THE GEOMETRIC CONTEXT OF THE NODALSEQUENCE: SURFACESOF REVOLUTION

Uzy Smilansky, Sven Gnutzmann and Panos Karageorge

Nodal Week Workshop, April 4th, 2006

Department of Physics of Complex Systems

Weizmann Institute of Science

Department of Mathematics, University of Bristol

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STATEMENT OF GENERAL PROBLEM

• Can one count the shape (up to scaling) of a separable(thus, integrable) surface?

nodal sequence → shape : one to one?

• separable surfaces (2-manifolds): elliptic billiards, certainpolygonic billiards, surfaces of revolution, Liouville surfaces

• moving from spectral sequence to nodal sequence: numberof nodal domains of wavefunction of a certain state

• no direct reference to spectrum (only its ordering)• nodal sequence is a sequence of natural numbers (scale

invariant)

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GEODESICFLOWS ON SURFACES

• configuration space,M: analytic Riemannian surface

• in local coordinates(q1, q2),

gq =∑

αβ

gαβ(q)dqα ⊗ dqβ

inner product on tangent planes,〈., .〉q := gq(., .)

• Lagrangian

L(q, v) :=1

2〈v, v〉q

• classical orbits: geodesic curves

Lagrange equations ⇐⇒ geodesic equations

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SIMPLE SURFACESOF REVOLUTION (SSR)

• M generated by rotation aboutx-axis off : [−1, 1] → R

• one ‘dimension’ ofM fixed: ||SN || = 2

• restrictions on generating curve:• f 2 analytic (avoid singularities atN,S)• f(x)2 = a±(1 ∓ x) + o(1 ∓ x) as |x| → 1− (∂M = ∅)• f is convex (single critical point-maximum)

• induced metric tensor is smooth, depends only onf(cylindrical coordinates,(q1, q2) = (x, ϕ))

g =(

1 + f ′(x)2)

dx2 + f(x)2dϕ2

• M satisfies Twist Hypothesis (TH)

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SIMPLE SURFACESOF REVOLUTION (SSR)

• one-parameter family of SSR: ellipsoids of revolution

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CLASSICAL DYNAMICS ON SSR

• flow onM integrable and separable• system is autonomous and rotationally invariant

• energyE, projection of angular momentum along axis ofrotationm

• OR: E and Clairaut integralc = m√2E

= f(x) sinα

• for givenE andm, motion is restricted between meridiansγ± (caustics) corresponding tox± (turning points forx-motion)

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CLASSICAL DYNAMICS ON SSR

• integrability⇒ invariant torii, local action-angle coordinates

(θ1, θ2, I1, I2)

• simplicity conditions⇒ a.a. coordinates are global

I1 =1

2π

∮

pϕdϕ = m, I2 =1

2π

∮

pxdx = n

•

n =1

π

∫ x+

x−

√

Ef(x)2 −m2

√

1 + f ′(x)2

f(x)dx

• turning points:Ef(x±)2 −m2 = 0

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CLASSICAL DYNAMICS ON SSR

• Hamiltonian homogeneous of order 2

h(λn, λm) = λ2h(n,m), λ ∈ R+

• TH:∂2h

∂n∂m(I) 6= 0 ∀(n,m)

• TH ⇒ ΓE := {(n,m) ∈ R2+|h(n,m) = E} on (positive)

action plane either convex or concave (monotonic)• h(n,m) = E inverted to given as function ofm

dn

dm(m) < 0,m ∈ [0,mmax]

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Γ CURVE

• due to homogeneity of hamiltonianΓ := Γ1 is the basicgeometric object of the formalism

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QUANTUM DYNAMICS ON SSR

• system is quantum integrable and separable

• ∆g onL2(M, µg) is self-adjoint, positive definite

• LB operator in l.c.

