) = 0}

is defined by s = k(k − 1)/2 homogeneous affine polynomials Pk,1, . . . , Pk,s . Thedefinition of a homogeneous affine polynomial Pi ∈ R[x1, . . . , xn] implies thefollowing expression for its initial exponent

expPi = (ei1, . . . , ein), where eij = 0 or 1.

We observe that the set determining the remainder of the division of a smoothfunction by affine polynomials satisfies the

�0 =⋃

(i1,...,ir )

{exp<(i1...ir )},

where <(i1...ij ) is the coordinate space defined by the equations xir = 0, for certain1 � ir � n. We will proceed inductively on n ∈ N and obviously the assertionis true for n = 1. The division of the germ of a smooth function f by analyticfunctions is reduced to the formal division according to [14]. Let the formal powerseries f of f at the origin of R

n be written as

f (x) =∑

1�i�s

Pi(x)Qi(x) + R(x),

where exp R ∈ �0. If Qi , R are the germs of smooth functions and Qi, R theirformal series, then, applying Theorem 3, we obtain

f (x) =∑

1�i�s

Pi(x)Qi(x) + R(x).


Consider the intersection of the variety V (P1, . . . , Ps) with a coordinate hyper-plane contained in �0, say {x1 = 0}. Let x′ = (x2, . . . , xn), l ∈ N, then

R(x) = R(0, x′) + xl1Rl(x)

as well as one that for an affine polynomial πi , i = 1, . . . , s,

Pi(x′) = Pi(0, x

′) = Pi(x) − πi(x′)x1.

Observe then that V (P1, . . . , Ps)∩{x1 = 0} has a dense complement in V (P1, . . . ,

Ps). The inductive hypothesis suggests that

R(0, x′) =∑

1�i�s

Pi (x′)Qi(x) =

∑1�i�s

Pi(x)Qi (x) + S(x)x1.

In consequence, we obtain that

f (x) =∑

1�i�s

Pi(x)Qi(x) + S(x)x1,

where exp S ∈ �0. Proceeding in this way and following the approach of [20],we annihilate the remainder by successively removing the coordinates in �0 andapplying Lojasiewicz’s theorem (cf. [13, 14]), we pass from the formal division tothe smooth division. ✷

The above result leads to a proof of a de Rham–Saito type of division lemmain the smooth context like the one used in the proof of Theorem 2. In the analyticcontext, this property could be obtained from the classical theorems on the divisionof differential forms, cf. [15, 17, 24].

LEMMA 5. Let

αy(x) =∑

1�i�n

ai(x, y) dxi

be the germ a p-parameter family of 1-forms at the origin in Rn × R

p, n > 3,whose coefficients are affine polynomials ai ∈ R[x1, . . . , xn, y1, . . . , yp] and

S(αy) = {x ∈ Rn | a1(x, y) = · · · = an(x, y) = 0}

with codim S(αy) � 3. If β is the germ of a smooth p-parameter family of 2-form(resp. 1-form) at the origin in R

n × Rp satisfying the compatibility condition α ∧

β = 0, then there exists a germ at the origin of Rn × R

p of a smooth p-parameterfamily of 1-form θy (resp. function λy) such that β = θ ∧ α (resp. β = λα).

Proof. We start with the case of 1-forms and we consider

β(x, y) =∑

1�i�n

bi(x, y) dxi


expressed in local coordinates (x1, . . . , xn, y1, . . . , yp) at the origin of Rn × R

p.Evidently, the compatibility condition is expressed as

biaj − bjai = 0, i �= j. (0)

We introduce the auxiliary parameters ξ1, . . . , ξn and setting

aj (x, y, ξ) =∑i �=j

ai(x, y)ξi and bj (x, y, ξ) =∑i �=j

bi(x, y)ξi ,

and summing (0) with respect to i, we obtain aj bj − bj aj = 0. The codimensionassumption on S(αy) and the fact that the coefficients are affine polynomials assertthat, if

V (aiy) = {x ∈ Rn | ai(x, y) = 0},

we could always select the parameters in such a way that V (aiy) ∩ V (aiy) has adense complement in V (aiy). This holds since the level set for an affine polyno-mial is a conical hypersurface through the origin with singularities that lead to aconical nowhere dense subset which is easily proved by induction on the numberof variables. The preceding lemma on the smooth division by an affine ideal assertsthat there exist real smooth functions λi on R

n × Rp such that bi = λiai . Setting

this back in (0), we obtain that (λj − λi)aj ai = 0 and λi = λj = λ, yieldingthe conclusion that β = λα. The case of 2-forms is treated by induction on thedimension n. Let

β(x, y) =∑

1�i<j<�n

bij (x, y) dxi ∧ dxj

be expressed in local coordinates (x1, . . . , xn, y1, . . . , yp) at the origin of Rn ×R

p.For n = 3, the affine polynomials are a combination of

Pi(x) = xi, Pij (x) = xixj , i �= j, P (x) = x1x2x3

and the compatibility condition is expressed as

a1b23 − a2b13 + a3b12 = 0. (1)

The varieties V (aiy) are either hyperplanes or quadratic cones and the codimensionassumption implies that V (a1y)∩V (a2y)∩V (a3y) always has a dense complementin V (a1y) ∩ V (a2y), V (a2y) ∩ V (a3y), V (a1y) ∩ V (a3y). The preceding lemma onthe smooth division by an ideal generated by affine polynomials implies that

b12 = h12a1 + g12a2, b13 = h13a1 + g13a3, b23 = h23a2 + g23a3. (2)

Substituting in (1), we obtain

(h12 + g23)a1a3 + (h13 + h23)a1a2 + (g13 + g12)a2a3 = 0.


The same arguments lead to

h12 + g23 = u123a2, h13 + h23 = u132a3, g13 + g12 = u213a1 (3)

and this in turn leads to

u123 + u132 + u213 = 0. (4)

Now plugging (3), (4) in (1) and setting

θ = −g12 dx1 + h12 dx2 + (h13 − u213a3) dx3,

we obtain β = θ ∧ α.Assume now that this is true up to order n− 1. Then we split the coordinates in

a manner that maintains the codimenension assumption and we embody the othercoordinate in the parameters. We then set

β = βn−1 + η ∧ dxn,

where

βn−1 =∑

1�i<j<n

bij dxi ∧ dxj and η =∑

1�i<n

cindxi

and, accordingly,

α = αn−1 + αn ∧ dxn,

where

αn−1 =∑

1�i<n

ai dxi.

The compatibility condition is now expressed as

βn−1 ∧ αn−1 + (anβn−1 − η ∧ αn−1) ∧ dxn = 0

and this is evidently reduced to

βn−1 ∧ αn−1 = 0 (5)

and

anβn−1 − η ∧ αn−1 = 0. (6)

The inductive hypothesis suggests that

βn−1 = θn−1 ∧ αn−1

which we set in (5) and obtain

(η − anθn−1) ∧ αn−1 = 0.


The latter by the division of the 1-forms suggests that

η = anθn−1 + f αn−1

and setting

θn = θn−1 + f dxn

we obtain that β = θn ∧ α. ✷

3. Generic Aspect for Singular Lagrangian Systems

3.1. Let M be the n-dimensional configuration space of a mechanical system withLagrangian function L: TM → R and consider the Legendre transformationbetween TM and T ∗M expressed as

L(x, v) = (x, dvL),

where dvL denote the vertical derivative. The equations of motion of the Lagrangiansystem (M,L) are expressed intrinsically by the Lagrangian vector field XL givenby the equation

iXL!L + dEL = 0,

where !L is the Lagrangian form defined as the pullback of the standard sym-plectic form of T ∗M through the Legendre transformation and the energy functionexpressed as

EL(x, v) =n∑

i=1

vi∂viL(x, v) − L(x, v).

The Lagrangian vector field XL is defined outside of the critical locus of the Legen-dre transformation, which coincides with the degeneracy locus "(!L) of the La-grangian form, i.e. points where its kernel is not trivial. The projection of this locusgives the singularities of the equations of motion on M; these singularities arecalled removable when the Lagrangian vector field extends smoothly everywhereon TM.

A Lagrangian system (M,L) is called generic if the differential of the cor-responding Legendre transformation is transverse to the Legendre stratificationof the Legendre bundle E(TM), cf. [22]. In this case, a stratification in smoothsubmanifolds is induced on the tangent bundle TM:

"k,s(!L) = {(x, v) ∈ TM | dim ker(x,v) !L = k, dim ker(x,v) DL = s},0 � k, s � n,

such that

"k,s(!L) =⋃i,j∈N

"k+i,s+j (!L).


The versal unfolding lemma for the generic Legendre sections proved in [22], aftera direct calculation, leads to the local models of the stratified degeneracy locus

"(!L) =⋃

(k,s) �=(0,0)

"k,s(!L).

More precisely, the strata of this locus are defined locally, in an appropriate chart,by quadratic-affine polynomials, i.e. homogeneous polynomials, without multiplefactors, each factor of which is expressed by fixing all variables but one as

Aix2i + Bixi + Ci with

Ai, Bi, Ci ∈ R[x1, . . . , xi−1, xi+1, . . . , xn], i = 1, . . . , n.

These quadratic-affine polynomials generate the ideal of smooth functions thatvanish on the strata of "(!L) in the ring of germs of smooth functions; in otherwords, the degeneracy locus possesses the differentiable Nullstellensatz ([22]). Thefollowing theorem is proved in an analogous way to Theorem 2; its proof is givenexplicitly in [22] in the general context of the singular variational problems.

THEOREM 6. The singularities of an n-dimensional generic Lagrangian system(M,L) are removable if and only if, at the points of the first singular stratum"1,1(!L), there holds !n−1

L ∧ dEL = 0. In this case, the projection of the integralscurves of the Lagrangian vector field gives the motion on M and, in particular, thetrajectories with initial conditions on a stratum "k,s(!L), k, s ∈ N, remain alwaysin this stratum.

Remark. This situation is illustrated in the following example which could beseen as a generalization of the celebrated Weierstrass example of a variational prob-lem having no trivial extrema. Let L(x, x′, ξ, ξ ′) be a Lagrangian on the tangentbundle of R

2 where (x, x′, ξ, ξ ′) denote the coordinates in R4 and assume that

dL(0) = 0, ∂2ξ L(0) = 0, ∂2

ξξ ′L(0) = 0,

∂2ξ ′L(0) �= 0, ∂3

ξ L(0) �= 0.

Clearly, the Lagrangian satisfies the classical transversality condition and, as aconsequence, the corresponding Legendre transformation satisfies this conditionat the origin of R

4 as well. The Malgrange preparation theorem and Morse Lemmasuggest that, in suitable local coordinates, we could take

L(x, x′, ξ, ξ ′) = ξ 3 + ξ ′2 + a(x.x′)ξ + b(x, x′)

with a, b smooth functions and the singular locus of the corresponding Legendretransformation is given by ξ = 0. The admissibility condition given by the abovetheorem implies the following expression for the energy function

EL(x, x′, ξ, ξ ′) = 2ξ 3 + ξ ′2 − b(x′).


The corresponding Lagrangian vector field is

XL = ξ∂

∂x+ ξ ′ ∂

∂x′ − 1

2b′(x′)

∂

∂ξ ′

and the projection of its integral curves gives the solution of this problem on R2.

3.2. The case of the Lagrangian systems (M,L) with the Lagrangian defined througha quadratic form Q on the manifold M,

L(x, v) = 12Q(x)(v, v) − V (π(v)),

where V is a potential function on M and π : TM → M the canonical projection,presents certain generic particularities for Q transverse to the natural stratificationof the fibering of quadratic forms on M. The critical locus of the correspondingLegendre transformation is projected on the degeneracy locus "(Q) consistingof the points where the kernel of the quadratic form is not trivial. This locus isstratified in smooth submanifolds

"k(Q) = {x ∈ M | dim kerx Q = k}, 0 < k � n,

of codimension k(k + 1)/2, if nonempty, and their closure satisfies the frontiercondition. On the complement of "(Q) a unique connection of vanishing torsionassociated with Q is defined, the Levi-Civita connection ∇. Let XQ(M) be theset of smooth vector fields that are tangent to the strata of "(Q) and assume,in addition, that the kernel of Q is transverse to the tangent space of "1(Q) atevery point of an open and dense set contained in "1(Q). Then, the Levi-Civitaconnection extends to a smooth map

∇: XQ(M) × XQ(M) → XQ(M)

if and only if "(Q) is autoparallel. In this case, on every stratum "k(Q) thequadratic form Q induces a pseudo-Riemannian metric Qk and the above mapinduces the Levi-Civita connection associated with Qk, 0 � k � n. The classi-cal situation concerning the geodesics on a regular pseudo-Riemannian manifoldcan be generalized in this singular context and thus we obtain the behavior of thegeodesics in the stratified singular locus, cf. [22].

Acknowledgement

This work was partially supported by the ‘C. Caratheodory’ Research Programmeof the University of Patras, Greece.

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323

An Electrostatics Approach to theDetermination of Extremal Measures

JEAN MEINGUETUniversité Catholique de Louvain, Institut Mathématique, Chemin du Cyclotron 2,B-1348 Louvain-la-Neuve, Belgium. e-mail: [email protected]

(Received: 20 February 2001)

Abstract. One of the most important aspects of the minimal energy (or induced equilibrium) prob-lem in the presence of an external field – sometimes referred to as the Gauss variation problem – isthe determination of the support of its solution (the so-called extremal measure associated with thefield). A simple electrostatic interpretation is presented here, which is apparently new and anywaysuggests a novel, rather systematic approach to the solution. By way of illustration, the classicalresults for Jacobi, Laguerre and Freud weights are explicitly recovered by this alternative method.

Mathematics Subject Classifications (2000): 31A15, 31A25, 78A30.

Key words: logarithmic potential, external field, Gauss variation problem, electrostatic interpreta-tion.

1. Introduction

Mathematicians (and physicists!) generally ‘know’ Dirichlet’s principle. They arelikely less familiar with the related W. Thomson (Lord Kelvin) principle (in elec-trostatics) and its special case called the Gauss variation problem (or forced equi-librium problem), which is the problem of minimizing – in the presence of a givenexternal field – the ‘energy’ associated with any sourceless (or solenoidal) vectorfield in the outer region bounded by a given closed set (the so-called ‘conductor’,supposed once for all to be ‘perfect’) over which a positive (electric) charge ofprescribed amount is to be distributed so as to reach equilibrium (see, e.g., [5], pp.43–44, 55–57, or [2], pp. 46, 51).

As a matter of fact, the underlying potential theory needed in the following isthe theory of logarithmic potentials with external fields, whose interaction withapproximation-theoretical techniques and problems in the complex plane or on thereal line proved extremely fruitful in recent years. As is well known, a point chargein the plane is ‘equivalent’ to a uniformly distributed charge on a straight line– perpendicular to the plane – in R3, such (positive or negative) point chargesrepelling or attracting each other according to an inverse distance law (well known

324 JEAN MEINGUET

consequence of Coulomb’s law). Gauss’s variation problem then becomes that ofminimizing the (weighted) energy integral

IQ(µ) :=∫�

∫�

log1

|z − t| dµ(z) dµ(t) + 2∫�

Q dµ, (1)

where the minimum is taken over all positive unit charge distributions (i.e., positiveunit Borel measures) µ carried by the conductor � (i.e., supp(µ) ⊆ �) while Q

(defined on � and real-valued) is the so-called external field (strictly speaking, sucha scalar ‘field’ Q is a potential). It is known – see, e.g., [6], pp. 26–33, for the basictheorem and its detailed mathematical proof – that, under rather weak conditionsof ‘admissibility’ on Q, there exists a unique solution µQ (called equilibrium orextremal measure associated with Q) of this optimization problem, which is suchthat the relation

Uµ(z) :=∫�

log1

|z − t| dµ(t) = −Q(z) + FQ, z ∈ SQ, (2)

holds quasi-everywhere (i.e., possibly up to a set of zero logarithmic capacity),where SQ := supp(µQ) is compact of positive capacity and FQ is the so-calledmodified Robin constant for Q. It should be stressed that a most glaring differencewith the classical equilibrium problem (for which Q = 0) is that SQ need notcoincide with the outer boundary of � and, in fact, can be an arbitrary subset of �,possibly with positive area.

Determining SQ is therefore one of the most important aspects of the energyproblem (or minimization of (1)). To find the extremal measure, it then remains tosolve Dirichlet problems (for the Laplace equation and the essential boundary con-ditions (2)) and to launch the classical recovery machinery (e.g., the Sokhotskyi–Plemelj formula for arcs and its integrated version known as the Stieltjes–Perroninversion formula of Cauchy transforms). As discovered by Mhaskar–Saff in theeighties, determining SQ amounts to minimizing over the set of possible supportsthe (quasi-everywhere) constant value FQ of the extremal potential. It is surpris-ing that such an obviously hard problem can be solved explicitly under suitableconvexity assumptions (satisfied by the important weights w := e−Q of Jacobi,Laguerre, and Freud), SQ being then an interval whose endpoints can be obtainedby solving a (simple) integral equation.

The main goal of this paper is to present (in Section 2) a novel, rather systematicapproach to the determination of SQ. This mathematical method can be regarded asa modern example of ‘physical mathematics’ in the sense of Sommerfeld; it is in-deed motivated by an apparently new electrostatic interpretation. By applying thisalternative method, we will rediscover rather automatically the ‘classical results’mentioned above (see Sections 3, 4 and 5).

EXTREMAL MEASURES VIA ELECTROSTATICS 325

2. A Physically Oriented Approach

By way of constructive illustration, we will consider here the simple, two-dimen-sional physical picture: a (perfect) conductor in vacuum, say the finite segment� := [−1, 1] in the extended complex z-plane C, is subjected to the electrostaticfield of potential

Q(z) := λ log1

|z − a| , z = x + iy, (3)

due to an electric charge λ > 0 located at an exterior point, say a > 1.• Suppose first that the conductor � is grounded (i.e., connected to earth), which

means that it may acquire whatever charges are necessary to enable it to remain atthe same potential (zero, by convention). The resulting potential thus created (thatis, (3) in the presence of the grounded �) is classically – up to the proportionalityfactor λ – the Green function of the complement of � (the so-called cut plane) withpole at a, viz.,

g(z, a) = log

∣∣∣∣1 − φ(a)φ(z)

φ(z) − φ(a)

∣∣∣∣, φ(z) := z +√z2 − 1 (4)

(see, e.g., [6], p. 110). It should be noted once for all that any expression like√z2 − 1 is to be understood as the branch that behaves like z near infinity, so

that w = φ(z) is simply the inverse of the well-known Joukowski conformal mapz = (1/2)(w + 1/w) of the exterior of the unit disk (in the φ-plane) onto thecomplement of �. It follows in particular that the circle (in the φ-plane):

φ(z) = φ(a)eiθ , −π � θ � π,

corresponds to the ellipse (in the z-plane):

z = a cos θ + i√a2 − 1 sin θ,

with foci at z = ±1, and semiaxes a,√a2 − 1, whose polar representation can be

written in the form

ρ = a2 − 1

a + cos �= a − cos θ, (5)

where ρ denotes the distance to the pole (of polar coordinates) z = 1, � is the ‘trueanomaly’ and θ is the ‘eccentric anomaly’ (these terms are borrowed from celestialmechanics).

The distribution µ of the charge that is induced (by electrostatic influence) onthe grounded conductor � by the point charge λ > 0 at a > 1 or, equivalently,−λ times the so-called balayage measure of the Dirac point mass at a onto � (see,e.g., [6], pp. 81–82), is given by

dµ(x) := − λ

π

∂

∂ng(x, a) dx

= − λ

π

∣∣∣∣√a2 − 1

(a − x)√

1 − x2

∣∣∣∣ dx, x ∈ [−1, 1], (6a)

326 JEAN MEINGUET

where dx is the arc length on �, and ∂/∂n denotes differentiation in the directionof the inner normal with respect to the complement of � (as a matter of fact,for obvious symmetry reasons, n may denote here either the upper or the lowernormal). The concrete expression on the right in (6a) is most important for thefollowing; it is found by taking the limit of the real part of (−iλ/π) times thederivative of the analytic function

log1 − φ(a)φ(z)

φ(z) − φ(a)

as z tends to x ∈ (−1, 1) from the upper half-plane, while keeping in mind that (forcontinuation reasons)

√1 − x2 is positive for y = 0+ (resp. negative for y = 0−)

and that√a2 − 1 is positive for a > 1 (but negative for a < −1, see Section 3).

With the change of variable x = cos θ , (6a) takes the simpler form

dµ(cos θ) = − λ

2π

∣∣∣∣√a2 − 1

a − cos θ

∣∣∣∣ dθ, θ ∈ [−π, π ], (6b)

where dθ denotes – throughout the whole paper – arc measure on the unit circle;in view of (5), the corresponding density (or Radon–Nikodym derivative)dµ(cos θ)/dθ of the induced charge has a nice geometric interpretation. As isclassically expected (see, e.g., [4], p. 230), for any grounded conductor occupying abounded region in the presence of a point charge, the density of the induced chargewill never change sign; more precisely, the total mass of the distribution µ is −λ

(this can be verified by explicit integration), while

minθ

dµ(cos θ)

dθ= − λ

2π

√a + 1

a − 1, (7)

this minimal value being attained for θ = 0.• Suppose now that the conductor � is insulated (i.e., imbedded in vacuum). If

a positive unit charge is placed on it in the absence of any external field, then itsequilibrium distribution µ0 (i.e., the unique positive unit Borel measure minimizingthe energy integral (1) where Q = 0) is known to be the arcsine distribution

dµ0(x) = 1

π√

1 − x2dx, x ∈ [−1, 1], (8a)

or, equivalently,

dµ0(cos θ) = 1

2πdθ, θ ∈ [−π, π ], (8b)

that is, the normalized arc measure. The constant value F0 assumed by its (loga-rithmic) potential on � (the so-called Robin constant) is clearly

F0 := log1

cap(�)= log 2, (9)


the logarithmic capacity cap(�) of a finite segment being notoriously equal toone-fourth its length.

