Quantum ergodicity. Nalini Anantharaman. 22 août Université de Strasbourg

Quantum ergodicity Nalini Anantharaman Université de Strasbourg 22 août 2016

I. Quantum ergodicity on manifolds. II. QE on (discrete) graphs : regular graphs. III. QE on graphs : other models, perspectives.

Figure: A few eigenfunctions of the Bunimovich billiard (Heller, 89).

A survey of QE on manifolds M a compact riemannian manifold, of dimension d. ψ k = λ k ψ k ψ k L 2 (M) = 1 in the limit λ k +. We study the weak limits of the probability measures on M, ψ k (x) 2 dvol(x) λ k +.

Most results could be generalized to more general Schrödinger equations ( 2 2 + V (x) ) ψ (x) = E ψ (x), in the semiclassical limit 0, E E 0 R. This question is linked with the ergodic theory for the geodesic flow / billiard flow / Hamiltonian flow. Hence the name quantum ergodicity.

Let (ψ k ) k N be an orthonormal basis of L 2 (M), with QE theorem (simplified) : ψ k = λ k ψ k, λ k λ k+1. Theorem (Shnirelman 74, Zelditch 85, Colin de Verdière 85) Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a C 0 (M). Then 1 N(λ) λ k λ M a(x) ψ k (x) 2 dvol(x) M a(x)dvol(x) 2 0. λ

Equivalently, there exists a subset S N of density 1, such that a(x) ψ k (x) 2 dvol(x) a(x)dvol(x). M k S k + M

Equivalently, there exists a subset S N of density 1, such that a(x) ψ k (x) 2 dvol(x) a(x)dvol(x). M k S k + M Equivalently, in the weak topology. ψ k (x) 2 dvol(x) k S dvol(x) k +

The full statement uses analysis on phase space, i.e. T M = {(x, ξ), x M, ξ T x M}. For a = a(x, ξ) a reasonable function on T M, we can define an operator on L 2 (M), a(x, D x ) (D x = 1 i x) Say a S 0 (T M) if a is smooth and 0-homogeneous in ξ (i.e. a is a smooth fn on the sphere bundle SM).

A few rules of pseudodifferential calculus M = R d, a = a(x, ξ) : R d R d C. Say a S k (T R d ) if a is smooth and k-homogeneous in ξ. Define a(x, D x )u = 1 (2π) d a (x, ξ) e iξ x û(ξ)dξ. Proposition (i) If k = 0, a(x, D x ) extends to a bounded operator on L 2 (R d ) ; more generally, a(x, D x ) sends H s to H s k. (ii) [a(x, D x ), b(x, D x )] = {a, b} (x, D x ) + l.o.t. In particular for b(x, D x ) =, [ ( ), ξ a(x, Dx )] = ξ xa (x, D x ) + l.o.t.

For a S 0 (T M), we consider ψ k = λ k ψ k, λ k λ k+1. ψ k, a(x, D x )ψ k L 2 (M).

For a S 0 (T M), we consider ψ k = λ k ψ k, λ k λ k+1. ψ k, a(x, D x )ψ k L 2 (M). This amounts to M a(x) ψ k(x) 2 dvol(x) if a = a(x).

Let (ψ k ) k N be an orthonormal basis of L 2 (M), with QE theorem : ψ k = λ k ψ k, λ k λ k+1. Theorem (Shnirelman, Zelditch, Colin de Verdière) Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a(x, ξ) S 0 (T M). Then 1 N(λ) ψ k, a(x, D x )ψ k L 2 (M) λ k λ ξ =1 2 a(x, ξ)dxdξ 0.

Recall : the geodesic flow φ t : SM SM (t R) is the flow generated by the vector field ξ ξ x. Ergodicity of the geodesic flow means that the only φ t -invariant L 2 functions on SM are the constant functions. Equivalently : the only L 2 -functions a on SM such that ξ ξ xa = 0 are the constant functions.