∆g = − 1√

|g|∑

αβ

∂

∂qα

(

√

|g|gαβ ∂

∂qβ

)

where|g| := det(gαβ)

• Schr̈odinger equation,∆gΨ = EΨ

• real, countable spectrum,0 = E1 < E2 ≤ E2 ≤ ...→ ∞• good quantum numbers: action variables(n,m) ∈ N0 × Z

• usual quantum numbers of spherical harmonics: c.t.(n,m) 7→ (l,m) = (n− |m|,m)

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QUANTUM DYNAMICS ON SSR

• metric rotational invariance→ rotational degeneracyEn,−m = Enm

• Bleher - 1993: spectral counting function for SSR,

N(E) =1

4πArea(M)E + E

1

4R(E)

whereR(E) :=

∑

γ∈Π

Aγ sin(`γ√E + σγ)

Π set of nontrivial, periodic (oriented) geodesics ofM• Zelditch - 2004: one can hear the shape of a SSR

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NODAL SEQUENCE

• separability⇒ Ψ(q) = ψ(x)eimϕ,m ∈ Z

• interested in nodal domains: real wavefunctionsψ(x) cosmϕ or ψ(x) sinmϕ

• number of nodal domains,νnm = (n+ 1)(2|m| + δm0)

• order spectrum: counting index

(n,m) 7→ k = N (n,m) := #{E ∈ Spec(∆g)|E ≤ Enm}

with N (n,m) = N(Enm)

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NODAL SEQUENCE

• from spectral sequence to nodal sequence: ordered numberof nodal domains

{νk}∞k=1, ν1 = 1 < νk, k > 1

• ordering arbitrariness in degeneracy classes• for any counting realization

• Courant - 1923:νk ≤ k, k ∈ N

• Pleijel - 1956: for planar billiards,

lim supk→∞νk

k≤

(

2j01

)2

= 0.69166... < 1

• distribution of normalized nodal sequence

ξk :=νk

k∈ (0, 1]

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NODAL DOMAIN NUMBER DISTRIBUTION

• chose counting convention,k = N (n,m)

• for Ig(E) := [E, (1 + g)E] with E, g > 0,

PIg(E)(ξ) :=1

NIg(E)

∑

k:Ek∈Ig(E)

δ(ξ − ξk)

• Blum, Gnutzmann, Smilansky - 2002: distribution has aclassical limitlimE→∞ PIg(E)(ξ) =: P (ξ)

• universal characteristics for separable systems• inverse square root singularity at someξmax ∈ (0, 1)• P (ξ) ≡ 0, ξ ≥ ξmax

• in d-surfaces,P (ξ) ∼ (ξmax − ξ)−d−3

2 as ξ → ξmax − 0

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L IMITING NODAL DISTRIBUTION

• limiting nodal distribution

P (ξ) =1

2A

∫

Γ

δ(ξ − m(s)n(s)

A )ds

whereA = Area{h(I) < 1} is positive quadrant

• ∫

[0,1]P (ξ)dξ = 1

• geometric interpretation:

• n(s)m(s): area of rectangles with one vertex on theorigin, one on(m(s),m(s)), and rest two onn andmaxes.

• probability density of scaled (by2A) areas of rectangleswith one point onΓ.

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L IMITING NODAL DISTRIBUTION

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L IMITING NODAL DISTRIBUTION

• by transforming backs 7→ m, we get

P (ξ) =1

4A

∫ mmax

0

∣

∣

∣n(m)−m · n′(m)

∣

∣

∣δ(ξ − m · n(m)

A )dm

P (ξ) = Θ(ξmax − ξ)∑

m: nmA

=ξ

1

4

∣

∣

∣

n(m) −mn′(m)

n(m) +mn′(m)

∣

∣

∣

and

P (ξ) =1

2Θ(ξmax − ξ)ξ

d

dξlog

m−(ξ)

m+(ξ)

wherem±(ξ) : m · n(m)/A ∈ (0, 1], andm−(ξ) → m+(ξ)asξ → ξmax

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L IMITING DISTRIBUTION: DERIVATION

• in the semiclassical limit (E → ∞)

PIg(E)(ξ) =1

2gAE

∫

R2+

χIg(E)(h(n,m))×

×δ(ξ − ξnm)dndm+O(1√E

)

whereξnm = νnm

N (n,m)