• Suppose finally that the conductor � is insulated in the field of potential (3).In view of (7), it is clear that

dµQ := dµ + λ

√a + 1

a − 1dµ0 = λ

2π

(−

√a2 − 1

a − cos θ+

√a + 1

a − 1

)dθ (10)

is the unique (nonnegative and absolutely continuous with respect to θ) equilibriumdistribution of charges over � that minimizes its potential; indeed, the definition(10) amounts simply to adding to the signed measure dµ (whose logarithmic po-tential plus the external field has the constant value 0 on �) the smallest multipleC dµ0 of the positive measure dµ0 (whose logarithmic potential on � has the con-stant value F0) that makes the resulting measure dµQ nonnegative, its logarithmicpotential on �, namely, the constant CF0 with C := λ

√(a + 1)/(a − 1), being

therefore as small as possible. Provided that

λ

(− 1 +

√a + 1

a − 1

)= 1, (11)

which simply means that the total mass of (10) over its support � is 1 (or equiv-alently, that the charge placed on � is λ + 1), the distribution µQ is nothing butthe extremal measure minimizing (1) for Q defined by (3) (after all, the potentialin electricity and magnetism is identical with potential energy per unit charge,see, e.g., [4], p. 53). The (logarithmic) potential FQ of � corresponding to thedistribution (10) ‘normalized’ by (11) – that is, the modified Robin constant for Q(see [6], p. 27) – is thus clearly

FQ = (λ + 1) log 2. (12)

As is easily verified (by an elementary computation detailed in [6], p. 46), thepotentials (9) and (12) satisfy the important relation

FQ = F0 +∫�

Q dµ0 (13)

according to which −FQ is the so-called ‘F -functional’ of Mhaskar–Saff (see [6],Chap. IV) whose maximization (over the set of possible supports) is achieved bythe support SQ of the extremal measure µQ; an alternative proof of (13) followsfrom the successive relations implied by

Q(x) + Uµ(x) = 0, x ∈ [−1, 1],∫�

dµ = −λ,

where Uµ denotes the logarithmic potential of the induced measure (6a), viz.:∫�

Q dµ0 = −∫�

Uµ dµ0 = −∫�

Uµ0 dµ = −F0

∫�

dµ

= λF0 = (1 + λ)F0 − F0 = FQ − F0

328 JEAN MEINGUET

(the only nontrivial equality sign is the second one, which is justified by the Fubini–Tonelli Theorem).

• It is just a simple exercise to rewrite our results about the extremal measurein a more familiar form (see, e.g., [7], p. 773). As a matter of fact, we have only tochange the interval [−1, a] into [−1, 1] via the affine transformation

ξ = 2x + (1 − a)

1 + a(14)

without modifying the ratio of the fixed charge of amount λ > 0 at the pointx = a > 1 (resp. ϑ ∈ (0, 1) at ξ = 1) to the continuous charge of amount 1(resp. 1 −ϑ) to be distributed on [−1, 1] (resp. on its image by (14)) so as to reachequilibrium. The latter condition, viz.,

λ = ϑ

1 − ϑ, (15)

combined with the normalization relation (11), yields

a = 1 + ϑ2

1 − ϑ2,

so that the actual support of the continuous charge 1 − ϑ on the ξ -axis (i.e., theimage of [−1, 1] by (14)) is

SQ = [−1, ξ0], ξ0 := 3 − a

1 + a= 1 − 2ϑ2. (16)

As to the distribution of this charge on SQ, it readily follows from (10) – alwaysnormalized by (11) – by the affine transformation (14), owing to formulas (15),(16). It turns out that the associated Jacobian has a remarkable form, viz.,

dξ

dθ= 1

2

√(ξ + 1)(ξ0 − ξ), −1 � ξ � ξ0,

where the factor 1/2 is due to the fact that we must integrate twice along cuts if weintegrate once over the unit circle. Hence, the final result

dµQ(ξ) = 1

π(1 − ϑ)

√(ξ + 1)(ξ0 − ξ)

1 − ξ 2dξ, −1 � ξ � ξ0, (17)

which concludes our alternative treatment of the simplest example of explicit de-termination of an extremal measure that is considered in [6] (see pp. 205–206,243), that is, the application entitled ‘Incomplete Polynomials of Lorentz’ (note,however, that our ξ is to be identified with −t in the last formula of Example 5.3on p. 243 in [6]).


3. The Extremal Measure for Jacobi Weights

This quite natural generalization of the physical picture considered in Section 2corresponds to the replacement of (3) by the electrostatic field of potential

Q(z) := λ1 log1

|z − a1| + λ2 log1

|z − a2|, (18)

the electric charge λ1 > 0 (resp. λ2 > 0) being located at a point outside theconductor � := [−1, 1], say a1 > 1 (resp. a2 < −1).

• Since the complement of � possesses an explicitly known Green function,namely (4), the potential of the total field created by the charges in (18) and thecountercharges induced by influence on the conductor � supposed to be groundedis classically given by the associated Green potential, viz.,

V (z) := λ1g(z, a1) + λ2g(z, a2) (19)

(see, e.g., [6], p. 124). In view of (6a), (6b) and (19), the distribution µ of the chargethat is induced by (18) on the grounded conductor � is given by

dµ(cos θ) = − 1

2π

(λ1

√a2

1 − 1

a1 − cos θ+ λ2

∣∣√a22 − 1

∣∣|a2| + cos θ

)dθ,

θ ∈ [−π, π ]. (20)

The total mass of this distribution is clearly −λ1 − λ2, while

C := −2π minθ

dµ(cos θ)

dθ= −2π min

(dµ

dθ(−1),

dµ

dθ(1)

)(21)

immediately follows from the convexity with respect to the variable cos θ of theparenthesized expression in (20) (its second derivative is indeed positive over[−1, 1]).

• Suppose now that the conductor � is insulated in the field of potential (18). Itis clear that

dµQ := dµ + C dµ0, with definitions (8b), (20) and (21), (22a)

is the unique (nonnegative and absolutely continuous with respect to θ) equilibriumdistribution of charges over � that minimizes its potential. However, to have achance to solve eventually the underlying Gauss variation problem or, equivalently,to minimize the potential FQ of � corresponding to the extremal measure µQ (oftotal mass 1!) in the presence of the external field of potential (18), the pointsa1 > 1 and a2 < −1 must be such that the minimal value (21) is as great aspossible. This requires of the two expressions on the right in (21) to be equal (theyvary indeed in opposite directions as either a1 > 1 or a2 < −1 varies), theircommon value being necessarily

C = 1 + λ1 + λ2 (22b)

330 JEAN MEINGUET

(since the total mass of (22a) over � must be 1, while the total mass of (20) is−λ1 − λ2). In other words, the following equations must be satisfied:

ϑ1

√a1 − 1

a1 + 1+ ϑ2

√|a2| + 1

|a2| − 1= 1, (23a)

ϑ1

√a1 + 1

a1 − 1+ ϑ2

√|a2| − 1

|a2| + 1= 1, (23b)

where

ϑ1 := λ1

1 + λ1 + λ2, ϑ2 := λ2

1 + λ1 + λ2. (23c)

This is equivalent to the apparently simpler nonlinear system for a1, a2:

ϑ1 =√a2

1 − 1

a1 − a2, ϑ2 =

∣∣√a22 − 1

∣∣a1 − a2

, a1 > 1, a2 < −1, (24a)

whose (unique) solution is

a1 = 1 + ϑ21 − ϑ2

2√#

, a2 = −1 + ϑ22 − ϑ2

1√#

, (24b)

where

# := [1 − (ϑ1 + ϑ2)2][1 − (ϑ1 − ϑ2)

2], (24c)

as it can be shown by somewhat lengthy (though elementary) computations.• To rewrite the extremal measure in a more familiar form (see, e.g., [7], pp. 772–

774), it remains only to change the interval [a2, a1] into [−1, 1] via the affinetransformation

ξ = 2x − (a1 + a2)

a1 − a2(25)

without modifying the ratios of the fixed charge of amount λ1 > 0 at x = a1 > 1(resp. ϑ1 ∈ (0, 1) at ξ = 1) and of the fixed charge of amount λ2 > 0 at x =a2 < −1 (resp. ϑ2 ∈ (0, 1) at ξ = −1) to the continuous charge of amount 1 (resp.1 −ϑ1 −ϑ2) to be distributed on [−1, 1] (resp. on its image by (25)) so as to reachequilibrium; in fact, the conditions

λ1 = ϑ1

1 − ϑ1 − ϑ2, λ2 = ϑ2

1 − ϑ1 − ϑ2and

1 + λ1 + λ2 = 1

1 − ϑ1 − ϑ2(26)


are trivially equivalent to (23c). It follows that the actual support of the continuouscharge 1 − ϑ1 − ϑ2 on the ξ -axis (i.e., the image of [−1, 1] by (25)) is

SQ = [ξ2, ξ1], (27a)

where

ξ1 = ϑ22 − ϑ2

1 + √#, ξ2 = ϑ2

2 − ϑ21 − √

#, with definition (24c). (27b)

As to the distribution of this continuous charge on SQ, it follows from (22) bythe affine transformation (25) – owing to formulas (23c), (24), (26), (27) – the finalresult being

dµQ(ξ) = 1

π(1 − ϑ1 − ϑ2)

√(ξ − ξ2)(ξ1 − ξ)

1 − ξ 2dξ, ξ2 � ξ � ξ1, (28)

in accordance with, e.g., [6] (see pp. 207 and 241).

4. The Extremal Measure for Laguerre Weights

A crucial step in the approach presented in this paper is the determination of theelectrostatic potential outside the grounded conductor � (i.e., any given compactset of C, of positive capacity) in the presence of the given external field. In theapplications considered so far, this fundamental influence problem could be solvedreadily owing to the explicit knowledge of the Green function (of the outer domainrelative to �). On the other hand, in the remaining applications, where the externalfield is defined directly (at least in part) rather than via given external charges,this crucial step actually requires solving explicitly a Dirichlet boundary valueproblem.

In the Laguerre case, the external field has for potential

Q(z) := λ z + s log1

|z − a| , λ > 0, s � 0, a < −1. (29)

Unlike the second term, which is of the type considered before (i.e., potential of acharge s � 0 located at a given point a < −1), the first term is not created by acharge but rather by a dipole at infinity (of axis 0x and of moment λ); though this‘physical’ interpretation may prove interesting (see, e.g., [1], p. 35), we will notexploit it here, essentially because it does not hold for non-uniform fields such asthe one considered in Section 5.

• Now suppose that the conductor � := [−1, 1] is grounded and subjected tothe field of potential (29). The potential of the total field thus created outside � isclearly the sum of three terms: the Green potential of the charge s located at thepoint a (i.e., s times the Green function (4), where φ(a) := a − |√a2 − 1| sincea < −1), the external field of potential λx, and the solution h(z) of the exterior

332 JEAN MEINGUET

Dirichlet problem:

#h(z) = 0, z �∈ [−1, 1],h(z) bounded as |z| → ∞ (i.e., regularity at infinity),

h(x) = −λx, x ∈ [−1, 1].It turns out that the conformal transplant of h under the Joukowski mapping, viz.,

H(w) := h

(1

2

(w + 1

w

)), w = |w|eiθ with |w| � 1, θ ∈ [−π, π ] (30)

can be obtained readily by separating the variables in the transplanted exteriorDirichlet problem:

#H(w) = 0, |w| � 1, (31a)

H(w) bounded as |w| → ∞, (31b)

H(eiθ ) = −λ cos θ, θ ∈ [−π, π ]. (31c)

Indeed, if we transform #H to polar coordinates |w|, θ , we get for the solutionH(w) of (31a) the general form

A0 + B0 log |w| +∑k �=0

Ak cos kθ + Bk sin kθ

|w|k ;

now the condition (31b) of regularity at infinity (see, e.g., [4], p. 248) impliesB0 = 0 and Ak = Bk = 0 for all negative integers k; the Dirichlet condition (31c)thus reduces to

∞∑k=0

(Ak cos kθ + Bk sin kθ) = −λ cos θ, θ ∈ [−π, π ],

which finally yields

A1 = −λ, B1 = 0, Ak = Bk = 0 for k �= 1.

In view of (30), the required potential of the total field created outside the groundedconductor � by the external field of potential (29) is given by

V (z) = s log

∣∣∣∣1 − φ(a)φ(z)

φ(z) − φ(a)

∣∣∣∣ + λ z − λ 1

φ(z). (32)

According to the classical definition

dµ(x) := − 1

π

∂

∂nV (x) dx, x ∈ [−1, 1],

we get from (32) the explicit expression

dµ(cos θ) = − 1

2π

(s|√a2 − 1||a| + cos θ

+ λ cos θ

)dθ, θ ∈ [−π, π ], (33)


for the distribution µ of the charge induced by (29) on the grounded conductor �.The remarkable relation (21) holds again (the parenthesized function of cos θ in(33) is indeed convex over [−1, 1]), so that we get explicitly

C = max

(s

√|a| + 1

|a| − 1− λ, s

√|a| − 1

|a| + 1+ λ

). (34)


dµQ := dµ + C dµ0, with definitions (8b), (33) and (34), (35)

is the unique (nonnegative and absolutely continuous with respect to θ) equilibriumdistribution of charges over � that minimizes its potential, for any given values ofthe parameters λ > 0, s � 0, a < −1. It turns out that a further minimizationof this potential is automatically achieved if the point a < −1 is such that the twoexpressions on the right in (34) are equal (they vary indeed in opposite directions asa varies); owing to this condition, which amounts to s = λ|√a2 − 1|, (34) reducesto C = λ|a| = √

λ2 + s2. But the total mass of (35) over � must be 1, while thetotal mass of (33) is −s, so that necessarily C = s + 1; all these relations finallyimply

a = − s + 1√2s + 1

, (36)

λ = √2s + 1. (37)

• To rewrite the extremal measure in a more familiar form (see, e.g., [6], pp.208 and 243), it remains only to change the interval [a, 1] into [0, ξ1] (where ξ1 isany finite positive number) via the affine transformation

ξ = ξ1x − a

1 − a(38)

without modifying the ratio of the fixed charge of amount s > 0 at the point x = a

defined by (36) (resp. ϑ ∈ (0, 1) at ξ = 0) to the continuous charge of amount 1(resp. 1−ϑ) to be distributed on [−1, 1] (resp. its image SQ by (38)) so as to reachequilibrium. It follows that the actual support of the continuous charge

1 − ϑ := 1

s + 1(39)

on the ξ -axis is

SQ = [ξ2, ξ1], (40a)

where

ξ2

s + 1 − √2s + 1

= ξ1

s + 1 + √2s + 1

=: 1

*. (40b)

334 JEAN MEINGUET

It should be noticed that, up to an additive (unimportant!) constant, the potential(29) on the positive ξ -axis has the simple expression *ξ − s log ξ , which dependson two independent parameters * > 0, s � 0 (remember that λ was eliminated by(37), for the sake of normalization). As to the distribution of the continuous charge(39) on SQ, it readily follows from (35) – normalized by (36), (37) – by the affinetransformation (38) and formulas (39), (40), the final result being

dµQ(ξ) = *

π

√(ξ − ξ2)(ξ1 − ξ)

ξdξ, ξ2 � ξ � ξ1, (41)

in accordance with [6] (see p. 243).

5. The Extremal Measure for Freud Weights

The external field – to which the standard conductor � := [−1, 1] is subjected –has now for potential

Q(z) := c|x|λ, c > 0, λ > 0 (and x := z). (42)

Unlike the ‘physical’ fields considered before, it is thus directly defined by itsmathematical expression rather than via given external electric charges (or dipoles).

• If the conductor � is grounded, the potential of the total field thus createdoutside � is naturally obtained by adding to (42) the solution h(z) of the exteriorDirichlet problem:

#h(z) = 0, z �∈ [−1, 1],h(z) bounded as |z| → ∞ (i.e., regularity at infinity),

h(x) = −c|x|λ, x ∈ [−1, 1].Here again, the conformal transplant H of h under the Joukowski mapping, whichis the function defined by (30), can be obtained readily by separating the variablesin the transplanted exterior Dirichlet problem:

#H(w) = 0, |w| � 1, (43a)

H(w) bounded as |w| → ∞, (43b)

H(eiθ ) = −c| cos θ |λ, θ ∈ [−π, π ]. (43c)

Indeed, if we transform #H to polar coordinates |w|, θ , we get for any solution of(43a, b) the Fourier series representation

H(w) = −c

∞∑′

k=0

A2kcos 2kθ

|w|2k ,

the Dirichlet boundary condition (43c) reducing to∞∑′

k=0

A2k cos 2kθ = | cos θ |λ, θ ∈ [−π, π ] (44a)


(where the prime affecting the summation symbol means that the first term is to betaken with half weight), or equivalently, to

A2k := 4

π

∫ π2

0(cos θ)λ cos 2kθ dθ

= 1

2λ−1

,(λ+1)

,(λ/2+k+1),(λ/2−k+1)(44b)

(see, e.g., [8], p. 263, Example 40). Owing to the reflection formula of Euler forthe gamma function, this expression of A2k can be rewritten in the form

A2k = ,(λ+1) sin (πλ/2)

2λ−1π(−1)k+1,(k−λ/2)

,(k+λ/2)

1

k+λ/2, (44c)

which yields (via Stirling’s formula) the asymptotic formula

A2k ∼ ,(λ+1) sin(πλ/2)

2λ−1π(−1)k+1 1

kλ+1as k → ∞; (44d)

Weierstrass’s test is thus applicable, so that the Fourier series (44a) of | cos θ |λconverges uniformly and absolutely to its generating function.

Since the potential V (z) of the total field created outside the grounded conduc-tor � by the external field of potential (42) has for conformal transplant (under theJoukowski mapping)

V(w) = −c

∞∑′

k=0

A2kcos 2kθ

|w|2k + c

2λ

(|w|+ 1

|w|)λ

| cos θ |λ, |w| � 1,

the distribution µ of the charge induced on � is apparently given by

dµ(cos θ) = − 1

2πlim

|w|→1+∂V(w)

∂|w| dθ = − c

π

∞∑k=1

kA2k cos 2kθ dθ. (45)

The total mass of µ is evidently 0 (since the lines of force of the field of potential(42) are parallel to the x-axis), while

C := −2π minθ

dµ(cos θ)

dθ= 2c

∞∑k=1

kA2k, (46)

this minimal value being attained for θ = 0 mod π – were it simply for ‘physical’reasons (logical interpretation of the underlying problem of electrostatic influence)– whereas

maxθ

dµ(cos θ)

dθ= − c

π

∞∑k=1

(−1)kkA2k > 0

336 JEAN MEINGUET

is attained for θ = π/2 mod π and is finite or not according as λ > 1 or not (thisfollows from the properties of A2k mentioned above).

It should be stressed that the trigonometric series in (45) is actually an Abelsum; however, by virtue of classical tests (substantially due to Abel) exploitingthe properties (44c), (44d) of the A2k’s, this series is convergent (except for θ =±π/2 mod 2π , whenever λ � 1), necessarily to its Abel sum. It turns out that thesum of the series in (46) can be found as an Abel sum by an explicit (but lengthy)computation, the final result being

∞∑k=1

kA2k = λ

π

∫ π2

0(cos θ)λ dθ = ,(λ/2+1/2)

,(λ/2),(1/2); (47)

the last expression is simply (44b) for k = 0, rewritten by means of Legendre’sduplication formula (see, e.g., [8], p. 240). Rather than give complementary details,we deem it preferable to describe briefly an alternative approach to (47), based onthe modern theory of generalized functions or distributions. Consider the classicalFourier series expansion

∞∑k=1

cos kθ

k= − log |2 sin(θ/2)| (48)

whose generating function goes out of bound at θ = 0 mod 2π , while being in-tegrable in the Lebesgue sense over the fundamental period interval (−π, π). Itis easily proved (see [3], p. 30) that the Fourier series in (48) converges in thesense of generalized functions to the function on the right-hand side, so that it maybe differentiated term-by-term (in the distributional sense) any number of times,which yields in particular the distributional result:

∞∑k=1

k cos θ = (log |2 sin(θ/2)|)′′; (49a)

by techniques that are standard in the theory of distributions (see, e.g., [3], p. 65,for similar results), we are led to concrete definitions of the second distributionalderivative – denoted by the symbol ′′ – in (49a), viz.,

〈(log |2 sin(θ/2)|)′′,-(θ)〉= −

∫ π

0

cos(θ/2)

2 sin(θ/2)

[d

dθ-(θ) + d

dθ-(−θ)

]dθ

= −∫ π

0

1

4 sin2(θ/2)[-(θ) + -(−θ) − 2-(0)] dθ, (49b)

where 〈·, ·〉 is the duality bracket between the dual topological vector spaces ofperiodic test functions -(θ) (i.e., infinitely differentiable functions of period 2π )and periodic distributions (of period 2π ). The result (49b) can be extended by


continuity to the function | cos(θ/2)|λ, which indeed can be regarded as the limitof a uniformly convergent sequence of test functions; it is readily verified thatthe values taken on this function by the two forms of the accordingly extendeddistribution in (49b) are simply π times the first two expressions in (47), whichidentity is again rigorously established.


dµQ := dµ + C dµ0, with definitions (8b), (45) and (46), (50)

is the unique (nonnegative and absolutely continuous with respect to θ) equilibriumdistribution of charges over � that minimizes its potential, for any given values ofthe parameters c > 0, λ > 0. But the total mass of (50) over � must be 1, whilethe total mass of (45) is 0, so that necessarily

C = 1, or equivalently, 1/c = 2∞∑k=1

kA2k. (51)

• To rewrite these results in a more familiar form (see [6], pp. 204 and 238), itremains only to change the interval [−1, 1] into

SQ = [−a, a], a > 0, (52a)

via the linear substitution ξ = ax. SQ is the support of the extremal measure µQ

relative to the external potential

γ |ξ |λ, γ > 0, (52b)

if and only if

a = γ −1/λc1/λ, where c :=√π ,(λ/2)

2,(λ/2+1/2), (52c)

as it follows from (51) in view of (47).

References

1. Bergman, S. and Schiffer, M.: Kernel Functions and Elliptic Differential Equations in Mathe-matical Physics, Academic Press, New York, 1953.

2. Gårding, L.: The Dirichlet problem, Math. Intelligencer 2 (1980), 43–53.3. Gel’fand, I. M. and Shilov, G. E.: Generalized Functions, Vol. I, Academic Press, New York,

1964.4. Kellogg, O. D.: Foundations of Potential Theory, Springer, Berlin, 1929.5. Pólya, G. and Szego, G.: Isoparametric Inequalities in Mathematical Physics, Ann. of Math.

Studies 27, Princeton Univ. Press, 1951.6. Saff, E. B. and Totik, V.: Logarithmic Potentials with External Fields, Grundlehren Math. Wiss.

316, Springer-Verlag, Berlin, 1997.7. Saff, E. B., Ullman, J. L. and Varga, R. S.: Incomplete polynomials: an electrostatics approach, In

E. W. Cheney (ed.), Approximation Theory III, Academic Press, San Diego, 1980, pp. 769–782.8. Whittaker, E. T. and Watson, G. N.: A Course of Modern Analysis, 4th edn, Cambridge Univ.

Press, London, 1927.


339

Universality in Orthogonal and SymplecticInvariant Matrix Models with Quartic Potential

ALEXANDRE STOJANOVICUniversité Paris 7, Institut de Mathématiques, Physique-mathématique et géométrie, case 7012,2, place Jussieu, 75251 Paris cedex 05, France

(Received: 10 March 2000)

Abstract. In this work, we develop an orthogonal-polynomials approach for random matrices withorthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in the-oretical physics, which are a generalization of Gaussian random matrices. The representation of thecorrelation functions in these matrix models, via the technique of quaternion determinants, makesuse of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas forGaussian random matrices, which essentially express them in terms of the reproducing kernel of thetheory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of thelocal statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models withtwo-band quartic potential by using the asymptotics given by Bleher and Its for the correspondingorthogonal polynomials.