Idea of proof. For any bounded operator K on L 2 (M), define the quantum variance Var λ (K) = 1 N(λ) ψ k, Kψ k L 2 2 (M). λ k λ

Idea of proof. For any bounded operator K on L 2 (M), define the quantum variance Var λ (K) = 1 N(λ) ψ k, Kψ k L 2 2 (M). λ k λ The proof start from the trivial observation that Var λ ([f ( ), K]) = 0 for any K, for any real function f. [f ( ), K] = f ( )K Kf ( ).

The proof start from the trivial observation that for any K. Var λ ([, K]) = 0 In addition, if K = a(x, D x ) is a pseudodifferential operator with a S 0 (T M), then where [, a(x, D x )] = b(x, D x ) b(x, ξ) = ( ) ξ ξ xa (x, ξ) + r(x, ξ) where r is 1-homogeneous in ξ and ξ ξ x is the derivative along the geodesic flow.

This implies that Var λ ( ( ξ ξ xa)(x, D x ) ) 0. λ +

This implies that Var λ ( ( ξ ξ xa)(x, D x ) ) 0. λ + If the geodesic flow is ergodic, this implies (density argument) Var λ (a(x, D x )) 0 λ + if a has zero mean.

This implies that Var λ ( ( ξ ξ xa)(x, D x ) ) 0. λ + In addition, Var λ (a(x, D x )) C ξ =1 a(x, ξ) 2 dxdξ. If the geodesic flow is ergodic, this implies (density argument) Var λ (a(x, D x )) 0 λ + if a has zero mean.

[Arnd Bäcker]

Introduction. Examples / counter-examples The Quantum Unique Ergodicity conjecture

Hassell s result on stadium billiards (2008) For almost every stadium, there exists a sequence of eigenfunctions that does not equidistribute over phase space.

The flat torus T d = R d /2πZ d Consider the special example φ k (x) = e ik.x, k Z d, φ k = k 2 φ k. These are joint eigenfunctions of, D x1,..., D xd.

The flat torus T d = R d /2πZ d Consider the special example φ k (x) = e ik.x, k Z d, φ k = k 2 φ k. These are joint eigenfunctions of, D x1,..., D xd. Consider the limit k +, k/ k ξ 0. One shows easily that φ k, a(x, D x )φ k L 2 (T d ) a(x, ξ 0 )dx. x T d QE does not hold for the eigenfunctions (φ k ).

The sphere. The eigenfunctions of S 2 are spherical harmonics, i.e. restrictions to S 2 R 3 of harmonic homogeneous polynomials in 3 variables. Harmonic homogeneous polynomials of degree l give rise to eigenfunctions of S 2 for the eigenvalue l(l + 1) (the dimension of the eigenspace is 2l + 1).

S 2 commutes ( with the infinitesimal rotations J 12 = 1 i x 1 x 2 x 2 x 1 ). The basis (Y m l ) l 0, m l of joint eigenfunctions of S d and J 12 cannot satisfy QE.

Random eigenbases (Zelditch) The set of bases of eigenfunctions of S 2 is in bijection with U(1) U(3) U(5) U(7).... Theorem (Zelditch 92, VanDerKam 97). Almost every choice of eigenbasis satisfies QE (even QUE).

Efns concentrating on a closed geodesic One can easily exhibit sequences of eigenfunctions concentrating on an equator. From that, Jakobson and Zelditch deduced that any convex combination of equators can be achieved as a limit of eigenfunctions.

The flat torus, R d /2πZ d revisited φ λ (x) = k Z d, k 2 =λ c k e ik.x. (Jakobson, Bourgain 97, Jaffard 90, A-Fermanian-Macià, A-Macià 2012) It s not possible for a sequence of eigenfunctions to concentrate on a closed geodesic. Assume φ λ (x) 2 dx ν(dx) λ + Then ν is absolutely continuous, i.e. it has a density ν(dx) = ρ(x)dx. Moreover, for any non-empty open Ω T d, there exists c(ω) > 0 such that ν(ω) c(ω).