• semiclassical EBK energies:Escnm = h(n+ 1

2,m)

• N (n,m) = N(Enm) ≈ 2AEnm ≈ 2AEscnm

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L IMITING DISTRIBUTION: DERIVATION

• rescale action plane(n,m) 7→√E(n,m), and use

homogeneity of hamiltonian

PIg(E)(ξ) =1

2gA

∫

R2+

χIg(1)(h(n,m))×

×δ(ξ − nm

Ah(n,m))dndm+O(

1√E

)

• change variables,(n,m) 7→ (E , s) with

dE = ωndn+ ωmdm, ds =1

√

ω2n + ω2

m

(−ωmdn+ ωndm)

ωn := ∂h∂n, ωm := ∂h

∂m; ∂(E,s)

∂(m,n)= 1

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L IMITING DISTRIBUTION: DERIVATION

• limiting distribution exists, independent of spectral interval,

limE→∞

PIg(E)(ξ) =1

2A

∫

Γ

δ(ξ − m(s) · n(s)

A )ds =: P (ξ)

• complementary integrated distribution,

ΠIg(E)(ξ) := 1 −∫ ξ

0

PIg(E)(t)dt

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THE CASE OF THE SPHERE

• for the sphereξmax = 12

andP (ξ) = 12Θ(1

2− ξ) 1√

2−ξ

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GEOMETRIC INFORMATION

• for m = 0, semiclassical energiesEscn = π2

`2f

n2

PIg(E)(ξ) = P 0Ig(E)(ξ) + P̃Ig(E)(ξ) +O(

1

E)

where

P 0Ig(E)(ξ) =

1

2gAE∑

n−≤n≤n+

δ(ξ −`2f

2π2An+ 1

n2) = O(

1√E

)

andn− = [`f

π

√E], n+ = [

`f

π

√

(1 + g)E]

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GEOMETRIC INFORMATION

• complementary integrated distribution

Π0Ig(E)(ξ) = 1 − 1

2gAE∑

n−≤n≤n+

Θ(ξ −`2f

2π2A1

n) +O(

1

E)

• by Poisson’s summation theorem,

Π0Ig(E)(ξ) = 1 − 1

2gAE

∫ n+

n−

Θ(ξ −`2f

2π2A1

n)dn+O(

1

E)

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GEOMETRIC INFORMATION

• semiclassical asymptotics of nodal domain distributionreveals reveals the length of the generating curve

Π0Ig(E)(ξ) = 1 − 1

2gAE( `2f

2π2A1

ξ− [

`fπ

√E]

)

×

×Θ(1

ξ−2π2A

`2f[`fπ

√E])Θ(

2π2A`2f

[`fπ

√

(1 + g)E]−1

ξ)+O(

1

E)

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GEOMETRIC INFORMATION

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GEOMETRIC INFORMATION

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ELLIPSOIDS OF REVOLUTION

• generated by a semi-ellipse of eccentricityε > 0,

f(x) = ε√

1 − x2, x ∈ [−1, 1]

• 0 < ε < 1: prolate (cigar)• ε > 1: oblate (flattened on poles)• ε = 1: sphere

• hamiltonian: implicit function of action variables, in termsof complete elliptic integrals

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ELLIPSOIDS AS DEFORMEDSPHERES

• sphere highly degenerate due to high symmetry:deformation to ellipsoid→ lift of some degeneracies

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ELLIPSOIDS OF REVOLUTION: OSCILLATING

PART OF SPECTRAL COUNTING FUNCTION

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ONE CAN COUNT THE SHAPE OF AN ELLIP-SOID

• for ellipsoids one only has to look at the classical limit• ξmax is monotonic in the eccentricityε:

dξmax

dε(ε) < 0, ε > 0

• the limiting distribution determines the geometry of theellipsoid

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ONE CAN COUNT THE SHAPE OF AN ELLIP-SOID

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FUTURE PERSPECTIVES

• proof of uniqueness of inversion for ellipsoids• obtain geometric information about other separable surfaces

• convex surfaces of revolution with non-empty boundary• Zoll surfaces• Liouville surfaces

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