Mathematics Subject Classifications (2000): 15A52, 42C05, 33C45.

Key words: random matrices, eigenvalues, correlation function, universality conjecture, orthogonalpolynomials.

1. Introduction

In this paper, we consider three types of matrix models defined in the followingterms. Let Enβ be the real vector space, respectively, of real symmetric n × n

matrices for β = 1, complex Hermitian n × n matrices for β = 2, and quaternionreal self-dual n× n matrices for β = 4. The probability measure which defines thematrix models on each space Enβ , β = 1, 2 or 4, is given by

Pnβ(dM) = 1

Znβ

exp(−nTrV (M))dM, M ∈ Enβ, (1.1)

where Znβ is the normalization constant, dM the Lebesgue measure on the con-sidered space Enβ , and V the potential, which is a real polynomial of even degree,denoted by d + 1 � 2, with a positive leading coefficient. With this convention, dis the degree of V ′ the derivative of V . Gaussian random matrices correspond tothe case where the potential V (λ) is proportional to λ2. The trace in the expressionof the probability measure (1.1) shows that the law is invariant under the action by

340 ALEXANDRE STOJANOVIC

conjugation of the orthogonal group for β = 1, the unitary group for β = 2, andthe unitary symplectic group for β = 4. Thus, the eigenvectors are uniformly dis-tributed and we are only interested in the behaviour of the eigenvalues. Accordingto [10, 11], all the eigenvalues (λ1, . . . , λn) of the random matrice (1.1) are realand their probability density with respect to the Lebesgue measure on R

n is givenby

p(n)β (λ1, . . . , λn) = 1

Qnβ

∏1�j<k�n

|λj − λk|βn∏

j=1

exp(−nV (λj )), (1.2)

where Qnβ is the new normalization constant. The k-point correlation functions ofthe probability distribution (1.2) are defined by

p(n)kβ (λ1, . . . , λk)

=∫(λk+1,...,λn)∈Rn−k

p(n)β (λ1, . . . , λk, λk+1, . . . , λn) dλk+1 . . . dλn, (1.3)

with k ∈ {1, . . . , n}. For the case β = 2, the expression of the correlation func-tions via the orthogonal polynomials technique is quite simple to obtain, whereas itcauses trouble in the cases β = 1 or 4. However, in the particular case of Gaussianrandom matrices, F. Dyson and M. Mehta (see [10, 11] and references therein)have developed the technique of quaternion determinants to express the correlationfunctions (1.3) in terms of matrix kernels. The elements of a matrix kernel areexpressed using Hermite polynomials which naturally appear for Gaussian randommatrices.

Let us recall some useful facts for this work, from the theory developed byDyson and Mehta. A complex quaternion q is defined as a linear complex combi-nation of the four 2 × 2 following complex matrices

1 =(

1 00 1

), e1 =

(0 −11 0

),

e2 =(

0 −i−i 0

), e3 =

(i 00 −i

),

where i is a complex number such that i2 = −1. Thus, a quaternion can berepresented as a 2 × 2 complex matrix and we denote

q = q(0) + q(1)e1 + q(2)e2 + q(3)e3, with q(j) ∈ C. (1.4)

If the q(j) are real, the quaternion is real, q(0) is the scalar part, and∑3

j=1 q(j)ej

the purely quaternion part. The dual quaternion of (1.4) is defined by

q = q(0) −3∑

j=1

q(j)ej .

ORTHOGONAL AND SYMPLECTIC INVARIANT MATRIX MODELS 341

Let Qn = (qjk)1�j,k�n be an n × n quaternion matrix, the dual quaternion matrixis defined by

Qn = (q ′jk

)1�j,k�n

, with q ′jk = qkj , for j, k ∈ {1, . . . , n}.

A quaternion matrix is self-dual if Qn = Qn. The quaternion determinant of Qn isdefined by

QdetQn

=∑σ∈Sn

ε(σ )∏

all the cycles (j1 → j2 → · · · jr → j1) of

the decomposition in disjoint cycles of σ

(qj1j2qj2j3...qjr j1)(0), (1.5)

where Sn is the symmetric group of order n, ε(σ ) is the signature of the permutationσ , and the (0) in the high index means that we can take the scalar part. An n × n

quaternion matrix can be viewed as a 2n × 2n complex matrix, denoted C(Qn),and from this notation we have the following result: The quaternion matrix Qn isself-dual if and only if ZC(Qn) is antisymmetric complex, where Z is the 2n× 2nblock-diagonal matrix with n diagonal blocks

(0 1

−1 0

)and, in this case, we have

QdetQn = Pf(ZC(Qn)) and detC(Qn) = (QdetQn)2,

where Pf is the Pfaffian of an antisymmetric complex matrix of even order.Recently, in [16], C. Tracy and H. Widom obtained general formulas for matrix

kernels by using the method of generating functional (see [9] and the appendixA.17 of [11]). Their results can be quite easily reformulated with quaternion deter-minants (see the preprint [13] for the details). Thus, we recall their results adaptedto the matrix models (1.1), we have to distinguish the cases β = 1 and β = 4. Wedefine the function

ε: x �→ ε(x) = 1/2 if x > 0,

0 if x = 0,−1/2 if x < 0.

Let f and g be two complex-valued functions defined on R, the convolution withthe function ε is defined by

(ε � f )(λ) =∫

R

ε(λ− µ)f (µ) dµ

and we have the property∫R

g(ε � f ) = −∫

R

f (ε � g).

We note f ′ for the derivative of f .


The case β = 1.Let (π(n)

! (λ))!∈N be the orthogonal polynomials on R with respect to the weightexp(−2nV (λ)). They satisfy∫

R

π(n)! (λ)π(n)

m (λ)e−2nV (λ) dλ = δ!m, !,m ∈ N,

where δ!m is the Kronecker symbol. Let us denote

ψ!(λ) = π(n)! (λ)e−nV (λ), ! ∈ N, (1.6)

as the associated system of orthonormal functions of L2(R,C) and

Kn(λ,µ) =n−1∑j=0

ψj(λ)ψj(µ), (1.7)

the reproducing kernel (in order to simplify the notations, we do not indicate then-dependence in superscripts). Let us define the coefficients

ajk =∫

R

(ε � ψj)ψk and cjk =∫

R

ψ ′jψk, j, k ∈ N. (1.8)

PROPOSITION 1.1. For even n, consider the n × n matrix A = (ajk)0�j,k�n−1

which is real antisymmetric invertible. Let B = (bjk)0�j,k�n−1 be the inverse of A.Following [16], let us define

sn(λ, µ) =n−1∑j,k=0

bjkψj (λ)(ε � ψk)(µ) and αn(λ) = 0. (1.9)

For odd n, consider the (n+ 1)× (n+ 1) matrix A = (a′jk)0�j,k�n, where

a′jk = ajk if j, k ∈ {0, . . . , n− 1},a′nj = −a′

jn =∫

R

ψj(λ)dλ if j ∈ {0, . . . , n− 1}, (1.10)

a′nn = 0.

Then A is real antisymmetric invertible. Let B = (bjk)0�j,k�n be the inverse of A.Let us define

sn(λ, µ) =n−1∑j,k=0

bjkψj (λ)(ε � ψk)(µ) and αn(λ) =n−1∑j=0

bjnψj (λ). (1.11)

Now, we define the elements of a matrix kernel for any integer n by

Sn(λ, µ) = sn(λ, µ)+ αn(λ), STn (λ, µ) = Sn(µ, λ), (1.12)

Dn(λ,µ) = −∂sn

∂µ(λ,µ) and

Jn(λ, µ) = ελ � Sn(λ, µ)− ε � αn(µ)− ε(λ− µ), (1.13)


where ελ indicates that the convolution by the function ε is made with respect tothe variable λ. The matrix kernel equals to

σn1(λ, µ) =(Sn(λ, µ) Dn(λ,µ)

Jn(λ, µ) STn (λ, µ)

). (1.14)

Then, the correlation functions (1.3) are given by

p(n)

k1 (λ1, . . . , λk)

= (n− k)!n! Qdet(σn1(λp, λq))1�p,q�k, k ∈ {1, . . . , n}. (1.15)

The case β = 4.Let (π(n)

! (λ))!∈N be the orthogonal polynomials on R with respect to the weightexp(−nV (λ)). They satisfy∫

R

π(n)

! (λ)π(n)m (λ)e−nV (λ) dλ = δ!m, !,m ∈ N.

Let us denote

ψ!(λ) = π(n)! (λ)e− nV (λ)

2 , ! ∈ N, (1.16)

as the associated system of orthonormal functions of L2(R,C) and

K2n+d(λ, µ) =2n+d−1∑j=0

ψj(λ)ψj(µ), (1.17)

the reproducing kernel. Let us define the coefficients

ajk =∫

R

ψjψ′k and cjk =

∫R

ψj(ε � ψk), j, k ∈ N. (1.18)

PROPOSITION 1.2. The 2n× 2n matrix A = (ajk)0�j,k�2n−1 is real antisymmet-ric invertible. Let B = (bjk)0�j,k�2n−1 be the inverse of A. Following [16], let usdefine

Sn(λ, µ) =2n−1∑j,k=0

bjkψ′j (λ)ψk(µ), STn (λ, µ) = Sn(µ, λ),

Dn(λ,µ) = −∂Sn

∂µ(λ,µ) and In(λ, µ) = ελ � Sn(λ, µ).

(1.19)

We define the matrix kernel

σn4(λ, µ) =(Sn(λ, µ) Dn(λ,µ)

In(λ, µ) STn (λ, µ)

). (1.20)


Then, the correlation functions (1.3) are given by

p(n)k4 (λ1, . . . , λk)

= (n− k)!n! Qdet(σn4(λp, λq))1�p,q�k, k ∈ {1, . . . , n}. (1.21)

The matrix kernels are interesting when they are expressed in terms of orthogo-nal polynomials and notably in functions of the reproducing kernel, like in the caseof the unitary invariant matrix model (the case β = 2 in (1.2)) or like in the case ofGaussain random matrices, see Chapters 6 and 7 of [11]. Because it allows to usethe asymptotic formulas of orthogonal polynomials, via the Christoffel–Darbouxformula, to study the asymptotic spectral behaviour of the random matrices whenn → ∞. Theorems 2.1 and 2.2 give new formulas for matrix kernels, generalizingthe known formulas for Gaussian random matrices. More precisely, the elementsof the matrix kernels are composed of a principal part, determining the asymptoticbehaviour in different asymptotic regimes, which is simply expressed in termsof the reproducing kernel and a negligible part, which is a finite sum with an n-independent number of terms which we have to estimate. In order to obtain theseformulas, the polynomial aspect of the potential V is very important, because itimplies that the infinite matrix of the derivation operator in the basis of orthonor-mal functions is multi-diagonal with d diagonals along each sides of the principaldiagonal and the method of computation is entirely based on this property. Let usremark that H. Widom [17] has obtained expressions for matrix kernels in terms ofthe reproducing kernel for more general classes of random matrices.

As an application, in Theorems 2.3 and 2.4 we prove, by using the formulasgiven in Theorems 2.1 and 2.2, the universality of the local statistics of eigenvalues,both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given in [2] for the correspondingorthogonal polynomials.

Let us denote ρn(λ) = p(n)

1β (λ) the one-point correlation function, then thedensity of states ρ(λ) is defined by the limit (if it exists in an appropriate topology,which is here the topology of pointwise convergence)

limn→+∞ ρn(λ) = ρ(λ) (1.22)

and the spectrum is the support of the density of states. The study of this quantityis relative to the global regime. Now, we recall the definitions of two quantitiesrelative to the local regime. Let us recall that the probability Rn(5) that an interval5 does not contain any eigenvalue is given by

Rn(5) =n∑

!=0

(−1)!

!!∫5!

Qdet(σnβ(λp, λq))1�p,q�! dλ1 . . . dλ!. (1.23)


The local statistic of eigenvalues in the bulk of the spectrum is defined by the limitof Rn(5), with

5 =(λ0, λ0 + v

nρn(λ0)

),

where λ0 is such that ρ(λ0) �= 0 and v is a positive fixed real. The local statistic ofeigenvalues at the edge of the spectrum is defined by the limit of Rn(5), with

5 =(λ0 − w

cn2/3, λ0 + v

cn2/3

),

where λ0 is a point at the edge of the spectrum and v,w and c are positive fixedreals.

The local statistic of eigenvalues in the bulk of the spectrum for Gaussian ran-dom matrices is given in [9, 11] (and see the references therein). The local statisticof eigenvalues at the edge of the spectrum for Gaussian orthogonal and symplecticinvariant matrix models was first derived in [15]. The universality means that thesequantities do not depend on a particuliar potential V but depend on the groupinvariance of matrix models only. Theorems 2.3 and 2.4 establish such results formatrix models with two-band quartic potential. Another work in which the localstatistics of eigenvalues both in the bulk and at the edge of the spectrum are givenis [7]. Finally, we point out that since the limiting scaled correlation functions(both in the bulk and at the edge of the spectrum) are proved for matrix modelswith two-band quartic potential to be identical with those of Gaussian orthogonaland symplectic invariant matrix models, that such quantities as level-spacing dis-tributions and the largest eigenvalue distributions will therefore also be identical(after proper normalization) with Gaussian random matrices (see again [15]).

2. Results

THEOREM 2.1 (the case β = 1). For even n, let us define, with (1.8),

sjk =n−1∑

!=k−daj!c!k, j, k ∈ {n− d, . . . , n− 1}. (2.1)

The d × d matrix D = (sjk)n−d�j,k�n−1 is invertible. Let D−1 = (tjk)n−d�j,k�n−1

be its inverse. Let us define the coefficients

gjk = −cjk +n−1∑

!=n−d

(cj! −

n−d−1∑m=j−d

cjmsm!

)t!k,

j, k ∈ {n− d, . . . , n− 1}, (2.2)

then the d × d matrix G = (gjk)n−d�j,k�n−1 is antisymmetric.


For odd n, let us define, with (1.8) and (1.10),

sjk =n∑

!=k−da′j!c!k, j, k ∈ {n− d, . . . , n}. (2.3)

The (d + 1) × (d + 1) matrix D = (sjk)n−d�j,k�n is invertible. Let D−1 =(tjk)n−d�j,k�n be its inverse. Let us define the coefficients

gjk = −cjk +n∑

!=n−d

(cj! −

n−d−1∑m=j−d

cjmsm!

)t!k, j, k ∈ {n− d, . . . , n}, (2.4)

then the (d + 1)× (d + 1) matrix G = (gjk)n−d�j,k�n is antisymmetric.Now, for any integer n, the functions (1.9) and (1.11) are given by

sn(λ, µ) = Kn(λ,µ)− εµ � Hn(λ,µ)+ εµ � Gn(λ,µ), (2.5)

where

Gn(λ,µ) =n−1∑

j,k=n−dgjkψj (λ)ψk(µ), (2.6)

Hn(λ,µ) =n+d−1∑k=n

n−1∑j=k−d

cjkψj (λ)ψk(µ). (2.7)

And, for even n, αn(λ) = 0, whereas for odd n, we have

αn(λ) =n−1∑

j=n−d(cjn + gjn)ψj (λ) with cjn + gjn = −

n−1∑!=n−d

cn!t!j . (2.8)

THEOREM 2.2 (the case β = 4). Let us define, with (1.18),

sjk =2n−1∑!=j−d

aj!c!k, j, k ∈ {2n− d, . . . , 2n + d − 1}. (2.9)

The d×d matrixD = (sjk)2n−d�j,k�2n−1 is invertible. LetD−1 = (tjk)2n−d�j,k�2n−1

be its inverse. Let us define the coefficients

gjk = ajk −2n−1∑!=k−d

(sj! +

2n−1∑m,p=2n−d

sjmtmp(δp! − sp!)

)a!k,

j, k ∈ {2n, . . . , 2n + d − 1}, (2.10)

then the d×d matrix G = (gjk)2n�j,k�2n+d−1 is antisymmetric. The function (1.19)is given by

Sn(λ, µ) = K2n+d(λ, µ)− εµ � Hn(λ,µ)+ εµ � Gn(λ,µ), (2.11)


where

Gn(λ,µ) =2n+d−1∑j,k=2n

gjkψj (λ)ψk(µ), (2.12)

Hn(λ,µ) = −2n+2d−1∑k=2n+d

2n+d−1∑j=k−d

ajkψj (λ)ψk(µ). (2.13)

Let us define the functions needed in the following theorems:

s(x) = sinπx

πx, a(x, y) = Ai(x)Ai′(y)− Ai′(x)Ai(y)

x − y

and

b(x, y) = 12Ai(x)(ε � Ai)(y),

where Ai is the Airy function (see [1]).

THEOREM 2.3 (the case β = 1). Let us consider the random matrix with orthog-onal invariant law (1.1), with

V (λ) = tλ2

4+ gλ4

8,

such that g > 0 and t < −2√g (two-band case), then we have the following

results.(i) The density of states (1.22) exists (the limit is taken in the sense of the

pointwise convergence on the real axis) and equals

ρ(λ) = g |λ|2π

√max(0, (Z2

1 − λ2)(λ2 − Z22)),

with

Z1 =(−t − 2

√g

g

)1/2

and Z2 =(−t + 2

√g

g

)1/2

.

(ii) The local statistic of eigenvalues in the bulk of the spectrum is given by

limn→+∞Rn

((λ0, λ0 + v

nρn(λ0)

))=

+∞∑!=0

(−1)!

!!∫ v

0· · ·∫ v

0Qdet(τ1(xp, xq))1�p,q�! dx1 . . . dx!,

with

τ1(x, y) =(

s(x − y) −s′(x − y)∫ x−y0 s(u)du − ε(x − y) s(x − y)

)


and where v is a positive real and λ0 is such that ρ(λ0) �= 0, i.e. λ0 ∈ (−Z2,−Z1)∪(Z1, Z2). More precisely, the limiting scaled correlation functions in the bulk of thespectrum is given by

limn→+∞

1

(nρn(λ0))kQdet

(σn1

(λ0 + xp

nρn(λ0), λ0 + xq

nρn(λ0)

))1�p,q�k

= Qdet(τ1(xp, xq))1�p,q�k

uniformly for (xp)1�p�k in any compact of Rk.

(iii) For the case of the local statistic of eigenvalues at the edge of the spectrum,we take the limit with n even or n odd. Then we have

limn→+∞Rn

((Zj − w

cjn2/3, Zj + v

cjn2/3

))=

+∞∑!=0

(−1)!

!!∫ v

−w· · ·∫ v

−wQdet(θ1(xp, xq))1�p,q�! dx1 . . . dx!,

where

θ1(x, y) = (a + b)(x, y) −∂(a + b)

∂y(x, y)∫ x

y(a + b)(u, y)du − ε(x − y) (a + b)(y, x)

,

if the limit is taken with n even or if the limit is taken with n odd and j = 1.Moreover, v, w are positive reals and cj = (−1)j21/3√gZj .

And if the limit is taken with n odd and j = 2, we have

θ1(x, y)

= (a + b)(x, y) + 1

2 Ai(x) −∂(a + b)

∂y(x, y)∫ x

y((a + b)(u, y) + 1

2Ai(u))du− ε(x − y) (a + b)(y, x) + 12Ai(y)

.

More precisely, the limiting scaled correlation functions at the edge of the spec-trum is given by

limn→+∞

1

(cjn2/3)k

Qdet

(σn1

(Zj + xp

cjn2/3, Zj + xq

cjn2/3

))1�p,q�k

= Qdet(θ1(xp, xq))1�p,q�k


THEOREM 2.4 (the case β = 4). Let us consider the random matrix with sym-plectic invariant law (1.1), with

V (λ) = tλ2

2+ gλ4

4,


such that g > 0 and t < −2√

2g (two-band case), then we have the followingresults.

(i) The density of states (1.22) exists (the limit is taken in the sense of thepointwise convergence on the real axis) and equals

ρ(λ) = g |λ|4π

√max(0, (Z2

1 − λ2)(λ2 − Z22)),

with

Z1 =(−t − 2

√2g

g

)1/2

and Z2 =(−t + 2

√2g

g

)1/2

.

(ii) The local statistic of eigenvalues in the bulk of the spectrum is given by

limn→+∞Rn

((λ0,λ0 + v

nρn(λ0)

))=

+∞∑!=0

(−1)!

!!∫ v

0· · ·∫ v

0Qdet(τ4(xp, xq))1�p,q�! dx1 . . . dx!,

with

τ4(x, y) =(

s(2(x − y)) −s′(2(x − y))∫ 2(x−y)0 s(u)du s(2(x − y))

),

and where v is a positive real and λ0 is such that ρ(λ0) �= 0, i.e. λ0 ∈ (−Z2,−Z1)∪(Z1, Z2).

More precisely, the limiting scaled correlation functions in the bulk of the spec-trum is given by

limn→+∞

1

(nρn(λ0))k

Qdet

(σn4

(λ0 + xp

nρn(λ0), λ0 + xq

nρn(λ0)

))1�p,q�k



(iii) The local statistic of eigenvalues at the edge of the spectrum is given by

limn→+∞Rn

((Zj − w

cjn2/3, Zj + v

cjn2/3

))=

+∞∑!=0

(−1)!

!!∫ v

−w· · ·∫ v

−wQdet(θ4(xp, xq))1�p,q�! dx1 . . . dx!,

where

θ4(x, y) = (a + b)(x, y) −∂(a + b)

∂y(x, y)∫ x

y(a + b)(u, y)du (a + b)(y, x)

,


and where v, w are positive reals and cj = (−1)j√

2gZj .More precisely, the limiting scaled correlation functions at the edge of the

spectrum is given by

limn→+∞

1

(cjn2/3)kQdet

(σn4

(Zj + xp

cjn2/3, Zj + xq

cjn2/3

))1�p,q�k

= Qdet(θ4(xp, xq))1�p,q�k


Remark 2.1. Using the result of the local statistics at the edge of the spectrum,we obtain the pointwise convergence to zero of the density of states at the pointsof the edge of the spectrum.

Remark 2.2. The formulas given by Theorems 2.1 and 2.2 allow us to recoverthe known formulas for Gaussian random matrices. Moreover, in this case, we cancompute exactly the coefficients (1.8), (1.10) and (1.18). In order to recover theformulas as they are given in [11], we have to take another scaling convention.