The flat torus, R d /2πZ d revisited φ λ (x) = c k e ik.x. k Z d, k 2 =λ (Jakobson, Bourgain 97, Jaffard 90, A-Fermanian-Macià, A-Macià 2012) It s not possible for a sequence of eigenfunctions to concentrate on a closed geodesic. Assume φ λ (x) 2 dx λ + ν(dx) Remark A full characterization of the set of limit measures is still missing.

The Quantum Unique Ergodicity conjecture QUE conjecture : Conjecture (Rudnick, Sarnak 94) On a negatively curved manifold, we have convergence of the whole sequence : ψ k, a(x, D x )ψ k L 2 (M) (x,ξ) SM a(x, ξ)dxdξ. Proven by E. Lindenstrauss in the special case of arithmetic congruence surfaces, for joint eigenfunctions of the Laplacian and the Hecke operators.

Entropy of eigenfunctions Assume ψ k, a(x, D x )ψ k L 2 (M) (x,ξ) SM a(x, ξ)dµ(x, ξ). If M has negative curvature, A-Nonnenmacher (06) proved that µ must have positive Kolmogorov-Sinai entropy.

µ h KS (µ) 0 has the following properties : A measure supported by a periodic trajectory has zero entropy. According to the Ruelle-Pesin inequality, h KS (µ) d 1 λ + k dµ, k=1 where the functions λ + 1 (ρ) λ+ 2 (ρ) λ+ d 1 (ρ) > 0 are the positive Lyapunov exponents of the flow. Equality iff µ is the Liouville measure (Ledrappier-Young). On a manifold of constant curvature 1, the inequality reads h KS (µ) d 1. The functional h KS is affine on the convex set of invariant probability measures.

Assume for all a ψ k, a(x, D x )ψ k L 2 (M) A (06) proved that h KS (µ) > 0. (x,ξ) SM a(x, ξ)dµ(x, ξ). In constant curvature 1, A-Nonnenmacher (06) proved the explicit bound h KS (µ) d 1 2. As a corollary, the support of µ has Hausdorff dimension d. Conjectured bound in variable curvature : h KS (µ) 1 2 d 1 λ + k dµ. k=1 (proven by G. Rivière for surfaces of non-positive curvature).

Brin and Katok s definition of entropy. For any time T > 0, introduce a distance on SM, d T (ρ, ρ ) = max t [ T /2,T /2] d(φt ρ, φ t ρ ), Denote by B T (ρ, ɛ) the ball of centre ρ and radius ɛ for the distance d T. Let µ be a φ t invariant probability measure on SM. Then, for µ-almost every ρ, the limit lim ɛ 0 lim inf 1 T + T log µ( B T (ρ, ɛ) ) def = h KS (µ, ρ) exists (local entropy). The Kolmogorov-Sinai entropy is the average of the local entropies : h KS (µ) = h KS (µ, ρ)dµ(ρ).

Open questions for negatively curved manifolds : Assume for all a, b, ψ k, a(x, D x )ψ k L 2 (M) M Is µ ergodic? (x,ξ) SM b(x) ψ k (x) 2 d Vol(x) ν = π µ M a(x, ξ)dµ(x, ξ). b(x)dν(x). can µ have ergodic components of zero entropy? for instance, can µ have a component carried on a closed geodesic? Is ν absolutely continuous? does it put positive mass on any open set?

Recent interest in local equidistribution M. Berry s semiclassical eigenfunction hypothesis. Q : let (ψ k ) be an ONB of eigenfunctions of. Let r k = r(λ k ) 0. Do we have 1 sup ψ x 0 k (x) 2 d Vol(x) 1 Vol(B(x 0, r k )) B(x 0,r k ) k 0? Hezari-Rivière, Han 2015 : negatively curved manyfolds, r k = ln λ α k, along a density 1 subsequence. Lester-Rudnick 2015, T d, d 2, r k = λ α k (α < 1 2(d 1) ), along a density 1 subsequence. Han 2015, T d, d 5, r k = λ α k eigenbases. (α < d 2 4d ), a.s. for random