The case β = 1. For the weight exp(−nV (λ)), we take here exp(− 12λ

2) (sinceV is an homogeneous polynomial, a simple change of variables allows us to passfrom one convention to an other). Here, the Hermite polynomials which appearare orthogonal with respect to the weight exp(−λ2). Then (see [14]) we have therelation

ψ ′p =

√p

2ψp−1 −

√p + 1

2ψp+1. (2.14)

Thus

cp,p+1 = −cp+1,p = −√p + 1

2,

the other coefficients being null. Multiplying (2.14) by ε � ψq and integrating, weget the relation

apq =√p − 1

pap−2,q −

√2

pδp−1,q for p > 0, q � 0. (2.15)

The parity of the weight implies that apq = 0, if p, q have the same parity. Further,by antisymmetry, we can suppose that p < q, then an elementary recurrence donewith (2.15) gives

apq =(p!q! 2q−p

)1/2((q − 1)/2)!(p/2)! ,


if p is even and q is odd, whereas apq = 0, if p is odd and q is even. Now, wecompute the coefficients a′

jn for even j , else they are null. Recall that (see [14]) theHermite polynomials are given by

πj(λ) = 1√√π2j j !

[j/2]∑!=0

(−1)!j !!!(j − 2!)!(2λ)

j−2!,

where [x] is the integer part of the real x. Moreover, we have∫R

λ2pe−λ2/2 dλ = √2π

(2p)!2pp! .

Thus, with even j , we get

a′nj =

∫R

ψj(λ) dλ =√

2π1/4√(2p)!2pp! .

Now, we can recover the known formulas. Since d = 1 and the coefficients gjk areantisymmetric, we have for the functions (2.6) and (2.7) that

Gn(λ,µ) = gn−1,n−1ψn−1(λ)ψn−1(µ) = 0

and

Hn(λ,µ) = −√n

2ψn−1(λ)(ε � ψn)(µ).

Further, for n odd, we compute the function (2.8), and we have

αn(λ) = (cn−1,n + gn−1,n)ψn−1(λ)

= −cn,n−1tn−1,n−1ψn−1(λ)

and, using (2.3), we find

tn−1,n−1 = 1

sn−1,n−1= 1

a′n−1,ncn,n−1

,

because the computation of the coefficients shows that an−1,n−2 = 0. Therefore,we get

αn(λ) = − 1

a′n−1,n

ψn−1(λ) = 1∫Rψn−1

ψn−1(λ)

and, since n− 1 is even, we verify that

(ε � ψn−1)(λ) =∫ λ

0ψn−1.

Thus, we recover exactly the formulas given in [11]. Now, we compute the valueof the term αn(λ) in the limit of the local statistics at the edge of the spectrum. We


have to rescale the weight in order to have the Hermite polynomials with respect toexp(−nλ2). Then, taking this new scaling into acount, the value of a′

nj computedabove gives∫

R

ψn−1 ∼ 23/4

√n, n → +∞,

and, by using the results of [14], we get

limn→+∞

1√2n2/3

αn

(√2 + 1√

2n2/3

)= 1

2Ai(x).

This is exactly the result we get in the case of the quartic potential at the point Z2.The nullity of this term at the point Z1 is a direct consequence of the presence ofthe sign (−1)[

n+s2 ] in the asymptotic formula of ψn+s , which reflects the two-band

structure of the spectrum.

The case β = 4. For the weight exp(−nV (λ)), we take here exp(−2λ2) and wehave the Hermite polynomials with respect to the weight exp(−2λ2). Adapting, thecomputations of the case β = 1, we get

Sn(λ, µ) = K2n+1(λ, µ)+ √2n+ 1ψ2n(λ)(ε � ψ2n+1)(µ).

Remark 2.3. In this remark, we discuss the general behaviour of the coefficients(2.2), (2.4) and (2.10) and the order of the different terms in (2.5) and (2.11), whenn → +∞.

The case β = 1, even n. In this case (and for more general classes of orthogo-nal polynomials, see e.g. [12]) we know that for the coefficients (1.8), we havecn+p,n+q = O(n), when n → +∞, for fixed p, q in Z. Further, we suppose (thisneeds a proof) for the coefficients (1.8) that an+p,n+q = O(1/n). We deduce that,for the coefficients (2.1) of the matrix D, we have sjk = O(1). Hence, we supposeat the very outset that detD ∼ c, with c > 0. Therefore, we have tjk = O(1). Underthese hypotheses, we can conclude, for the coefficents (2.2), that gjk = O(n).Moreover, if we suppose that the orthonormal functions satisfy

‖ψn+p‖L∞(R) = O(n1/6) and ‖ε � ψn+p‖L∞(R) = O(1/√n),

we deduce that the terms (2.7) ε � Hn(λ,µ) and (2.6) ε � Gn(λ,µ) are O(n2/3)

uniformly in λ,µ, whereas the reproducing kernel Kn(λ,µ) is of the order n, whichis dominant with respect to O(n2/3).

The case β = 1, odd n. In this case, we suppose, moreover for the coefficients(1.10), that a′

n+p,n = O(1/√n). Thus, we deduce for the coefficients (2.3) that

sjk = O(√n). The behaviour of the coefficients tjk is more complicated. We begin

by studying the behaviour of detD. Let us define the line vectors

Lj = (σjk)n−d�k�n if j ∈ {n− d, . . . , n},


with

σjk =n−1∑

!=k−daj!c!k = O(1) if j ∈ {n− d, . . . , n− 1}

and σnk =n−1∑

!=k−da′n!c!k = O(

√n).

Moreover, we define

C = (cnk)n−d�k�n.

Then, the matrix D can be rewritten as

D = (Lj + a′jnC)n−d�j�n.

A multilinear development gives

detD = det(Lj )n−d�j�n +n−1∑

j=n−da′jn det

...

Lj−1

C

Lj+1...

= O(n),

because Lj = O(1) if j ∈ {n− d, . . . , n− 1}, Ln = O(√n) and C = O(n). Thus,

we suppose that detD ∼ cn, with c > 0. Under this supplementary hypothesis, theexpressions given by Theorem 2.1 allow us to show that we have gjk = O(n), ifj, k < n and gjn = −gnj = −cjn + O(

√n), for the coefficients (2.4). Indeed, we

have

tjk = (−1)j+k

detDdkj with dkj = det(spq)n−d�p,q�n, p �=k, q �=j .

Let us define the line vectors

L(j)p = (σpq)n−d�q�n, q �=j if j, p ∈ {n− d, . . . , n}

and

C(j) = (cnq)n−d�q�n, q �=j if j ∈ {n− d, . . . , n}.Then, for k ∈ {n− d, . . . , n− 1}, a multilinear development gives

dkj = det(L(j)p

)n−d�p�n, p �=k +

+n−1∑

p=n−d, p �=ka′pn det

((1 − δsp)L

(j)s + δspC

(j))n−d�s�n, s �=k

= O(n) (2.16)


and

dnj = det(L(j)p

)n−d�p�n−1 +

+n−1∑

p=n−da′pn det

((1 − δsp)L

(j)s + δspC

(j))n−d�s�n−1 = O(

√n).

Hence,

tjk = O(1) if j ∈ {n− d, . . . , n}, k ∈ {n− d, . . . , n− 1}and tjn = O(1/

√n).

Thus, we just have to proven−1∑

!=n−dsm!t!k =

{O(1) if k ∈ {n− d, . . . , n− 1},O(1/

√n) if k = n,

with m ∈ {n−2d, . . . , n−d−1}. Moreover, sm! = a′mncn!+O(1), because m < n.


!=n−dcn!t!k =

{O(

√n) if k ∈ {n− d, . . . , n− 1},

O(1) if k = n.

In fact, (2.8) shows that∑n−1

!=n−d cn!t!n = −(gnn + cnn) = 0. For k < n, accordingto (2.16), we have

t!k = (−1)!+k

detD

(O(

√n)+

n−1∑p=n−d, p �=k

a′pn det

((1 − δsp)L

(!)s +

+ δspC(!))n−d�s�n, s �=k

).


!=n−dcn!(−1)!+k det

((1 − δsp)L

(!)s + δspC

(!))n−d�s�n, s �=k = 0,

but this is the development with respect to the line indexed by k �= p of

det((1 − δsp)(1 − δsk)Ls + (δsp + δsk)C)n−d�s�n

which is null, because the line C appears twofold. Hence, we have the result onthe coefficients gjk . Therefore, for the function (2.8), we have αn(λ) = O(n2/3),uniformly in λ and we conclude as in the case of even n.

The case β = 4. This case is similar to the first case, we just have to exchange therole of the coefficients ajk and cjk given in (1.18).

Remark 2.4. We see that the main problem in the estimation of the coefficientsgjk is the division by detD, because it obliges us to compute the equivalent and


not just an estimate. In order to do this in the case of the quartic potential, wecompute the exact form of equivalents of the coefficients (1.8), (1.10) and (1.18),the explicit form of detD in terms of these coefficients, and then we can computethe equivalent of detD. But, despite the parity of the potential V and its low de-gree, the computations are long and fastidious. And, although there exist similarasymptotic formulas for orthogonal polynomials corresponding to general polyno-mial potential (see [5]), it seems impossible to use the same method to prove thebehaviour of detD. That is why we propose the following hypothetic way to findthe asymptotic behaviour of detD. We should use something like the variationalapproach, see [4, 8], because we have the following expression of detD in termsof normalization constants. Let us denote

Qr,β(w) =∫

· · ·∫

Rr

∏1�j<k�r

|λj − λk|βr∏

j=1

w(λj) dλj ,

where, w is a positive function of one real variable. Then, for the case β = 1, evenn, we have

detD =(Qn

2 ,4(e−2nV )Qn,1(e−nV )

2n(n/2)!Qn,2(e−2nV )

)2

,

for the case β = 1, odd n,

detD =(n+ 1)

(Qn+1

2 ,4(e−2nV )Qn,1(e−nV )

)2

22n(((n+ 1)/2)!)2Qn,2(e−2nV )Qn+1,2(e−2nV ),

and for the case β = 4

detD =(Qn,4(e−nV )Q2n,1(e−nV/2)

22nn!Q2n,2(e−nV )

)2

.

Results from [8] show that the first term of the asymptotic developement of thenormalization constant is en

2×constant. It is not sufficient here, thus we need for otherterms.

3. Proofs of the Results

We give only the proof of Theorem 2.1 for even n; the cases β = 1, odd n, andβ = 4 are based on the same principles but they are technically more complicated(see [13] for details). The proof of Theorem 2.4 is similar to the proof of Theorem2.3.

Proof of Theorem 2.1. We suppose n is even.

First step: By using the orthogonality property and the polynomial aspect of V , wefind the relations

ψ ′j =

j+d∑k=j−d,k�0

cjkψk, j ∈ N, (3.1)


and

nV ′ψj =j−1∑

k=j−d,k�0

cjkψk −j+d∑

k=j+1

cjkψk, j ∈ N. (3.2)

Moreover, starting from the relation δjk = ∫Rψjψk, integrating by parts, and using

(3.1), we get the identity

δjk =k+d∑

!=k−d,!�0

aj!c!k, j, k ∈ N. (3.3)

Second step: We compute the matrix product AC in terms of coefficients (2.1) byusing (3.3). In terms of block matrices, we have

AC =(In−d E

0 D

)(3.4)

with

D = (s!k)n−d�!,k�n−1 and E = (s!k)0�!�n−d−1,n−d�k�n−1. (3.5)

Third step: We compute (AC)−1. According to [16], we know that A and C areinvertible, hence the matrix D (3.5) is invertible too. A block-matrix computationusing (3.4) gives us

(AC)−1 = In +(

0 −ED−1

0 D−1 − Id

)with D−1 − Id = (t!k − δ!k)n−d�k,!�n−1 . (3.6)

The coefficients are equal to

t!k − δ!k =n−1∑

m=n−d(δ!m − s!m)tmk, !, k ∈ {n− d, . . . , n− 1}.

Thus, (3.6) becomes

(AC)−1 = In + (0 F)

with F =(

n−1∑m=n−d

(δ!m − s!m)tmk

)0�!�n−1,n−d�k�n−1

(3.7)

Fourth step: We compute B = A−1. We use B = C(AC)−1 and (3.7). Since B andC are antisymmetric, we get

B = C +(

0n−d 00 G

)with G = (gjk)n−d�j,k�n−1 (3.8)


and

gjk =n−1∑m=0

cjm

(n−1∑

!=n−d(δm! − sm!)t!k

). (3.9)

Fifth step: We simplify the expression of the coefficients (3.9) to get (2.2). We have

gjk =n−1∑

m=j−dcjm

(n−1∑


)

=n−d−1∑m=j−d

cjm

(n−1∑


)+

n−1∑!,m=n−d

cjm(δm! − sm!)t!k

=n−d−1∑m=j−d

cjm

(−

n−1∑!=n−d

sm!t!k

)+

n−1∑m=n−d

cjm(tmk − δmk)

= −cjk +n−1∑

!=n−d

(cj! −

n−d−1∑m=j−d

cjmsm!

)t!k.

Sixth step: We compute Dn(λ,µ). (1.9), (1.13) and (3.8) give us

Dn(λ,µ) = −n−1∑j,k=0

cjkψj (λ)ψk(µ)−n−1∑

j,k=n−dgjkψj (λ)ψk(µ). (3.10)

Let us denote

kn(λ, µ) =n−1∑j=0

π(n)j (λ)π

(n)j (µ).

We have, with (1.7),

−n−1∑j,k=0

cjkψj (λ)ψk(µ)

= −1

2

∫ (∂Kn

∂ν(λ, ν)Kn(ν, µ)−Kn(λ, ν)

∂Kn

∂ν(ν, µ)

)dν

= −1

2

∫R

(∂kn

∂ν(λ, ν)kn(ν, µ)− kn(λ, ν)

∂kn

∂ν(ν, µ)

)e−n(V (λ)+V (µ)+2V (ν)) dν

= 1

2

(∂kn

∂λ(λ,µ)− ∂kn

∂µ(λ,µ)

)e−n(V (λ)+V (µ)),

because, by using the orthognality property, we get∫R

π(n)′j (ν)kn(ν, µ)e

−2nV (ν) dν = π(n)′j (µ), for j ∈ {0, . . . , n− 1}.


Then (3.10), with (2.6), becomes

Dn(λ,µ) = 1

2

(∂kn


∂µ(λ,µ)

)e−n(V (λ)+V (µ)) −Gn(λ,µ)

and we get the result with (2.7) because

1

2

(∂kn


∂µ(λ,µ)

)e−n(V (λ)+V (µ))

= −1

2

(∂Kn

∂λ(λ,µ)− ∂Kn

∂µ(λ,µ)

)+ 1

2(nV ′(λ)− nV ′(µ))Kn(λ,µ)

= −∂Kn

∂µ(λ,µ)+ 1

2

(∂Kn

∂λ(λ,µ)+ ∂Kn

∂µ(λ,µ)

)e−n(V (λ)+V (µ)) +

+ 1

2(nV ′(λ)− nV ′(µ))Kn(λ,µ)

= −∂Kn

∂µ(λ,µ)+Hn(λ,µ).

Since, by using (3.1), we have

∂Kn

∂λ(λ,µ)+ ∂Kn

∂µ(λ,µ)

= −n+d−1∑j=n

n−1∑k=j−d

cjkψj (λ)ψk(µ)+n+d−1∑k=n

n−1∑j=k−d

cjkψj (λ)ψk(µ)

and, by using (3.2), we have

(nV ′(λ)− nV ′(µ))Kn(λ,µ)

=n+d−1∑j=n

n−1∑k=j−d

cjkψj (λ)ψk(µ)+n+d−1∑k=n

n−1∑j=k−d

cjkψj (λ)ψk(µ).

Proof of Theorem 2.3. Let us denote, with (2.6) and (2.7)

Mn(λ,µ) = Gn(λ,µ)−Hn(λ,µ). (3.11)

Thus, according to Lemmas 4.1 and 4.3, (3.11) has the form (here, in fact, we haved = 3)

Mn(λ,µ) =d−1∑

p,q=−dm(n)pqψn+p(λ)ψn+q(µ), with m(n)

pq = O(n). (3.12)

And, for odd n, with (4.8), for (2.8), we have

αn(λ) =−1∑

p=−dbn+p,nψn+p(λ), with bn+p,n = O(

√n). (3.13)


(i) First, we show the existence of the density of states. According to (1.15) and(2.5), we have

ρn(λ) = 1

nSn(λ, λ) = 1

nKn(λ, λ)+ 1

n(εµ � Mn)(λ, λ)+ 1

nαn(λ). (3.14)

Moreover, with (4.3), (3.12) and (3.13), we have the estimates

1

n|(εµ � Mn)(λ, µ)|

�d∑

p,q=−d

1

n

∣∣m(n)pq

∣∣‖ψn+p‖L∞(R)‖ε � ψn+q‖L∞(R) = O(n−1/3) (3.15)

and

1

n|αn(λ)| �

−1∑p=−d

1

n|bn+p,n|‖ψn+p‖L∞(R) = O(n−1/3). (3.16)

Thus, according to (3.14), (3.15) and (3.16), the result follows from the limit

limn→+∞ ρn(λ) = lim

n→+∞1

nKn(λ, λ),

which is proved in [2].(ii) Secondly, we compute the limits of the elements of the matrix kernel in the

local limit inside the spectrum. Let λ0 be such that ρ(λ0) �= 0, let us denote

xn = x

nρn(λ0)and yn = y

nρn(λ0).

Therefore, for each λ0, there exists a positive real δ such that, if x and y arecompact, then λ0 + xn and λ0 + yn are in the part

P = (−Z2 + δ,−Z1 − δ) ∪ (Z1 + δ, Z2 − δ).

The estimates are given uniformly with respect to P and by Lemma 4.1, we knowthat ‖ψn+s‖L∞(P ) = O(1). According to [2], we know that

limn→+∞

1

nρn(λ0)Kn(λ0 + xn, λ0 + yn) = s(x − y), (3.17)

uniformly with respect to (x, y) belonging to a compact of R2. Therefore, accord-

ing to (3.15), (3.16) and (2.5), we get

limn→+∞

1

nρn(λ0)Sn(λ0 + xn, λ0 + yn) = s(x − y).


The results of [2] allow us to derive (3.17) and thus we get

limn→+∞

−1

(nρn(λ0))2

∂Kn

∂µ(λ0 + xn, λ0 + yn)

= limn→+∞

−1

nρn(λ0)

∂Kn

∂y(λ0 + xn, λ0 + yn)

= −s′(x − y).

Since, with (4.3) and (3.12), we have

1

(nρn(λ0))2Mn(λ0 + xn, λ0 + yn) = O(1/n),

we finally obtain

limn→+∞

1

(nρn(λ0))2Dn(λ0 + xn, λ0 + yn) = −s′(x − y).

Now, we show that

limn→+∞ Jn(λ0 + xn, λ0 + yn) =

∫ x−y

0s(u) du − ε(x − y). (3.18)

In order to do this, we use the antisymmetry, which gives Jn(λ, λ) = 0 for allλ ∈ R. Thus, we compute the limit of

Jn(λ0 + xn, λ0 + yn)− Jn(λ0 + yn, λ0 + yn).

Let f be a function and we have the property

(ελ � f )(λ) =∫ λ

−∞f (µ) dµ − 1

2

∫ +∞

−∞f (µ) dµ.

Thus, we have

(ελ � Kn)(λ0 + xn, λ0 + yn)− (ελ � Kn)(λ0 + yn, λ0 + yn)

=∫ λ0+xn

−∞Kn(ν, λ0 + yn) dν −

∫ λ0+yn

−∞Kn(ν, λ0 + yn) dν

=∫ λ0+xn

λ0+ynKn(ν, λ0 + yn) dν

= 1

nρn(λ0)

∫ x

y

Kn

(λ0 + u

nρn(λ0), λ0 + y

nρn(λ0)

)du.

Hence,

limn→+∞((ελ � Kn)(λ0 + xn, λ0 + yn)− (ελ � Kn)(λ0 + yn, λ0 + yn))

=∫ x

y

s(u− y) du =∫ x−y

0s(u) du.


Moreover, we have

ε(λ0 + xn − λ0 − yn) = ε(x − y).

Finally, with (4.3) and (3.12), we get

|ελ � (εµ � Mn)(λ0 + xn, λ0 + yn)− ελ � (εµ � Mn)(λ0 + yn, λ0 + yn)|

�d∑

p,q=−d

∣∣m(n)pq

∣∣|ε � ψn+p(λ0 + xn)−

− ε � ψn+p(λ0 + yn)||(ε � ψn+q)(λ0 + yn)|

�d∑

p,q=−dO(

√n)|ε � ψn+p(λ0 + xn)− ε � ψn+p(λ0 + yn)| = O(1/

√n)

because

|ε � ψn+p(λ0 + xn)− ε � ψn+p(λ0 + yn)| =∣∣∣∣∫ λ0+xn

λ0+ynψn+p

∣∣∣∣ = O(1/n),

and, similarly, we have

|(ε � αn)(λ0 + xn)− (ε � αn)(λ0 + yn)| = O(1/√n),

hence we get (3.18).(iii) We see that the matrix kernel has a limit with an inhomogeneous normal-

ization. Nevertheless, we have only to show that

limn→+∞

1

(nρn(λ0))kQdet

(σn1

(λ0 + xp

nρn(λ0), λ0 + xq

nρn(λ0)

))1�p,q�k


uniformly with respect to (xp)1�p�k belonging to any compact of Rk. We remark

that the antidiagonal terms of the matrix kernel in (1.5) are always multiplied bythemselves in equal numbers. This is only a simple combinatorial result on thealgebraic expression (1.5) of the quaternion determinant.

(iv) Finally, we just have to prove that we can pass to the limit in the expression

Rn

((λ0, λ0 + s

nρn(λ0)

))=

n∑!=0

(−1)!

!!∫ s

0· · ·∫ s

0Qdet

(σn1

(λ0 + xp

nρn(λ0), λ0 +

+ xq

nρn(λ0)

))1�p,q�!

dx1 . . . dx!,


where

σn1

(λ0 + xp

nρn(λ0), λ0 + xq

nρn(λ0)

)is the matrix kernel with its inhomogeneous normalization and where the derivationand integration are made with respect to the variables xp. Let us denote

d(n)! = (−1)!

!!∫ s

0· · ·∫ s

0Qdet

(σn1

(λ0 + xp

nρn(λ0), λ0 + xq

nρn(λ0)

))1�p,q�!

×× dx1 . . . dx!,

if 0 � ! � n and d(n)! = 0 if ! > n. Then, we have

Rn

((λ0, λ0 + s

nρn(λ0)

))=

+∞∑!=0

d(n)! .

Further, for all ! ∈ N, we have

limn→+∞ d

(n)! = d!,

where

d! = (−1)!

!!∫ s

0· · ·∫ s

0Qdet(τ1(xp, xq))1�p,q�! dx1 . . . dx!.

Moreover, since the convergence of

limn→+∞ σn1

(λ0 + x

nρn(λ0), λ0 + y

nρn(λ0)

)= τ1(x, y)

is uniform on (0, s)2 and that the elements of τ1(x, y) are bounded on (0, s), wededuce that the elements of

σn1

(λ0 + x

nρn(λ0), λ0 + y

nρn(λ0)

)are bounded by a constant C > 0. Thus, by using the Hadamard inequality, we get∣∣∣∣Qdet

(σn1

(λ0 + xp

nρn(λ0), λ0 + xq

nρn(λ0)

))1�p,q�!

∣∣∣∣ � (2!C2)!/2,

hence∣∣d(n)!

∣∣ � s!

!! (2!C2)!/2.

The right-hand side in the inequality is the general term of a convergent series,therefore the theorem of dominated convergence gives the result.


(v) For the case at the edge of the spectrum, we proceed essentially as inside thespectrum. By the result given in [2], we know that

limn→∞

1

cjn2/3Kn

(Zj + x

cjn2/3, Zj + y

cjn2/3

)= a(x, y).

First, we will show that the complementary terms in the expression of the matrixkernel, which are expressed in terms of Gn(λ,µ), are negligible. But here thesituation is more complicated, because the norm of uniform convergence of ψn+son a neighbourhood of the edge of the spectrum is only O(n1/6) (see Lemma 4.1).We use the antisymmetry of the coefficients gjk. We have

Gn(λ,µ) = gn−3,n−2(ψn−3(λ)ψn−2(µ)− ψn−2(λ)ψn−3(µ))++ gn−2,n−1(ψn−2(λ)ψn−1(µ)− ψn−1(λ)ψn−2(µ)).

In order to get

‖Gn‖L∞((Zj−δ,Zj+δ)2) = O(n7/6),

since gjk = O(n), it suffices to prove that

‖ψn+p ⊗ ψn+q − ψn+q ⊗ ψn+p‖L∞((Zj−δ,Zj+δ)2) = O(n1/6). (3.19)

Moreover, from the analysis of the asymptotic formulas done in [13], with δ apositive real, we have, uniformly in λ ∈ (Zj − δ, Zj + δ),

ψn+s(λ) = (−1)σ0n1/6√gλ√|C′(λ)|Ai(n2/3C(λ))+ O(1), (3.20)

where σ0 = (2 − j)[(n+ s)/2] and

C(λ) = (−1)j (λ− Zj )

(∫ 1

0

√yf (Zj + (λ− Zj)y) dy

)2/3

,

with

f (x) = 3

4gx

√(−1)k(x + Z)(x2 − Z2

k ).

The fact that the principal part in (3.20) does not depend on s, except for a signwhich can be factorized, gives the result (3.19). In conclusion, we have

1

(cjn2/3)2

‖Gn(λ,µ)‖L∞((Zj−δ,Zj+δ)2) = O(n−1/6).

The same arguments allow us to prove that

1

cjn2/3

‖εµ � Gn(λ,µ)‖L∞((Zj−δ,Zj+δ)2) = O(n−1/6)


and

‖ελ � εµ � (Gn(λ,µ)−Gn(µ,µ))‖L∞((Zj−δ,Zj+δ)2) = O(n−1/3).

Now, we will study the contributions of the complementary terms in the expressionof the matrix kernel, which are expressed in terms of Hn(λ,µ). From the proof ofTheorem 2.1 we see that Hn(λ,µ) = 1

2(αn(λ, µ)+ βn(λ,µ)), where

αn(λ,µ) = ∂Kn

∂λ(λ,µ)+ ∂Kn

∂µ(λ,µ)

= −n+d−1∑j=n

n−1∑k=j−d

cjk(ψj (λ)ψk(µ)+ ψk(λ)ψj(µ))

and

βn(λ,µ) = (nV ′(λ)− nV ′(µ))Kn(λ,µ)

=n+d−1∑j=n

n−1∑k=j−d

cjk(ψj (λ)ψk(µ)− ψk(λ)ψj(µ)).

Thus, we see that βn(λ,µ) has an antisymmetric form similar to the form ofGn(λ,µ), so the same arguments prove that βn(λ,µ) is negligible. Therefore, acontribution is only given by αn(λ,µ). First, we compute this contribution to theterm Dn(λ,µ) in the matrix kernel. We have

limn→+∞

1

(cjn2/3)2αn

(Zj + x

cjn2/3, Zj + y

cjn2/3

)= lim

n→+∞1

cjn2/3

(∂Kn

∂x

(Zj + x

cjn2/3, Zj + y

cjn2/3

)+

+ ∂Kn

∂y

(Zj + x

cjn2/3, Zj + y

cjn2/3

))= ∂a

∂x(x, y) + ∂a

∂y(x, y).

By using the relation Ai′′(x) = xAi(x), the value of the limit becomes

∂a

∂x(x, y) + ∂a

∂y(x, y) = −Ai(x)Ai(y).

In conclusion, we obtain

limn→+∞

1

(cjn2/3)2Dn

(Zj + x

cjn2/3, Zj + y

cjn2/3

)= −∂a

∂y(x, y) − 1

2Ai(x)Ai(y).


Now, we compute the contribution to the term Sn(λ, µ) in the matrix kernel. Wehave

εµ � αn(λ, µ) = 1

2

(∫ µ

−∞αn(λ, ν) dν −

∫ +∞

µ

αn(λ, ν) dν

),

where

λ = Zj + x

cjn2/3and µ = Zj + y

cjn2/3.

When ν is outside a neighbourhood of the edge of the spectrum, this means that νis in the complementary P of the part

[−Z2 − δ,−Z2 + δ] ∪ [−Z1 − δ,−Z1 + δ] ∪ [Z1 − δ, Z1 + δ]∪[Z2 − δ, Z2 + δ],

then, using the fact that∫Pψn+s = O(n−1) (see the proof of Lemma 4.2 of [13])

and Lemma 4.1, the form of αn(λ,µ) shows that the restriction of the integralabove to the part P is estimated by O(n) × O(n1/6) × O(n−1) = O(n1/6). Now,we will show that when ν is in a neighbourhood of a point B of the edge of thespectrum different of Zj , this means that B ∈ {−Z2,−Z1, Z1, Z2}−{Zj}, then theabove integral restricted to such a neighbourhood is negligible. We have to estimate∫ B+δ

B−δ

(∂Kn

∂λ

(Zj + x

cjn2/3, ν

)+ ∂Kn

∂ν

(Zj + x

cjn2/3, ν

))dν

=∫ B+δ

B−δ∂Kn

∂λ

(Zj + x

cjn2/3, ν

)dν+

+Kn

(Zj + x

cjn2/3, B + δ

)−Kn

(Zj + x

cjn2/3, B − δ

).

By using the Christoffel–Darboux formula, we see that the absolute value of thedenominator is bounded from below by |B − Zj |/2 > 0 (for δ small enough andn sufficiently large). It is this fact which allows us to estimate correctly the abovequantity. Thus, we get

Kn

(Zj + x

cjn2/3, B ± δ

)

= √Rn+1

ψn+1(Zj + x

cj n2/3

)ψn(B ± δ)− ψn

(Zj + x

cj n2/3

)ψn+1(B ± δ)

Zj + x

cj n2/3 − (B ± δ)

= O(1)O(n1/6)O(n1/6) = O(n1/3),

where√Rn+1 = O(1) is a coefficient of the three-terms recurrence relation of

orthogonal polynomials (see [13] for details) and, in the same way,∫ B+δ

B−δ

√Rn+1

ψn+1(Zj + x

cj n2/3

)ψn(ν)− ψn

(Zj + x

cj n2/3

)ψn+1(ν)(

Zj + x

cj n2/3 − ν

)2 dν = O(n1/3).


We also have∫ B+δ

B−δ

√Rn+1

ψ ′n+1

(Zj + x

cj n2/3

)ψn(ν)− ψ ′

n

(Zj + x

cj n2/3

)ψn+1(ν)

Zj + x

cj n2/3 − νdν = O(n1/3),

because

ψ ′n+s

(Zj + x

cjn2/3

)= (−1)σ0

n5/6√gZj√|C′(Zj )|Ai′(x)+ O(n1/6) = O(n5/6)

and ∫ B+δ

B−δψn+s(ν)

Zj + x

cj n2/3 − νdν = O(n−1/2),

since the bounded function in front of ψn+s , which is O(1), does not change theestimate in the proof of Lemma 4.2 of [13]. Therefore, we have

εµ � αn

(Zj + x

cjn2/3, Zj + y

cjn2/3

)= 1

2

(∫ Zj+ y

cj n2/3

Zj−δ−∫ Zj+δ

Zj+ y

cj n2/3

)×

×(∂Kn

∂λ

(Zj + x

cjn2/3, ν

)+ ∂Kn

∂ν

(Zj + x

cjn2/3, ν

))dν + O(n1/3)

= 1

2

(∫ y

−δcj n2/3−∫ +δcj n2/3

y

)×(∂Kn

∂x

(Zj + x

cjn2/3, Zj + t

cjn2/3

)+

+ ∂Kn

∂t

(Zj + x

cjn2/3, Zj + t

cjn2/3

))dt + O(n1/3)

hence, dividing by cjn2/3 and taking the limit, we get

limn→+∞

1

cjn2/3εµ � αn

(Zj + x

cjn2/3, Zj + y

cjn2/3

)= 1

2

(∫ y

−∞−∫ +∞

y

)(−Ai(x)Ai(t)) dt

= −Ai(x)(ε � Ai)(y).

To conclude, we obtain

limn→+∞

1

cjn2/3Sn

(Zj + x

cjn2/3, Zj + y

cjn2/3

)= a(x, y) + 1

2Ai(x)(ε � Ai)(y).


By similar arguments, we compute

limn→+∞

(In

(Zj + x

cjn2/3, Zj + y

cjn2/3

)− In

(Zj + y

cjn2/3, Zj + y

cjn2/3

))=∫ x

y

a(s, y) ds + 1

2

∫ x

y

Ai(s) ds(ε � Ai)(y).

For the case of odd n, we have to compute the contribution of the term αn(λ). Asin the case of Gaussian random matrices, this term contributes. We have

αn(λ) = (cn−3,n + gn−3,n)ψn−3(λ)+ (cn−1,n + gn−1,n)ψn−1(λ)

and the computations give

cn−3,n + gn−3,n = − 1

αn(cn,n−3(1 − an−1,n+2cn+2,n−1)),

cn−1,n + gn−1,n = − 1

αn(cn,n−1 + cn,n−3an−3,n+2cn+2,n−1),

where αn is given in the proof of Lemma 4.3. Hence, we get (see [13] for thedetails)

cn−3,n + gn−3,n = √n(−t)1/4 (1 + u)1/4

2√

2+ O(n1/6)

and

cn−1,n + gn−1,n = √n(−t)1/4 (1 + u)1/4

2√

2+ O(n1/6).

Moreover, we have

ψn+s(λ) = (−1)σ0n1/6√gλ√|C′(λ)|Ai(n2/3C(λ))+ O(1),

where

σ0 = (2 − j)

[n+ s

2

], C(Zj) = 0 and C′(Zj ) = cj .

Thus, we get

1

cjn2/3ψn+s

(Zj + x

cjn2/3

)= (−1)σ0

√n

√gZj

cj√|cj |

Ai

(n2/3

(C′(Zj )

cjn2/3

x + O(n−4/3)

))+ O(n−2/3)

= (−1)σ0

√2n(−t)1/4(1 + (−1)ju)1/4

Ai(x)+ O(n−2/3).


At the point Z1, we conclude that

1

c1n2/3αn

(Z1 + x

c1n2/3

)= 1

4((−1)[

n−32 ] + (−1)[

n−12 ])

(1 + u

1 − u

)1/4

Ai(x)+ O(n−1/6) = O(n−1/6)

and, at the point Z2, we have

1

c2n2/3αn

(Z2 + x

c2n2/3

)= 1

2Ai(x)+ O(n−1/6).

Moreover, we have

(ε � αn)

(Zj + x

cjn2/3

)− (ε � αn)

(Zj + y

cjn2/3

)= 1

cjn2/3

∫ x

y

αn

(Zj + s

cjn2/3

)ds,

which allows us to conclude because the convergence of the integrand is uniformin the variable s belonging to any compact.

4. Auxiliary Results

In this part, we give the results on the asymptotics behaviour of the different coef-ficients which allow us to prove Theorems 2.3 and 2.4 on the asymptotic regime ofthe matrix model with two-band quartic potential. The proofs of Lemmas 4.1 and4.2 are based on the asymptotic formulas of [2] for orthonormal functions, then weuse the standard results on asymptotics expansions of integrals of [3] and classicalresults on the Airy function given in [1] (see [13] for details).

LEMMA 4.1 (the case β = 1). We define the parameter u = −2√g/t ∈ ]0, 1[.

For n → +∞ and p fixed in Z, we have

cn+p,n+p−1 = n

4

√−t(√1 + u+ (−1)n+p√

1 − u)+ O(1), (4.1)

cn+p,n+p−3 = n

4

√−t(√1 + u− (−1)n+p√

1 − u)+ O(1). (4.2)

Moreover, we have

‖ψn+p‖L∞(J ) = O(n1/6), ‖ψn+p‖L∞(J c) = O(1)

and ‖ε � ψn+p‖L∞(R) = O(n−1/2), (4.3)

where

J = (Z1 − δ, Z1 + δ) ∪ (Z2 − δ, Z2 + δ),


δ is a positive real, and J c is the complementary of J in R. For p fixed in Z, suchthat n+ p is even (else a′

n,n+p = 0), we have, when n → +∞,

a′n,n+p =

√2

n

1

(−t)1/4

((−1)

[n+p

2

](1 − u)1/4

+ 1

(1 + u)1/4

)+ O

(1

n5/6

),

where [x] is the integer part of the real x. And we have for n → +∞

an+p,n+q = 1

n√−t

1

2π

∫ π

0

( √1 + u

1 + u cos v

sin(q − p)v/2

sin v/2−

− (−1)n+p√

1 − u

1 + u cos v

cos(q − p)v/2

cos v/2

)dv +

+ 1

n√−t

(−1)n+p

2

((−1)([

n+p2 ]+[ n+q2 ])

(1 − u)1/2+ 1

(1 + u)1/2

)+

+ 1

n√−t

(−1)n+p(−1)[n+s

2 ]

(1 − u2)1/4+ O

(1

n4/3

),

where s = p, if n+ p is even and s = q, if n+ q is even and with p, q fixed in Z,of different parity (else an+p,n+q = 0).

LEMMA 4.2 (the case β = 4). Here the parameter u = −2√g/t belongs to the

interval ]0, 1/√

2[. For n → +∞ and p fixed in Z, we have

a2n+p,2n+p−1 = −n

4

√−2t(√

1 + √2u+ (−1)2n+p

√1 − √

2u)+ O(1), (4.4)

a2n+p,2n+p−3 = −n

4

√−2t(√

1 + √2u− (−1)2n+p

√1 − √

2u)+ O(1). (4.5)

Moreover, we have

‖ψ2n+p‖L∞(J ) = O(n1/6), ‖ψ2n+p‖L∞(J c) = O(1)

and (4.6)

‖ε � ψ2n+p‖L∞(R) = O(n−1/2),

where

J = (Z1 − δ, Z1 + δ) ∪ (Z2 − δ, Z2 + δ),

δ is a positive real and J c is the complementary of J in R. And we have, forn → +∞,

c2n+p,2n+q

= − 1

n√−2t

(−1)n+p

2

((−1)([

2n+p2 ]+[ 2n+q

2 ])

(1 − √2u)1/2

+ 1

(1 + √2u)1/2

)+


+ 1

n√−2t

1

2π

∫ π

0

( √1 + √

2u

1 + √2u cos v

sin(p − q)v/2

sin v/2+

+ (−1)2n+p√

1 − √2u

1 + √2u cos v

cos(q − p)v/2

cos v/2

)dv−

− 1

n√−2t

(−1)2n+p(−1)[2n+s

2 ]

(1 − 2u2)1/4+ O

(1

n4/3

),

where s = p, if 2n + p is even and s = q, if 2n + q is even and with p, q fixed inZ, of different parity (else c2n+p,2n+q = 0).

LEMMA 4.3. (the case β = 1). For the coefficients (2.2) and (2.4), when n →+∞, we have

gjk = O(n), j, k ∈ {n− 3, n − 2, n− 1}. (4.7)

And, when n is odd, for the coefficients (2.4), we have

gjn = −gnj = −cjn + O(√n), j ∈ {n− 3, n− 2, n − 1}. (4.8)

LEMMA 4.4. (the case β = 4). For the coefficients (2.10), when n → +∞, wehave

gjk = O(n), j, k ∈ {2n, 2n+ 1, 2n + 2}. (4.9)

Proof of Lemma 4.3. We have to distinguish the cases of even and odd n.We examine the case of n even. We follow the procedure explained in Remark

2.2. According to Lemma 4.1, for the coefficients (1.8), we have, that

cn+p,n+q = O(n) and an+p,n+q = O(1/n).

Hence, for the coefficients (2.1), we get sn+p,n+q = O(1). Thus, in view of theexpressions of the coefficients (2.2), it suffices to prove (4.7) that the coefficientst!k in the expressions (2.2) are O(1). Since the coefficients t!k are the elementsof the inverse of the matrix D = (sjk)n−d�j,k�n−1, the difficulty is to prove thatdetD ∼ c �= 0, as n → +∞, where c is a positive constant. In order to do this,we have to compute the exact expression of detD in terms of coefficients (1.8) anduse the asymptotic forms of these coefficients given by Lemma 4.1 to compute theasymptotic form of detD. We find

detD = (1 − an−2,n+1cn+1,n−2)(1 − an−3,ncn,n−3 − an−1,ncn,n−1 −−an−1,n+2cn+2,n−1 + an−3,ncn,n−3an−1,n+2cn+2,n−1 −−an−1,ncn,n−3an−3,n+2cn+2,n−1)

and, using Lemma 4.1, we get

detD = c + O

(1

n1/3

),


with

c =(

1

4

√1 + u+ √

1 − u

(1 + u)1/4√

1 − u

((1 + u)1/4 − (−1)[

n−22 ](1 − u)1/4))2

> 0,

where u is the parameter of Lemma 4.1.Now we examine the case of n odd. As we see in Remark 2.2, the asymptotic

orders of the coefficients are different because of the presence of the coefficients(1.10) for which we have a′

n,n+p = O(1/√n). Hence, for the coefficients (2.3),

we get sn+p,n+q = O(√n). Here we have to prove that detD ∼ cn �= 0, when

n → +∞, where c is a positive constant. In order to do this, we have to computethe exact expression of detD in terms of coefficients (1.8) and (1.10) and usethe asymptotic forms of these coefficients given by Lemma 4.1 to compute theasymptotic form of detD. We find detD = αnγn, where

αn = 1 + (a′n−3,n − an−3,n)cn,n−3 + (a′

n−1,n − an−1,n)cn,n−1 −− an−1,n+2cn+2,n−1 + (a′

n−1,n − an−1,n)cn,n−3an−3,n+2cn+2,n−1 −− (a′

n−3,n − an−3,n)cn,n−3an−1,n+2cn+2,n−1

and

γn = a′n,n+3cn+3,n(an−2,n+1cn+1,n−2 − 1)−

− a′n,n+1(cn+1,n−2an−2,n+3cn+3,n + cn+1,n).

Thus, we see that we just have to prove that αn and γn have the following asymp-totic form c

√n, with c �= 0. By using Lemma 4.1, we get

αn = √n(−t)1/4c + O(n1/6),

with

c = −1√2

u2((1 + u)1/4 − (−1)[

n−12 ](1 − u)1/4

)√

1 − u2(√

1 + u− √1 − u)2

< 0

and

γn = √n(−t)1/4c′ + O(n1/6),

with

c′ = −1√2

(1 + u)1/4 − (−1)[n−1

2 ](1 − u)1/4

√1 − u2

< 0.

Proof of Lemma 4.4. We follow the same scheme. According to Lemma 4.2, forthe coefficients (1.18) we have that

a2n+p,2n+q = O(n) and c2n+p,2n+q = O(1/n).


Hence, for the coefficients (2.9), we get s2n+p,2n+q = O(1). Thus, in view ofthe expressions of the coefficients (2.10), it suffices to prove (4.9) that the coef-ficients t!k in the expressions (2.10) are O(1). Since the coefficients t!k are theelements of the inverse of the matrix D = (sjk)2n−d�j,k�2n−1, we have to provethat detD ∼ c �= 0, for n → +∞, where c is a positive constant. We computethe exact expression of detD in terms of coefficients (1.18) and use the asymptoticforms of these coefficients given by Lemma 4.2 to compute the asymptotic form ofdetD. We find

detD = (1 − a2n−2,2n+1c2n+1,2n−2)(1 − a2n−3,2nc2n,2n−3 − a2n−1,2nc2n,2n−1 −− a2n−1,2n+2c2n+2,2n−1 + a2n−3,2nc2n,2n−3a2n−1,2n+2c2n+2,2n−1 −− a2n−3,2nc2n,2n−1a2n−1,2n+2c2n+2,2n−3)

and with Lemma 4.2, we get

detD = c + O

(1

n1/3

),

with

c =(

1

4

√1 + √

2u+√

1 − √2u

(1 + √2u)1/4

√1 − √

2u

((1 + √

2u)1/4 + (−1)n(1 − √2u)1/4

))2

> 0,

where u is the parameter of Lemma 4.1.

Acknowledgement

The author is grateful to the referees and to Anne Boutet de Monvel for useful andcritical remarks.

References

1. Abramowitz, M. and Stegun, I. A. (eds): Handbook of Mathematical Functions, Dover, NewYork, 1968.

2. Bleher, P. and Its, A.: Semi-classical asymptotics of orthogonal polynomials, Riemann–Hilbertproblem and universality in the matrix model, Ann. of Math. 150 (1999), 185–266.

3. Bleistein, N. and Handelsman, R. A.: Asymptotic Expansions of Integrals, Dover, New York,1986.

4. Boutet de Monvel, A., Pastur, L. and Shcherbina, M.: On the statistical mechanics approach tothe random matrix theory: the integrated density of states, J. Statist. Phys. 79 (1995), 585–611.

5. Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X.: Asymptotics forpolynomial orthogonal with respect to varying exponential weight, Internat. Math. Res. Notes16 (1997), 759–782.

6. DiFrancesco, P., Ginsparg, P. and Zinn-Justin, J.: 2D gravity and random matrices, Phys. Rep.254 (1995), 1–133.

7. Forrester, P. J., Nagao, T. and Honner, G.: Correlations for the orthogonal-unitary andsymplectic-unitary transitions at the hard and soft edges, Nuclear Phys. B 553 [PM] (1999),601–643.


8. Johansson, K.: On fluctuation of eigenvalues of random Hermitian matrices, Duke Math. J. 91(1998), 151–204.

9. Mehta, M. L.: Random Matrices, and the Statistical Theory of Energy Levels, Academic Press,New York, 1967.

10. Mehta, M. L.: Matrix Theory, Selected Topics and Useful Results, Les Éditions de physique,France, 1989.

11. Mehta, M. L.: Random Matrices, Academic Press, New York, 1991.12. Pastur, L. and Shcherbina, M.: Universality of the local eigenvalue statistics for a class of

unitary invariant random matrix ensembles, J. Statist. Phys. 86 (1997), 109–147.13. Stojanovic, A.: Une approche par les polynômes orthogonaux pour des classes de matrices

aléatoires orthogonalement et symplectiquement invariantes : application à l’universalité dela statistique locale des valeurs propres, Preprint, www.physik.uni-bielefeld.de/bibos/preprints,00-01-06.

14. Szego, G.: Orthogonal Polynomials, Amer. Math. Soc., Providence, 1939.15. Tracy, C. A. and Widom, H.: Orthogonal and symplectic matrix ensembles, Comm. Math. Phys.

177 (1996), 727–754.16. Tracy, C. A. and Widom, H.: Correlation functions, cluster functions and spacing distributions

for random matrices, J. Statist. Phys. 92 (1998), 809–835.17. Widom, H.: On the relation between orthogonal, symplectic and unitary matrix ensembles,

J. Statist. Phys. 94 (1999), 347–364.


375

On the Solution of a Painlevé III Equation

HAROLD WIDOM�

Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A.e-mail: [email protected]

(Received: 23 November 2000)

Abstract. In a 1977 paper of B. M. McCoy, C. A. Tracy and T. T. Wu there appeared for the firsttime the solution of a Painlevé equation in terms of Fredholm determinants of integral operators.Their proof is quite complicated. We present here one which is more straightforward and makes useof recent work of the author and C. A. Tracy.

Mathematics Subject Classifications (2000): 34M65, 45B05, 35Q53.

Key words: Painlevé equation, Fredholm determinant, sinh-Gordon equation.

1. Introduction

In the 1977 paper of McCoy, Tracy and Wu [2] the following result was established.

THEOREM. Let

ψ(t) =∞∑n=0

2

2n + 1λ2n+1

∫ ∞

1· · ·

∫ ∞

1

2n+1∏j=1

e−tuj

uj + uj+1×

×[

2n+1∏j=1

(uj − 1

uj + 1

)α− 12 +

2n+1∏j=1

(uj − 1

uj + 1

)α+ 12]

dy1 . . . dy2n+1. (1)

Then ψ satisfies the equation

ψ ′′(t) + t−1ψ ′(t) = 12 sinh 2ψ + 2αt−1 sinhψ. (2)

This is a special case of the Painlevé III equation, and (1) gives a one-parameterfamily of solutions. It is clearly expressible in terms of the Fredholm determinantsof the kernels

e−tu

u + v

(u − 1

u + 1

)α± 12

acting on L2(1,∞). The proof in [2] is quite complicated, and the purpose of thisnote is to give a more straightforward one.

� Research supported by National Science Foundation grant DMS-9732687.

376 HAROLD WIDOM

First we give an equivalent formulation of the solution in terms of the kernel

K(x, y) = e−t (x+x−1)/2

x + y

∣∣∣∣x − 1

x + 1

∣∣∣∣2α

(3)

acting on L2(0,∞). This is the representation

ψ = log det

(I + λ

2K

)− log det

(I − λ

2K

), (4)

where K is the operator with kernel K(x, y). (This is very possibly known butseems not to have been written down in the literature before.)

To derive this second representation of ψ we make the changes of variableuj = (xj + x−1

j )/2 in the multiple integral in the first representation. Then√uj + 1

uj − 1= xj + 1

xj − 1, duj = 1

2(x2j − 1)

dxjx2j

,

and the integral becomes

1

22n+1

∫ ∞

1· · ·

∫ ∞

1

2n+1∏j=1

e−t (xj+x−1j )/2

(xj + xj+1)(xjxj+1 + 1)×

×[

2n+1∏j=1

(xj + 1)2 +2n+1∏j=1

(xj − 1)2

](xj − 1

xj + 1

)2α

dx1 . . . dx2n+1.

If we denote by K± the operators on L2(1,∞) with kernels

K±(x, y) = e−t (x+x−1)/2

(x + y)(xy + 1)(x ± 1)(y ± 1)

(x − 1

x + 1

)2α

,

then the multiple integral is equal to

trK2n+1+ + trK2n+1

−

and so

ψ = 2∞∑n=0

(λ/2)2n+1

2n + 1(trK2n+1

+ + trK2n+1− ).

Thus (4) will be a consequence of the following lemma:

LEMMA. For each m � 1, we have trKm+ + trKm− = trKm.Proof. Let f be an eigenfunction for K with eigenvalue λ,∫ ∞

0

e−t (x+x−1)/2

x + y

∣∣∣∣x − 1

x + 1

∣∣∣∣2α

f (y) dy = λf (x).

ON THE SOLUTION OF A PAINLEVE III EQUATION 377

Then the substitutions x → x−1, y → y−1 show that x−1f (x−1) is also aneigenfunction corresponding to the same eigenvalue. Hence, any eigenfunctioncan be written as the (orthogonal) sum of an ‘even’ eigenfunction f+ satisfyingx−1f+(x−1) = f+(x) and an ‘odd’ eigenfunction f− satisfying x−1f−(x−1) =−f−(x). (Of course, one of these is probably zero.) The change of variable y →y−1 and the relations y−1f±(y−1) = ±f±(y) show that∫ 1

0

e−t (x+x−1)/2

x + y

(1 − x

1 + x

)2α

f±(y) dy

= ±∫ ∞

1

e−t (x+x−1)/2

xy + 1

(x − 1

x + 1

)2α

f±(y) dy.

We deduce that f± are eigenfunctions corresponding to the eigenvalue λ for theoperators K± on L2(1,∞) with kernels

e−t (x+x−1)/2

[1

x + y± 1

xy + 1

](x − 1

x + 1

)2α

= K±(x, y),

and the statement of the lemma follows. ✷

2. Proof of the Theorem

Direct proofs of the fact that ψ as given by (4) satisfies the Painlevé equation whenα = 0 have already been given [1, 3]. We shall make use of some of the results of[3] here and therefore follow that paper’s notation, more or less. We give the proofin stages.

Part A. First, we introduce parameters r and s, define

E(x) =√λ

2e(rx+sx−1)/2

∣∣∣∣x − 1

x + 1

∣∣∣∣α, K(x, y) = E(x)E(y)

x + y,

and let K be the operator with this kernel K(x, y). (In the notation of [3], r =t1, s = t−1. The formulas we quote from there will be in terms of our parametersr and s. For convenience, we have changed the definition of K to conform to thenotation used there.) Define

ϕ(r, s) := log det(I + K) − log det(I − K).

Then ψ(t) = ϕ(−t/2,−t/2). We know from [3] that ϕ satisfies the sinh-Gordonequation

∂2ϕ

∂r∂s= 1

2 sinh 2ϕ. (5)

378 HAROLD WIDOM

In order to deduce (2) from this we must first find a connection between the r

and s derivatives of ϕ. (When α = 0 the determinants, and so also ϕ, dependonly on the product rs and (2) in this case is almost immediate.) To this end, weobserve that the determinants are unchanged if K(x, y) is replaced by K(x, y) :=sK(sx, sy). This is the same as replacing E(x) by

E(x) =√λ

2e(rsx+x−1)

∣∣∣∣sx − 1

sx + 1

∣∣∣∣α.Now

∂sE(x) =(rx + 2α

x

s2x2 − 1

)E(x),

which gives

∂sK(x, y) = rE(x)E(y) + 2αs2xy − 1

(s2x2 − 1)(s2y2 − 1)E(x)E(y).

(The denominator x+y in K(x, y) was cancelled by its occurrence also as a factorin both summands.) Hence

∂slog det(I + K)

= ∂s log det(I + K)

= tr(I + K)−1

[rE(x)E(y) + 2α

s2xy − 1

(s2x2 − 1)(s2y2 − 1)E(x)E(y)

].

(We abused the notation here by writing in the brackets the kernel of the operatorthat is meant.) Now we undo the variable change we made, which means we re-place x by x/s and y by y/s and divide by s in the expressions for the kernels, andwe obtain

∂slog det(I + K)

= tr(I + K)−1

[r

sE(x)E(y) + 2α

s

xy − 1

(x2 − 1)(y2 − 1)E(x)E(y)

].

If we had differentiated with respect to r without making a preliminary variablechange, we would have obtained

∂r log det(I + K) = tr(I + K)−1E(x)E(y).

Hence, we have shown that

s∂s log det(I + K) − r∂r log det(I + K)

= 2α tr(I + K)−1

[xy − 1

(x2 − 1)(y2 − 1)E(x)E(y)

].


Replacing K by −K and subtracting gives the relation (we use subscript notationfor derivatives)

rϕr − sϕs = 4α tr(I − K2)−1

[xy − 1

(x2 − 1)(y2 − 1)E(x)E(y)

]. (6)

This is the desired connection between the r and s derivatives of ϕ.

Part B. Our goal is to show that ψ(t) = ϕ(−t/2,−t/2) satisfies (2), but becauseof those awkward factors −1/2 we prefer to derive the equivalent equation

d2

dt2ϕ(t, t) + t−1 d

dtϕ(t, t) = 2 sinh 2ϕ(t, t) − 4αt−1 sinh ϕ(t, t). (7)

We use

d2

dt2ϕ(t, t) = 2ϕrs(t, t) + ϕrr(t, t) + ϕss(t, t),

(8)d

dtϕ(t, t) = ϕr(t, t) + ϕs(t, t).

Now we know that ϕ(r, s) satisfies the sinh-Gordon equation (5) so let us see whatidentity we have to derive. Set

T = tr(I − K2)−1

[xy − 1

(x2 − 1)(y2 − 1)E(x)E(y)

].

Differentiating (6) with respect to r and s gives

rϕrr + ϕr − sϕrs = 4αTr, rϕrs − sϕss − ϕs = 4αTs.

Therefore

−(r + s)ϕrs + rϕrr + sϕss + ϕr + ϕs = 4α(Tr − Ts),

(r + s)ϕrs + rϕrr + sϕss + ϕr + ϕs = 2(r + s)ϕrs + 4α(Tr − Ts).

Setting r = s = t and using (8), we get

td2

dt2ϕ(t, t) + d

dtϕ(t, t) = 4tϕrs(t, t) + 4α(Tr − Ts)(t, t).

Hence, by (5),

d2

dt2ϕ(t, t) + t−1 d

dtϕ(t, t) = 2 sinh 2ϕ(t, t) + 4αt−1(Tr − Ts)(t, t).

It follows that (7) is equivalent to

(Tr − Ts)(t, t) = − sinh ϕ(t, t). (9)

380 HAROLD WIDOM

Part C. The functions

Ei(x) = xiE(x), Fi(x) = Ei(x)

x2 − 1,

Qi = (I − K2)−1Ei, Pi = (I − K2)−1KEi

will arise in the computations leading to the identity (9). We have

Tr − Ts = tr(I − K2)−1

[xy − 1

(x2 − 1)(y2 − 1)(∂r − ∂s)E(x)E(y)

]+

+ tr(∂r − ∂s)(I − K2)−1

[xy − 1

(x2 − 1)(y2 − 1)E(x)E(y)

]. (10)

Now

(∂r − ∂s)E(x)E(y) = 12(x − x−1 + y − y−1)E(x)E(y),

which gives

xy − 1

(x2 − 1)(y2 − 1)(∂r − ∂s)E(x)E(y) = 1

2

(y − x−1

y2 − 1+ x − y−1

x2 − 1

)E(x)E(y).

Hence, the first summand in (10) equals

(Q0, F1) − (Q−1, F0). (11)

Next, using the notation a ⊗ b for the operator with kernel a(x)b(y), we have([3], p. 4)

(∂r − ∂s)(I − K2)−1 = 12 (P0 ⊗ Q0 + Q0 ⊗ P0 − P−1 ⊗ Q−1 + Q−1 ⊗ P−1),

and it follows that the second summand in (10) equals

(Q0, F1)(P0, F1) − (Q0, F0)(P0, F0)−− (Q−1, F1)(P−1, F1) + (Q−1, F0)(P−1, F0). (12)

We introduce notations for the various inner products:

Ui,j = (Qi, Fj ), Vi,j = (Pi, Fj ).

These are analogous to the inner products

ui,j = (Qi, Ej ), vi,j = (Pi, Ej )

which play a crucial role in [3] and will here, also. We have shown that

Tr − Ts = U0,1 − U−1,0 + U0,1V0,1 − U0,0V0,0 − U−1,1V−1,1 ++ U−1,0V−1,0. (13)


Part D. There are relations among the various quantities appearing on the right sideof (13). If we set

ui = u0,i , vi = v0,i , Ui = U0,i, Vi = V0,i,

then we have the recursion formulas

xQi(x) − Qi+1(x) = viQ0(x) − uiP0(x),

xPi(x) + Pi+1(x) = uiQ0(x) − viP0(x).

(The first is formula (9) of [3], the second is obtained similarly.) Taking innerproducts with Ej gives the formulas

ui,j+1 − ui+1,j = viuj − uivj , vi,j+1 + vi+1,j = uiuj − vivj (14)

of [3] and taking inner products with Fj gives the analogous formulas

Ui,j+1 − Ui+1,j = viUj − uiVj , Vi,j+1 + Vi+1,j = uiUj − viVj . (15)

Observe the special case i = j = −1 of the second part of (14):

u2−1 + 1 = (1 + v−1)

2. (16)

(The ui,j are symmetric in i and j .) In fact ([3], p. 8),

u−1 = sinh ϕ, 1 + v−1 = cosh ϕ.

We see from the above formulas that all the Ui,j and Vi,j may be expressedin terms of the Ui and Vi (with coefficients involving the ui and vi). But noticethat Fi+2 − Fi = Ei (here we use the form of Fi for the first time). This givesUi+2 − Ui = ui, Vi+2 − Vi = vi and using this also it is clear that everything canbe expressed in terms of the four unknown quantities U0, V0, U1 and V1 (and theui and vi). Using (16) also we compute that (13) equals

−v−1U1 + u−1V1 + u−1++ U1V1 − U0V0 − ((1 + v−1)U0 − u−1V0)(u−1U0 − (1 + v−1)V0)++ ((1 + v−1)U1 − u−1V1 − u−1)(u−1U1 − (1 + v−1)V1 − v−1). (17)

Now we are going to use, as we did before, the fact that conjugation by theunitary operator f (x) → x−1f (x−1) has the effect on K of interchanging r and s.Thus K is invariant under this conjugation when r = s. Since Ei is sent to E−i−1

and Fi to −F−i+1 we find that when r = s

U0 = −(Q−1, F−1) = −(1 + v−1)U0 + u−1V0,

U1 = −(Q−1, F0) = −(1 + v−1)U−1 + u−1V−1

= −(1 + v−1)(U1 − u−1) + u−1(V1 − v−1).

382 HAROLD WIDOM

From these we deduce that

V0 = u−1

v−1U0, V1 = u−1

v−1U1 − 1. (18)

Using these we find that when r = s (17) simplifies to

2u−1

v−1(U 2

0 − U 21 ).

Since u−1 = sinh ϕ, we have shown that the desired identity (9) is equivalent to

U1(t, t)2 − U0(t, t)

2 = 12v−1(t, t). (19)

Part E. Let us compute d/dt of both sides of (19). Of course, d/dtU0(t, t) = (∂r +∂s)U0(t, t), etc., so we begin by writing down these derivatives. We have

2(∂r + ∂s)Ei = Ei+1 + Ei−1, 2(∂r + ∂s)Fi = Fi+1 + Fi−1,

2(∂r + ∂s)(I − K2)−1

= 12 (P0 ⊗ Q0 + Q0 ⊗ P0 + P−1 ⊗ Q−1 + Q−1 ⊗ P−1).

(For the last, see [3], p. 4.) Using these we compute

2(∂r + ∂s)Ui

= 2Ui+1 + 2u0Vi + (1 + u2−1 + (1 + v−1)

2)Ui−1 − 2u−1(1 + v−1)2Vi−1.

Taking i = 0 and 1 and using (18) and (16) we find that when r = s

2(∂r + ∂s)U0 = 2u0u−1

v−1U0 − 2v−1U1,

2(∂r + ∂s)U1 = 2u0u−1

v−1U1 − 2v−1U0.

Hence,

(∂r + ∂s)(U21 − U 2

0 ) = 2u0u−1

v−1(U 2

1 − U 20 ). (20)

Now we compute in a similar way (cf., [3], p. 5)

2(∂r + ∂s)vi

= u−1u0 + v−1v0 + v0 + v−1,1 + u−1,−1u−1 + v−1,−1v−1 + v−2 + v−1,−1.

Again applying the operator f (x) → x−1f (x−1) we find that when r = s we have

u−1,−1 = u0, v−1,−1 = v0, v−2 = v−1,1,

so the above is

2(u−1u0 + v−1v0 + v0 + v−1,1).


Applying the second part of (14) with i = −1, j = 0 gives v0 + v−1,1 = u−1u0 −v−1v0, and so we have shown that, when r = s,

(∂r + ∂s)v−1 = 2u−1u0 = 2u0u−1

v−1v−1.

This relation and (20) show that U1(t, t)2 − U0(t, t)

2 and v−1(t, t) are equal upto a constant factor, and to deduce (19) it remains only to compute this factor.We do this by determining the asymptotics of both quantities as t → −∞. Forconvenience, we evaluate everything at r = s = −t and let t → +∞.

We have

v−1 = ((I − K2)−1KE,E−1

).

If we were to replace (I − K2)−1 by I we would be left with

(KE,E−1) =∫ ∞

0

∫ ∞

0

E(x)2E(y)2

x + yy−1 dy dx ∼ 1

2

(∫ ∞

0E(x)2 dx

)2

since the main contributions to the integrals come from neighborhoods of x =y = 1. It is an easy exercise to show that∫ ∞

0e−t (x+x−1)

∣∣∣∣x − 1

x + 1

∣∣∣∣2α

dx ∼ $(α + 12)2

−2αt−α− 1

2 e−2t , (21)

and so

(KE,E−1) ∼ λ2

4$(α + 1

2)22−4α−1t−2α−1 e−4t .

The error caused by our replacement of (I − K2)−1 by I is of smaller order ofmagnitude. This follows from the fact that the square of the L2 norm of E isO(t−α−1/2 e−2t ), as shown above, and, hence, so is the operator norm of K. Thusthe error, which equals ((I − K2)−1K3E,E−1), is O(t−4α−2 e−8t ). Therefore wehave shown

v−1 ∼ λ2

4$(α + 1

2)22−4α−1t−2α−1 e−4t .

Next,

U1 − U0 = ((I − K2)−1E, (x + 1)−1E

),

U1 + U0 = ((I − K2)−1E, (x − 1)−1E

)and U 2

1 − U 20 is the product of these. As before, replacing (I − K2)−1 by I in

each factor will not affect the first-order asymptotics of the product. After thisreplacement the first inner product becomes λ/2 times the integral in (21) but withan extra factor x + 1 in the denominator. Thus

U1 − U0 ∼ λ

2$(α + 1

2)2−2α−1t

−α− 12 e−2t .

384 HAROLD WIDOM

After the replacement the second inner product becomes λ/2 times∫ ∞

0e−t (x+x−1)

∣∣∣∣x − 1

x + 1

∣∣∣∣2α dx

x − 1.

This is a little trickier since when we make the variable change x = 1 + y tocompute the asympotics, we must use the second-order approximations x+x−1 →2 + y2 − y3 and (x + 1)−2α → 2−2α(1 − αy). But it is still straightforward and wefind that

U1 + U0 ∼ λ

2$(α + 1

2)2−2α−1t

−α− 12 e−2t .

Thus

U 21 − U 2

0 ∼ λ2

4$(α + 1

2 )22−4α−2t−2α−1 e−4t ∼ 1

2v−1.

We knew that (U 21 − U 2

0 )/v−1 is a constant and now we see that the constantequals 1/2. This establishes (19) and concludes the proof.

References

1. Bernard, D. and LeClair, A.: Differential equations for sinh-Gordon correlation functions at thefree fermion point, Nuclear Phys. B 426 [FS] (1994), 534–558.

2. McCoy, B. M., Tracy, C. A. and Wu, T. T.: Painlevé functions of the third kind, J. Math. Phys.18 (1977), 1058–1092.

3. Tracy, C. A. and Widom, H.: Fredholm determinants and the mKdV/sinh-Gordon hierarchies,Comm. Math. Phys. 179 (1996), 1–10.


385

Asymptotic Value Distribution for Solutionsof the Schrödinger Equation

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: 5 December 2000)

Abstract. We consider the Dirichlet Schrödinger operator T = −(d2/d x2)+V , acting in L2(0,∞),where V is an arbitrary locally integrable potential which gives rise to absolutely continuous spec-trum. Without any other restrictive assumptions on the potential V , the description of asymptoticsfor solutions of the Schrödinger equation is carried out within the context of the theory of valuedistribution for boundary values of analytic functions. The large x asymptotic behaviour of the solu-tion v(x, λ) of the equation Tf (x, λ) = λf (x, λ), for λ in the support of the absolutely continuouspart µa.c. of the spectral measure µ, is linked to the spectral properties of this measure which aredetermined by the boundary value of the Weyl–Titchmarsh m-function. Our main result (Theorem 1)shows that the value distribution for v′(N, λ)/v(N, λ) approaches the associated value distributionof the Herglotz function mN(z) in the limit N → ∞, where mN(z) is the Weyl–Titchmarsh m-function for the Schrödinger operator −(d2/d x2)+V acting in L2(N,∞), with Dirichlet boundarycondition at x = N . We will relate the analysis of spectral asymptotics for the absolutely continuouscomponent of Schrödinger operators to geometrical properties of the upper half-plane, viewed as ahyperbolic space.

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

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

1. Introduction

The principal aim of this paper is to provide a general analysis of the link be-tween asymptotics for solutions of the Schrödinger equation on the half-line, at realspectral parameter λ, and spectral properties, considered in relation to the Weyl–Titchmarsh m-function and its boundary values, for the associated Schrödingeroperator, in the case that this operator has absolutely continuous spectrum. Wedo not, here, restrict attention to the case in which there is purely absolutely con-tinuous spectrum – rather, we shall assume that there is an absolutely continuouscomponent to the spectrum, and we shall take λ to belong to a spectral support ofthe absolutely continuous component of the spectral measure.

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

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

A secondary aim of the paper will be to exhibit the degree to which the analysisof spectral asymptotics for the absolutely continuous component of Schrödingeroperators is related to, and dependent on, the geometrical properties of the complexupper half-plane, viewed as a hyperbolic space, and mappings of this space byanalytic functions. In particular, we shall show how the theory of value distributionfor boundary values of analytic functions provides a natural framework for thedescription of large x asymptotics for solutions of the Schrödinger equation. Theseideas can be linked, through the use of hyperbolic metric, to an analysis of anglesubtended, at points of C+ and their images under analytic maps, by Borel subsetsof R. We will outline several interesting byproducts of this work, including result-ing estimates of boundary behaviour and associated limiting value distribution forHerglotz functions.

We begin by establishing the notation and background for the study of spectraltheory and asymptotics for the Schrödinger operator on the half-line.

Let a potential function V (x), defined for 0 � x < ∞, be given, with V real-valued and integrable over bounded subintervals of [0,∞). We make no specialassumptions regarding the behaviour of V (x) in the limit as x → ∞.

We associate with V the differential expression τ = −(d2/d x2) + V . Thenτ may be used to define the self-adjoint operator T = −(d2/d x2) + V , actingin L2(0,∞) and subject to Dirichlet boundary condition at x = 0. Correspond-ingly, we may define the one-parameter family Tα = −(d2/d x2) + V of self-adjoint operators in L2(0,∞), subject for any α ∈ [0, π) to the boundary condition(cos α)f (0) + (sin α)f ′(0) = 0; we may thus identify T0 with T .

We are assuming here that the differential expression τ belongs to the limit-point case at infinity (see [1]), in which case no boundary condition at x = +∞ isrequired in order to define Tα as a self-adjoint operator. The alternative assumption,that τ belongs to the limit-circle case, is known to lead to purely discrete spectrumfor Tα (see [1]). Since we are concerned in this paper with the absolutely continuouspart of the spectrum, we need not allow for the possibility of limit-circle at infinity.

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 equation

−d2f (x, λ)

d x2+ V (x)f (x, λ) = λf (x, λ), (1)

−d2f (x, z)

d x2+ V (x)f (x, z) = zf (x, z), (2)

in each case taking 0 � x < ∞.In particular, define specific solutions uα(x, λ), vα(x, λ) for λ ∈ R (and corre-

spondingly uα(x, z), vα(x, z) in the case z ∈ C+) subject to initial conditions

uα(0, λ) = cos α, u′α(0, λ) = sinα,

vα(0, λ) = − sinα, v′α(0, λ) = cos α.

(3)

ASYMPTOTIC VALUE DISTRIBUTION 387

In the special case α = 0, we shall usually denote u0, v0 by u, v, respectively.Note that vα satisfies the boundary condition appropriate to Tα; in particular, ifvα(·, λ) ∈ L2(0,∞) then λ will be an eigenvalue of Tα .

For each α ∈ [0, π) and for all z ∈ C+, the Weyl–Titchmarsh m-function mα(·)is defined (assuming limit-point case at infinity) by the condition that

uα(·, z) + mα(z)vα(·, z) ∈ L2(0,∞). (4)

Then mα is a Herglotz function, that is analytic in the upper half-plane, with strictlypositive imaginary part. In the special case α = 0, we shall normally denote m0(z)

by m(z), in which case (4) becomes

u(·, z) + m(z)v(·, z) ∈ L2(0,∞).

An alternative characterisation of m is through the observation that m(z) = f ′(0, z)/f (0, z) for any (nontrivial) solution f (·, z) of Equation (2), such that f (·, z) ∈L2(0,∞).

In this paper, we shall in addition be interested in the m-function related to thedifferential expression τ = −(d2/d x2)+V where V (x) is defined on the truncatedinterval N � x < ∞, for any N > 0. Taking for simplicity the case of Dirichletboundary condition at x = N , we may define the self-adjoint operator T N =−(d2/d x2) + V acting in L2(N,∞), subject to boundary condition f (N) = 0.

Correspondingly, solutions uN(·, z), vN(·, z) of Equation (2) with Im z > 0,may be defined subject to initial conditions

uN(N, z) = 1, (uN)′(N, z) = 0,vN(N, z) = 0, (vN)′(N, z) = 1,

and the m-function mN(·) with Dirichlet boundary condition at x = N is deter-mined by the condition that

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

Note that mN(·) is the standard m-function for the Dirichlet Schrödinger operator−(d2/d x2)+V (x+N) acting in L2(0,∞). An alternative characterisation of mN(·)is through the observation that mN(z) = f ′(N, z)/f (N, z) for any (nontrivial)square-integrable solution f (·, z) of Equation (2). Since u(·, z) + m(z)v(·, z) isjust such a square-integrable solution, we can write explicitly

mN(z) = u′(N, z) + m(z)v′(N, z)

u(N, z) + m(z)v(N, z). (5)

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 ([2])

F(z) = a + bz +∫ ∞

−∞

(1

t − z− t

t2 + 1

)dρ(t), (6)


where ρ satisfies in addition the integrability condition∫ ∞−∞(t2 + 1)−1 dρ(t) < ∞.

The function ρ may then be used to define a Lebesgue–Stieltjes measure µ =dρ(t). Such a measure may be defined in particular for the m-function m(z), inwhich case the measure carries all of the spectral information for the Dirichletoperator T , in the sense that T is unitarily equivalent to the multiplication operatorg(λ) → λg(λ) in the Hilbert space L2(R; dρ). In that case the Lebesgue–Stieltjesmeasure µ = dρ is called the spectral measure of the operator T . We shall beconcerned with values of the spectral parameter λ which belong to the support ofthe absolutely continuous component µa.c. of the measure µ.

The main question to be addressed will be as follows: How can one describethe large x asymptotic behaviour of the solution v(x, λ) of Equation (1), for λ inthe spectral support of µa.c., in terms of spectral properties of this measure? Herewe would expect ‘spectral properties of this measure’ to include a prescription ofthe boundary value m+(λ) of the m-function, defined as m+(λ) = limε→0+ m(λ +iε), since m+(λ) determines both the density function 1/π Imm+(λ) of µa.c., and,through equations linking m(z) algebraically with other functions mα(z), the cor-responding density functions for the spectral measures of the Tα . We shall alsofind that the boundary value mN+(λ) of the ‘truncated’ m-function mN(z) plays animportant role in describing asymptotic behaviour of v(x, λ) for large x.

Our key result (Theorem 1) relating asymptotics to spectral properties relatesnot to the solution v(x, λ) directly, but to its logarithmic derivative q(x, λ), de-fined by q(x, λ) = v′(x, λ)/v(x, λ), where the prime denotes differentiation withrespect to x. In order to achieve maximum generality, that is without imposingrestrictive assumptions on the potential, the description of large x asymptotics hasto be carried out within the context of the theory of value distribution.

We shall explain more fully in Section 2 the background to the idea of valuedistribution for any real-valued (Lebesgue) measurable function F(λ). For sucha function F , value distribution assigns to any pair A, S of Borel subsets of R anonnegative (and possibly infinite) number M(A, S;F), given by M(A, S;F) =|A ∩ F−1(S)|, where | · | stands for Lebesgue measure. Thus M is the Lebesguemeasure of all λ ∈ A for which F(λ) ∈ S. We may now give a more precise state-ment to Theorem 1, in terms of asymptotic value distribution. Namely, the large x

value distribution, over λ, of the logarithmic derivative q(x, λ) = v′(x, λ)/v(x, λ)of the solution v of Equation (1) is given by the formula

limN→∞

{|{λ ∈ A; q(N, λ) ∈ S}| − 1

π

∫A

θ(mN+(λ), S) dλ

}= 0. (7)

Here A is an arbitrary subset of the essential support of µa.c., having finite measure,and θ(z, S), for any z ∈ C+, denotes the angle subtended at z by the subset S ofthe real z-axis.

We consider the asymptotic formula (7) to be unusual in several respects. Forexample,


(i) Equation (7) holds for arbitrary locally integrable potentials which give riseto absolutely continuous spectrum, and hence is a result of considerable generality.For certain special classes of potential, the result may be further qualified by moredetailed information regarding the dependence on λ and N of mN+(λ). The depen-dence of mx+(λ) on x is subject to a first-order Riccati-type differential equation,and solutions exhibiting the so-called δ-clustering property ([3]) may be used toprovide bounds for mN+(λ). Examples of classes of potentials which can be treatedin this way include the L1(0,∞) class, for which one has limN→∞ mN+(λ) = i

√λ

so that mN+(λ) may be replaced by i√λ in Equation (7), and potentials such as −kx2

(k � 0) which give rise to absolutely continuous spectrum over the whole of R

([4]). Recent developments in the theory of slowly decreasing potentials, satisfyingpower bounds of the form |V (x)| � const. × x−(δ+1/2) (δ > 0) as x → ∞, haveshown that for these potentials the large x behaviour of v(x, λ) is governed foralmost all λ > 0 by WKB-like asymptotics [5–8].

(ii) A second aspect of the asymptotic formula (7) to which we would drawattention appears to be completely new.

Notice first of all that q(N, λ) = v′(N, λ)/v(N, λ) may be determined from thesolution of the Schrödinger equation (1) on the interval 0 � x � N , and subjectto the appropriate initial conditions at x = 0. Hence, q(N, λ) is determined by thepotential function V over this finite interval [0, N].

On the other hand, one has the following prescription for the determination ofmN+(λ): let f (·, z), with Im z > 0, be any (nontrivial) solution of the Schrödingerequation (2) on the interval N � x < ∞, and belonging to L2(N,∞); thenset mN+(λ) = limε→0+ f ′(N, λ + iε)/f (N, λ + iε). Thus we see that mN+(λ) isdetermined by the potential function V across the semi-infinite interval (N,∞).It follows, then, that the asymptotic formula (7) expresses the convergence asN → ∞, for any choice of Borel sets A, S, of two analytic expressions to eachother, of which the first (|{λ ∈ A; q(N, λ) ∈ S}|) is dependent on the potentialfunction V only on the interval [0, N], and the second (1/π

∫Aθ(mN+(λ), S) dλ) is

dependent on the potential function V only on the interval (N,∞).The fact that these two intervals of definition of V do not overlap implies that

we have, in Equation (7), the convergence of two expressions which are, in somesense, ‘independent’ of each other. This will have important consequences for ourunderstanding of the theory of absolutely continuous spectrum for Schrödingeroperators and we will return to this point in Section 4.

(iii) A third aspect of the asymptotic formula (7) to which we draw attention isthe role in the description of asymptotics of ideas drawn from the theory of valuedistribution, and the link with angle subtended. These and other ideas are relatedto the geometry of hyperbolic space, about which we shall have more to say later.

(iv) Finally, we draw attention to the link which will become more apparentthrough the proof of Equation (7), between rate of convergence in (7) and regularityproperties of m+(λ).


The paper is organised as follows: In Definition 1 of Section 2, we make moreprecise the notion of a value distribution function M, and introduce as an impor-tant special case (Definition 2) the value distribution function associated with anygiven Herglotz function F . The value distribution function for F may be obtainedby integrating, with respect to the spectral parameter λ, the angle subtended by agiven set S at the boundary value point F+(λ), and is the limit of the correspondingintegral over the complex line Im z = δ, as δ approaches zero. Surprisingly, theconvergence of this limit is uniform over all Herglotz functions F and all sub-sets S of R (see Lemma 1), a result that has far-reaching consequences for valuedistribution and spectral theory.

In Section 3, we exhibit the connection between angle subtended and the hy-perbolic metric for C+. Instead of using hyperbolic metric directly, we rely on anestimate of separation γ , defined by Equation (15), of points in C+; γ is relatedto hyperbolic metric by Equation (16), and to angle subtended by Equation (17).In Lemma 3 we arrive at one of the key estimates of value distribution theory forsolutions of the Schrödinger equation, which demonstrates a precise bound for thelarge N convergence with respect to hyperbolic metric of the negative logarithmicderivative −f ′(N, z)/f (N, z) of solutions of Equation (2).

Theorem 1 of Section 4 is a statement of the main result of this paper, whichshows that the value distribution for v′(N, λ)/v(N, λ) approaches the associatedvalue distribution of the Herglotz function mN(·) in the limit as N → ∞. Theproof of this theorem relies heavily on the results of the previous sections, and thesequence of inequalities which make up the proof allow in many cases a preciseestimate of the rate of convergence.

In the final Section 5, we consider some applications of the theory to specificexamples of Schrödinger operators. These include examples of slowly decayingpotentials, potentials unbounded below at infinity and sparse potentials.

2. Value Distribution and Herglotz Functions

Given a (Lebesgue) measurable function f : R → R, the distribution of points(λ, f (λ)) of the graph of f may be described in terms of a measure M0, given forBorel subsets S of R

2 by

M0(S) = |{λ ∈ R; (λ, f (λ)) ∈ S}|.Here | · | stands for Lebesgue measure, and M0(S) is the measure of the set of all λsuch that the corresponding point (λ, f (λ)) of the graph of f belongs to the set S.

In the case that S = A × S is the product of a pair of Borel subsets A, S of R,we shall write M0(A × S) = M(A, S), where

M(A, S) = |{λ ∈ A;f (λ) ∈ S}| = |A ∩ f −1(S)|. (8)

The mapping M: (A, S) → M(A, S), which assigns an extended real nonnegativenumber to pairs of Borel subsets of R, 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 in particular the measure A → M(A, S) is absolutelycontinuous with respect to Lebesgue measure.

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.

It is important to note that not all distribution functions M are of the formM(A, S) = |A ∩ f −1(S)| for some real-valued measurable function f . (If such f

does exist, then property (iii) above is equivalent to the condition that |f −1(S)| = 0whenever |S| = 0.) There may be no function f for which M(A, S) describes thedistribution of values. Nevertheless, the definition of value distribution functionadopted here allows for the more general situation that M may describe a limitingvalue distribution for a sequence {fn} of functions, in the sense that M(A, S) =limn→∞ |A ∩ f −1

n (S)|. This wider use of terminology is especially appropriate inthe description of absolutely continuous spectra.

Properties (ii) and (iii) imply that any value distribution function M may berepresented in terms of families of measures {µy} (y ∈ R) and S → ω(λ, S)

(λ ∈ R) as

M(A, S) =∫A

ω(λ, S) dλ, M(A, S) =∫S

µy(A) dy (9)

with 0 � ω(λ, S) � 1; in the case M(A, S) = |A ∩ f −1(S)| we can take ω(·, S)to be the characteristic function of the set f −1(S).

The theory of value distribution for Herglotz functions starts from the observa-tion that, to any Herglotz function F , one may associate in a natural way a valuedistribution function M defined by the first of Equations (9), where ω is given by

ω(λ, S) = limε→0+

1

πθ(F (λ + iε), S), (10)

with θ(F (λ+ iε), S) again being the angle subtended at the point F(λ+ iε) of theupper half-plane by the subset S of the real z-axis.

DEFINITION 2. We shall refer to the function M, which is defined by M(A, S) =∫Aω(λ, S) dλ, with ω(λ, S) given by Equation (10), as the associated value distri-

bution function for the Herglotz function F . Since ω,M, and related functions aredependent on F , we shall often write ω(λ, S;F),M(A, S;F), and so on.


For complex argument z ∈ C+, we define ω(·, S;F) by ω(z, S;F) =1πθ(F (z), S), so that ω(λ, S;F) = limδ→0+ ω(λ + iδ, S;F).

For almost all λ ∈ R, 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 Im F+(λ) > 0,

(11)

where F+(λ) = limε→0+ F(λ+ iε) is the boundary value function for the Herglotzfunction F . It is known that F+(λ) exists for almost all λ ∈ R. In the particularcase that F+ is almost everywhere real, ω(·, S;F) is the characteristic function ofF−1

+ (S), and we have that in that case M(A, S;F) = |A ∩ F−1+ (S)|.

If M(A, S;F) = ∫Sµy(A) dy is the associated value distribution function for

a Herglotz function F , as in the second of Equations (9), we can give an explicitconstruction for the family of measures {µy}. (See Pearson [9]; our constructionhere differs in minor details from that of Pearson [9], but is essentially the same.)Define first of all a one-parameter family of Herglotz functions {Fy} by Fy(z) =(y − F(z))−1 (y ∈ R). Then the measure µy is just the spectral measure µ = dρdefined through Equation (6) by the Herglotz representation for the function Fy .

Although we have seen that both the measures ω(λ, ·) and µy in Equation (9)may be constructed explicitly in the case of a value distribution function definedby a Herglotz function, it may be difficult in practice to evaluate M(A, S;F)

through these integral formulae. This is because little is known a-priori regardingthe measures µy in the absence of detailed spectral information, and on the otherhand the determination of ω(λ, S;F) through Equation (10) requires knowledgeof the behaviour of the Herglotz function close to the real axis, where precisebounds are not easy to obtain. One can get round some of these difficulties bytranslating λ by a small increment iδ off the real axis, and making use of a re-markable estimate of the rate of convergence in the limit δ → 0+. Define first ofall a translated Herglotz function Fδ by Fδ(z) = F(z + iδ), with δ > 0, and setωδ(λ, S;F) = ω(λ, S;Fδ) = (1/π)θ(F (λ + iδ), S).

An application of the Lebesgue dominated convergence theorem shows that(assuming |A| < ∞)

M(A, S;F) = limδ→0+

∫A

ωδ(λ, S;F) dλ =∫A

ω(λ, S;F) dλ. (12)

The following lemma has the surprising consequence that, for any fixed A, thislimit is uniform over all Borel sets S, and over all Herglotz functions F .

LEMMA 1. Let A ⊂ R be a given Borel set, having finite Lebesgue measure. Thenthere exists a function EA: R → R (depending on A) such that the inequality∣∣∣∣

∫A

ωδ(λ, S;F) dλ −∫A

ω(λ, S;F) dλ

∣∣∣∣ � EA(δ) (13)


holds for any δ > 0, for any Borel set S ⊂ R, and for all Herglotz functionsF . Moreover, the function EA can be chosen such that EA(δ) is a nondecreasingfunction of δ, with limδ→0+ EA(δ) = 0.

Proof. A detailed proof of this result is given in Breimesser and Pearson, Geo-metrical aspects of spectral theory and value distribution for Herglotz functions, inpreparation (from now on, we refer to this as BP2000).

Remarks. The lemma shows that the convergence of the integral used in (12) todetermine the value distribution function M is uniform over S and F . An explicitbound in (13) is obtained by setting

EA(δ) = 1

π

∫A

θ(λ + iδ, Ac) dλ, (14)

where Ac is the complement of A. With this choice of function EA, (13) is optimalsince equality can be attained by taking S = Ac and F(z) = z. In the case thatA = (a, b) is an interval, we find

EA(δ) = 2(b − a)

πtan−1 δ

b − a+ δ

πlog

(1 + (b − a)2

δ2

),

which illustrates clearly in this case the convergence to zero of EA(δ) in the limitδ → 0+.

Lemma 1, taken together with Equation (12), enables us to carry out an estimate,to order EA(δ), of the value distribution function M(A, S;F), through an evalua-tion of the integral

∫Aωδ(λ, S;F) dλ. Here the integrand is given, apart from the

multiplicative constant 1/π , by the angle subtended by the set S at points F(λ+iδ),as λ is varied over A. In the following section, we shall see how such estimates ofangle subtended may be carried out, using the geometry of the upper half-plane asan example of hyperbolic space.

3. Angle Subtended and Hyperbolic Space

The angle subtended by a Borel set S ⊂ R at a point z ∈ C+ is given by

θ(z, S) =∫S

Im1

t − zdt.

For z1, z2 ∈ C+, θ(z1, S) will be close to θ(z2, S) provided z1 is close to z2, unlessz1 or z2 is close to the real axis. One can give a quantitative expression to this factby defining an estimate of separation γ (·, ·) of points in the upper half-plane, givenby

γ (z1, z2) = |z1 − z2|√Im z1

√Im z2

(z1, z2 ∈ C+). (15)


The estimate of separation γ is related to the hyperbolic distance d betweenpoints in C+ by the equation

γ (z1, z2) = 2 sinh

{d(z1, z2)

2

}. (16)

(For proof of this and related results see BP2000; note that some authors use 12d

instead of d in defining hyperbolic distance. See Anderson [10] for an elemen-tary introduction to the geometry of hyperbolic spaces, and Jost [11] for a moreadvanced treatment. See also Pommerenke [12].)

We shall find γ as an estimate of separation of points in C+ more convenientto use than hyperbolic distance; however it should be noted that although d(·, ·)satisfies the triangle inequality and therefore defines a metric, γ (·, ·) does not.

The connection between γ and angle subtended is given by the result (BP2000)that

γ (z1, z2) = supS

|θ(z1, S) − θ(z2, S)|√θ(z1, S)

√θ(z2, S)

, (17)

where the supremum is taken over all Borel subsets of R having strictly positiveLebesgue measure. We shall need two further properties of hyperbolic metric,which we state for convenience in terms of γ rather than d, in the following lemma.

LEMMA 2. Let M be a Möbius transformation, defined by

M(z) = az + b

cz + d(z ∈ C+),

with a, b, c, d real coefficients and ad −bc > 0. Then M leaves γ invariant, in thesense that

γ (M(z1),M(z2)) = γ (z1, z2). (18)

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

γ (F (z1), F (z2)) � γ (z1, z2). (19)

Proof. See BP2000. Möbius transformations with real coefficients and positivediscriminant ad − bc leave C+ invariant, and the invariance of the metric d undersuch transformations may be considered the defining property of hyperbolic metricfor C+.

To carry out an analysis of value distribution for solutions of the Schrödingerequation, we need to make use of the link between value distribution and angle sub-tended. As may be seen from Equation (17), this will involve estimates of the sepa-ration γ , as applied to solutions of the Schrödinger equation. The following lemmaprovides such an estimate, in the case that we consider the logarithmic derivativeof solutions of the Schrödinger equation (2), with complex spectral parameter.


LEMMA 3. Let u(·, z), v(·, z) be solutions of Equation (2), subject to initial con-ditions (3), with α = 0. Let m(1) be any constant such that Imm(1) � 0. Then, forany N > 0, and for all z ∈ C+, we have the estimate

γ

(−v′(N, z)

v(N, z),−u′(N, z) + m (1)v′(N, z)

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

)� 1√

I (I + 1), (20)

where the integral I is defined by

I (N, z) = (Im z)

∫ N

0Im( u(x, z)v(x, z)) dx. (21)

Proof. For simplicity of notation, we shall write u, v for the solutions u(·, z),v(·, z) respectively, and WN(f, g) = f (N)g′(N)− g(N)f ′(N) for the Wronskianof two functions f, g at x = N . We make use of the standard identities ([1])∫ N

0vv dx = 1

2iIm zWN(v, v );∫ N

0Im( uv) dx = − 1

2Im z{1 − ReWN(u, v )}.

(22)

We shall also need the identity

|W(u, v )|2 = 1 − W(u, u )W(v, v ), (23)

where W stands for the Wronskian; Equation (23) may be verified directly by set-ting 1 = W(u, v)W( u, v ) on the right-hand side, and writing out the Wronskiansexplicitly.

A straightforward calculation, using Equation (15) and with W(u, v) = 1,shows that at x = N we have

γ 2

(−v′

v,−u′ + m (1)v′

u + m (1)v

)= 1

−WN(v,v )WN (u+m(1)v,u+m(1)v )

4

.

For simplicity of notation we shall write just γ 2 for the left hand side of thisequation.

We consider first of all the case in which m(1) is real. The denominator on theright hand side is then of the form A + Bm(1) + C(m(1))2, where A � 0, C � 0and B is real. Hence the denominator is bounded below by A − (B2/4C), leadingto the estimate

γ 2 � − 4[WN(u, u )WN(v, v ) + (

ImWN(u, v ))2] . (24)

We can now use Equation (23) on subtracting (ReWN(u, v ))2 from both sides and

substituting for (ImWN(u, v ))2 in (24), to obtain

γ 2 � − 4

1 − (ReWN(u, v ))2= 1

I 2 + I,


where I = − 12(1 − ReWN(u, v )). Lemma 3 now follows, in the case m(1) real, by

using the second identity in Equation (22).In the case that

m(1) = Rem(1) + iY, with Y > 0,

there are additional terms

Y 2(WN(v, v ))2 + 2iY (ReWN(u, v ))WN(v, v )

in the square bracket in Equation (24). These terms may be taken into account bymaking the replacement u → u − iY v in this denominator. Equation (23) remainsvalid with u−iY v for u, and we may again obtain the estimate (20), with I replacedin this case by the integral

I ′ = (Im z)

∫ N

0Im(uv ) dx + Y

2iWN(v, v ).

Writing WN(v, v ) = 2iIm z∫ N

0 vv dx, we see that I ′ � I , and we have provedLemma 3 in the general case.

COROLLARY 1. With the same notation as in the statement of Lemma 3, 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, (25)

where convergence is uniform in m(1) for Imm(1) � 0 and uniform in z for z in anyfixed compact subset of C+.

Proof. Using (20) and (21), Equation (25) depends on the fact that

limN→∞

∫ N

0Im( uv) dx = +∞.

We show first of all that the ratio∫ N

0 Im( uv) dx/∫ N

0 vv dx is a nondecreasingfunction of N for N > 0.

Using the expression for these integrals in terms of Wronskians, the derivativewith respect to N of this ratio is given by

{(uv − vu )WN(v, v ) + vv(2 − WN(u, v ) − WN( u, v))}4(Im z)

(∫ N

0 vv dx)2 ,

where solutions u, v are evaluated at x = N .Using WN(u, v) = WN( u, v ) = 1, the numerator in this expression may be

simplified to −(v − v )2 = 4(Im v)2. Hence

d

dN

{∫ N

0 Im( uv) dx∫ N

0 vv dx

}=

(Im v(N, z)

)2

(Im z)(∫ N

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


from which it follows that the ratio of integrals cannot decrease with N . It followsthat the integral I (N, z) in Equation (21) is at least as large as const. × ∫ N

0 vv dx.

Moreover, standard arguments ([1]) imply that limN→∞∫ N

0 vv dx = +∞ in thelimit-point case, with divergence uniform over compact subsets of C+. The finalconclusion of the corollary is then a straightforward consequence.

4. Asymptotic Value Distribution for Absolutely Continuous Spectra

We are now ready to prove the main result regarding asymptotic value distributionin the limit N → ∞ of the logarithmic derivative q(N, λ) = v′(N, λ)/v(N, λ)

of the solution v(·, λ) of Equation (1). We have previously referred to this resultin Equation (7). As before, mN(z) is the Weyl–Titchmarsh m-function for theSchrödinger operator −d2/dx2 + V acting in L2(N,∞), with Dirichlet boundarycondition at x = N , and mN+(λ) is the boundary value of mN(z) as z approachesλ ∈ R. We let θ(z, S) stand for the angle subtended at z by a subset S of the realaxis. Lebesgue measure is denoted by | · |.THEOREM 1. Let A be a Borel subset of an essential support of the absolutelycontinuous part µa.c. of the spectral measure µ for the Dirichlet Schrödinger op-erator T = −d2/dx2 + V , acting in L2(0,∞); we suppose also that A has finiteLebesgue measure. Then we have, for any Borel subset S of R,

limN→∞

{∣∣∣∣{λ ∈ A; v

′(N, λ)

v(N, λ)∈ S

}∣∣∣∣ − 1

π

∫A

θ(mN+(λ), S) dλ

}= 0. (26)

Proof. Let m(z) denote the Weyl–Titchmarsh m-function for the differentialoperator T . An essential support for µa.c. is the set of all λ ∈ R at which theboundary value m+(λ) of m(z) exists with strictly positive imaginary part. Hencewe may assume without loss of generality that Imm+(λ) > 0 for all λ ∈ A.

Let p be a positive constant. We shall show that the expression within the curlybrackets is bounded in absolute value by 6p|A|, for all N sufficiently large.

We first define a partition A = A0 ∪A1 ∪A2 ∪ · · · ∪An of the set A into finitelymany (n+ 1 in number) disjoint subsets A0, A1, A2, . . . , An, having the propertiesthat |A0| � p|A|, with Aj bounded for j � 1. To each j � 1 we associate aconstant m(j) with Imm(j) > 0, such that

γ (m+(λ),m(j)) � p (all λ ∈ Aj , j � 1). (27)

The general idea behind the construction of such a partition of A, together withassociated constants m(j) (j = 1, 2, . . . , n), is as follows. Points λ ∈ A at which|λ| or |m+(λ)| are large, or at which Imm+(λ) is small, are put into the set A0.This leaves the set A\A0. The range of m+(λ), as λ runs over A\A0, is containedin a compact subset C of C+, and a partition C = C1 ∪ C2 ∪ · · · ∪ Cn of C into


disjoint subsets can be found such that, for all j = 1, 2, . . . , n, we have z1, z2 ∈Cj ⇒ γ (z1, z2) � p. (Thus each Cj must have diameter � p, as measured by γ .)Finally, take Aj = (A\A0)∩m−1+ (Cj ) and m(j) = m+(λj ) for any (fixed) λj ∈ Aj ,and (27) is satisfied.

Now let δ be a positive constant such that, for arbitrary Herglotz function F , forany Borel subset S of R, and for all j = 1, 2, . . . , n, we have the bound∣∣∣∣

∫Aj

ωδ(λ, S;F) dλ −∫Aj

ω(λ, S;F) dλ

∣∣∣∣ � p|Aj | (28)

(cf. Equation (13)). That such δ exists follows from Lemma 1. A-priori, we candefine δ separately for each value of j ; thus δ is a function of j . However, bytaking the minimum value of δ(j) as j runs from 1 to n, we may assume δ isindependent of j .

We shall complete the proof of Equation (26) by showing that, for j � 1,(i): 1

π

∫Aj

θ(mN+(λ), S) dλ is close to the integral

1

π

∫Aj

θ

(u′(N, λ) + m(j)v′(N, λ)

u(N, λ) + m(j)v(N, λ), S

)dλ,

and that(ii): |{λ ∈ Aj ; v′(N, λ)/v(N, λ) ∈ S}| is close to the same integral for all N

sufficiently large. We deal successively with (i) and (ii). Precise statements of (i)and (ii) are made in (29) and (31) below.

Proof of (i). From Equation (5), we have

mN+(λ) = u′(N, λ) + m+(λ)v′(N, λ)

u(N, λ) + m+(λ)v(N, λ).

Hence, for fixed N and λ, the mapping from m+(λ) to mN+(λ) is a Möbius trans-formation with real coefficients and discriminant uv′ − vu′ = 1. From Lemma 2,which asserts the invariance of γ (·, ·) under such Möbius transformations, we seethat the bound (27) implies that

γ

(mN

+(λ),u′(N, λ) + m(j)v′(N, λ)

u(N, λ) + m(j)v(N, λ)

)� p, for j � 1 and λ ∈ Aj .

We can now use Equation (17) to deduce a corresponding bound for angle sub-tended, namely∣∣∣∣θ(mN

+(λ), S) − θ

(u′(N, λ) + m(j)v′(N, λ)

u(N, λ) + m(j)v(N, λ), S

)∣∣∣∣ � πp,

and integration with respect to λ over Aj leads to the bound∣∣∣∣ 1

π

∫Aj

θ(mN+(λ), S) dλ − 1

π

∫Aj

θ

(u′(N, λ) + m(j)v′(N, λ)

u(N, λ) + m(j)v(N, λ), S

)dλ

∣∣∣∣ � p|Aj |,(29)


which holds for j = 1, 2, . . . , n.

Proof of (ii). For j � 1, define the subset Aδj of C+, consisting of all z ∈ C+

of the form z = λ + iδ, for λ ∈ Aj . Thus Aδj is the translation of Aj by distance

δ above the real z-axis. Since Aj is bounded, Aδj is contained in a compact subset

of C+. Hence the corollary to Lemma 3 implies that a positive constant N0 can befound such that, for j � 1, N � N0 and z ∈ Aδ

j , we have the bound

γ

(−v′(N, z)

v(N, z),−u′(N, z) + m (j)v′(N, z)

u(N, z) + m (j)v(N, z)

)� p. (30)

As in the case of the constant δ, we may choose N0 to be independent of j . Fol-lowing a similar argument to that in the proof of (i) above, we may convert the γ

estimate into an estimate of angle subtended. Setting z = λ + iδ and integratingwith respect to λ over Aj , we have the bound∣∣∣∣

∫Aj

1

πθ

(−v′(N, λ + iδ)

v(N, λ + iδ),−S

)dλ−

−∫Aj

1

πθ

(−u′(N, λ + iδ) + m (j)v′(N, λ + iδ)

u(N, λ + iδ) + m (j)v(N, λ + iδ),−S

)dλ

∣∣∣∣ � p|Aj |,

valid for j � 1 and N � N0. Here −S is the set −S = {λ ∈ R;−λ ∈ S}.Each of the two integrals above is of the form

∫Aj

ωδ(λ,−S;F) dλ for some F ;namely

F = F1 = −v′(N, ·)v(N, ·) ,

F = F2 = −u′(N, ·) + m (j)v′(N, ·)u(N, ·) + m (j)v(N, ·) .

Using (28) in each case, we may compare the difference between these two inte-grals with the corresponding difference in the limit δ → 0+. Noting that

limδ→0+

1

π

∫Aj

θ

(−v′(N, λ + iδ)

v(N, λ + iδ),−S

)dλ

=∣∣∣∣{λ ∈ Aj ;−v′(N, λ)

v(N, λ)∈ −S

}∣∣∣∣=

∣∣∣∣{λ ∈ Aj ; v

′(N, λ)

v(N, λ)∈ S

}∣∣∣∣,we arrive at the bound∣∣∣∣∣

∣∣∣∣{λ ∈ Aj ; v

′(N, λ)

v(N, λ)∈ S

}∣∣∣∣ − 1

π

∫Aj

θ

(−u′(N, λ) + m(j)v′(N, λ)

u(N, λ) + m(j)v(N, λ),−S

)dλ

∣∣∣∣∣� 3p|Aj |.


We may use the identity θ(−ω,−S) = θ( ω, S) to rewrite this inequality in theform ∣∣∣∣∣

∣∣∣∣{λ ∈ Aj ; v

′(N, λ)

v(N, λ)∈ S

}∣∣∣∣ − 1

π

∫Aj

θ

(u′(N, λ) + m(j)v′(N, λ)

u(N, λ) + m(j)v(N, λ), S

)dλ

∣∣∣∣∣� 3p|Aj |, (31)

which holds for all j � 1 and N � N0, and completes the proof of (ii).Combining inequalities (29) and (31) now yields, for j � 1 and N � N0,∣∣∣∣∣

∣∣∣∣{λ ∈ Aj ; v

′(N, λ)

v(N, λ)∈ S

}∣∣∣∣ − 1

π

∫Aj

θ(mN+(λ), S) dλ

∣∣∣∣∣ � 4p|Aj |. (32)

Noting that A0 was chosen such that |A0| � p|A|, we now have, for all N � N0,∣∣∣∣∣∣∣∣{λ ∈ A; v

′(N, λ)

v(N, λ)∈ S

}∣∣∣∣ − 1

π

∫A

θ(mN+(λ), S) dλ

∣∣∣∣�

n∑j=0

∣∣∣∣∣∣∣∣∣{λ ∈ Aj ; v

′(N, λ)

v(N, λ)∈ S

}∣∣∣∣ − 1

π

∫Aj

θ(mN+(λ), S) dλ

∣∣∣∣

� 2|A0| + 4pn∑

j=1

|Aj | � 6p|A|. (33)

Since p > 0 was arbitrary, we have verified Equation (26), and the theoremfollows.

Remarks. There is an interesting variant of the argument leading up to theinequality (31) in the proof of the theorem. For each j = 1, 2, . . . , n supposethat a function m(j)(z) can be found, analytic in the lower half-plane with positiveimaginary part, and such that

γ (m+(λ),m(j)− (λ)) � p (all λ ∈ Aj , j � 1),

where m(j)− (λ) is the lower boundary value of m(j)(z), defined by m

(j)− (λ) =

limε→0+ m(j)(λ − iε).Thus (27) corresponds to the special case in which each m(j) is a constant

function. Then (29) holds with m(j) replaced by m(j)− (λ) and (30) holds with m (j)

replaced by m (j)( z ). (Note that m (j)( z ) for Im z > 0 defines an analytic functionwith negative imaginary part.) We can then carry through the proof of (31) withagain m(j) replaced by m

(j)− (λ), and the conclusions (32) and (33) of the proof

follow as before. This more general variant of the argument used in the proof ofTheorem 1 is of particular interest in view of the possibility of taking m(j)(z) to bethe m-function for the zero potential, in which case m

(j)+ (λ) is both an upper and a

lower boundary value, for λ > 0.


On examination of the proof of Theorem 1, it will be found that all of thebounds are uniform over Borel subsets S of R. It follows that Equation (26) re-mains valid if we replace the single Borel set S by an arbitrary family {SN } ofBorel sets, parametrised by N . If the family {SN } is chosen in such a way that1/π

∫Aθ(mN+(λ), SN) dλ converges to a limit as N → ∞ (this can always be

done), then the measure |{λ ∈ A; v′(N, λ)/v(N, λ) ∈ SN }| will converge tothe same limit. We may then characterise the asymptotic value distribution ofv′(N, λ)/v(N, λ) by considering such families {SN } and their corresponding lim-iting measures.

5. Applications

More detailed applications of the results of this paper will be considered elsewhere.Here, as an illustration of the main ideas, we shall refer briefly to three impor-tant classes of potential, namely slowly decaying potentials, potentials unboundedbelow at infinity, and sparse potentials.

(i) Slowly decaying potentials. We consider in particular the class of potentialssubject to a pointwise bound |V (x)| � const. × x−α in the limit x → ∞, for someconstant α > 1/2 (see [5–8]). Such a bound implies, for almost all λ > 0, theexistence of a solution f (x, λ) of Equation (1), satisfying WKB-type asymptotics,and in particular having the property that limN→∞ f ′(N, λ)/f (N, λ) = i

√λ. From

the results of Amrein and Pearson in [3], the solutions f (x, λ) may be used toconstruct families of solutions of the associated Riccati equation which exhibitthe clustering property. (For definition of clustering and related ideas, see [3].)In particular, for almost all λ > 0, we have limN→∞ mN+(λ) = i

√λ, with the

consequence that, in the conclusion of Theorem 1, Equation (26) may be written,for Borel subsets A of R+ having finite Lebesgue measure,

limN→∞

∣∣∣∣{λ ∈ A; v

′(N, λ)

v(N, λ)∈ S

}∣∣∣∣ = 1

π

∫A

θ(i√λ, S) dλ.

(ii) Potentials unbounded at infinity. We consider, as an example of this type,the class of potentials Vβ(x) = −βx2 (β > 0). This class of potentials has beendiscussed in [3]. Proceeding along similar lines to (i) above, it may be shown thatfor all λ ∈ R we have mN+(λ) ∼ iβ1/2N in the limit N → ∞. As a consequence,defining the family {SN } of Borel sets by SN = NS ≡ {Nλ;λ ∈ S}, we have theasymptotic formula

limN→∞

∣∣∣∣{λ ∈ A; v

′(N, λ)

v(N, λ)∈ SN

}∣∣∣∣ = 1

π

∫A

θ(iβ1/2, S) dλ.

(Note also that v′(N, λ)/(Nv(N, λ)) converges in value distribution as N → ∞.)(iii) Sparse potentials. Applications to sparse potentials will be further dis-

cussed in BP2000. There is an extensive literature on this class of potentials; see[13] and other references contained therein. We say that V is a sparse potential if


there exists a sequence of intervals {(ak, bk)} ≡ {Ik}, having length lk = (bk − ak),such that lk → ∞ as k → ∞, and such that V (x) ≡ 0 for x ∈ Ik. Given such asequence of intervals, we have the following lemma (for proof, see BP2000):

LEMMA 4. Let V be a sparse potential, and {(ak, bk)} = {Ik} any correspondingsequence of intervals on which V = 0, with length lk → ∞. Then if A and S areBorel subsets of R, with A bounded, it follows that

limk→∞

1

π

∫A

θ(mak+ (λ), S) dλ = 1

π

∫A

θ(i√λ, S) dλ; (34)

limk→∞

∣∣∣∣{λ ∈ A; v

′(bk, λ)v(bk, λ)

∈ S

}∣∣∣∣ = 1

π

∫A

θ(i√λ,−S) dλ. (35)

Here mak(z) is the Dirichlet m-function for the interval [ak,∞). If mak+ (λ) is real,then (1/π)θ(mak+ (λ), S) is to be interpreted as the characteristic function of the set{λ;mak+ (λ) ∈ S}.

Equations (34) and (35) play a key role in understanding asymptotic value dis-tribution for Schrödinger operators with sparse potentials, and its implications forspectral theory. Since we could redefine the sequence of intervals {Ik}, replacingeach interval (ak, bk) in the sequence by a pair of intervals (ak, ck), (ck, bk), withck = (ak + bk)/2, Equations (34) and (35) remain valid with ak , respectively bk,replaced by ck on the left hand side. If A is a subset of R− and S is a subset of R+ forwhich the right hand side of (35) is nonzero, this implies that v′(ck, λ)/v(ck, λ) andm

ck+ (λ) should have different asymptotic value distribution, for λ ∈ A. On the otherhand, if in addition A is a subset of an essential support of the spectral measureµa.c., then according to Theorem 1, the asymptotic value distributions should bethe same. Hence, in fact there can be no absolutely continuous measure for λ < 0.

Equations (34) and (35), together with Theorem 1, may also be used to prove,for various 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 in thispaper can easily be extended to include such distributional potentials.) We can thendefine a sequence of intervals {Ik}, with Ik = (xk, xk+1), and let A ⊂ R+ be a sub-set of an essential support of µa.c.. Noting that θ(i

√λ, S) = θ(i

√λ,−S) for λ > 0,

Equations (34) and (35), together with Theorem 1, 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 intervals Ik.


References

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

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

3. Amrein, W. O. and Pearson, D. B.: Stability criteria for the Weyl m-function, Math. Phys. Anal.Geom. 1 (1998), 193–221.

4. Eastham, M. S. P.: The asymptotic form of the spectral function in Sturm-Liouville problemswith a large potential like −xc (c � 2), Proc. Roy. Soc. Edinburgh A 128 (1998), 37–45.

5. Kiselev, A.: Absolutely continuous spectrum of one-dimensional Schrödinger operators andJacobi matrices with slowly decreasing potentials, Comm. Math. Phys. 179 (1996), 377–400.

6. Christ, M., Kiselev, A. and Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Math. Res. Lett. 4(5) (1997),719–723.

7. Christ, M. and Kiselev, A.: Absolutely continuous spectrum of one-dimensional Schrödingeroperators with slowly decreasing potentials: some optimal results, J. Amer. Math. Soc. 11(1998), 771–797.

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10. Anderson, J. W.: Hyperbolic Geometry, Springer, London, 1999.11. Jost, J.: Compact Riemann Surfaces, Springer, Berlin, 1997.12. Pommerenke, C.: Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992.13. Simon, B. and Stolz, G.: Operators with singular continuous spectrum: sparse potentials, Proc.

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Mathematical Physics, Analysis and Geometry 3: 405–406, 2000. 405

Contents of Volume 3

Volume 3 No. 1 2000

A. KHORUNZHY and G. J. RODGERS / Asymptotic Distribution ofEigenvalues of Weakly Dilute Wishart Matrices 1–31

A. PANKOV and K. PFLÜGER / On Ground-Traveling Waves for theGeneralized Kadomtsev–Petviashvili Equations 33–47

BERNARD DECONINCK and HARVEY SEGUR / Pole Dynamics forElliptic Solutions of the Korteweg–deVries Equation 49–74

YU. AMINOV and A. SYM / On Bianchi and Bäcklund Transforma-tions of Two-Dimensional Surfaces in E4 75–89

AMÉDÉE DEBIARD and BERNARD GAVEAU / The Ground State ofCertain Coulomb Systems and Feynman–Kac Exponentials 91–100

Volume 3 No. 2 2000

M. M. KIPNIS / Periodic Ground State Configurations in a One-Dimensional Hubbard Model of Statistical Mechanics 101–115

P. AGRANOVICH / Polynomial Asymptotic Representation of Subhar-monic Functions in a Half-Plane 117–138

G. FELDER, Y. MARKOV, V. TARASOV and A. VARCHENKO /Differential Equations Compatible with KZ Equations 139–177

DMITRY SHEPELSKY / A Riemann–Hilbert Problem for Propagationof Electromagnetic Waves in an Inhomogeneous, Dispersive �

Waveguide 179–193

406 CONTENTS OF VOLUME 3

Volume 3 No. 3 2000

GIANLUCA GEMELLI / Second-Order Covariant Tensor Decomposi-tion in Curved Spacetime 195–216

ZIED AMMARI / Asymptotic Completeness for a Renormalized Non-relativistic Hamiltonian in Quantum Field Theory: The NelsonModel 217–285

I. K. KHANNA, V. SRINIVASA BHAGAVAN and M. N. SINGH /Generating Relations of the Hypergeometric Functions by theLie Group-Theoretic Method 287–303

Volume 3 No. 4 2000

SPYROS PNEVMATIKOS and DIMITRIS PLIAKIS / Gauge Fieldswith Generic Singularities 305–321

JEAN MEINGUET / An Electrostatics Approach to the Determinationof Extremal Measures 323–337

ALEXANDRE STOJANOVIC / Universality in Orthogonal and Sym-plectic Invariant Matrix Models with Quartic Potential 339–373

HAROLD WIDOM / On the Solution of a Painlevé III Equation 375–384

S. V. BREIMESSER and D. B. PEARSON / Asymptotic Value Distrib-ution for Solutions of the Schrödinger Equation 385–403

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