Quantum ergodicity Nalini Anantharaman Universit e de Strasbourg - PowerPoint PPT Presentation

QE on manifolds QE on discrete graphs Perspectives on random matrices Quantum ergodicity Nalini Anantharaman Universit´ e de Strasbourg 24 mars 2016

QE on manifolds QE on discrete graphs Perspectives on random matrices Quantum ergodicity on manifolds (comparing < 0 curvature, > 0 curvature and 0 curvature). QE on large graphs.

QE on manifolds QE on discrete graphs Perspectives on random matrices

QE on manifolds QE on discrete graphs Perspectives on random matrices Figure : A few eigenfunctions of the Bunimovich billiard (Heller, 89).

QE on manifolds QE on discrete graphs Perspectives on random matrices 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 d Vol ( x ) λ k − → + ∞ .

QE on manifolds QE on discrete graphs Perspectives on random matrices This question is linked with the ergodic theory for the geodesic flow / billiard flow. Hence the name quantum ergodicity.

QE on manifolds QE on discrete graphs Perspectives on random matrices Let ( ψ k ) k ∈ N be an orthonormal basis of L 2 ( M ), with − ∆ ψ k = λ k ψ k , λ k ≤ λ k +1 . QE theorem (simplified) : Theorem (Shnirelman 74, Zelditch 85, Colin de Verdi` ere 85) Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a ∈ C 0 ( M ) . Then 2 � � 1 � � � a ( x ) | ψ k ( x ) | 2 d Vol ( x ) − � � a ( x ) d Vol ( x ) − →∞ 0 . → � � N ( λ ) λ − � � M M λ k ≤ λ

QE on manifolds QE on discrete graphs Perspectives on random matrices Equivalently, there exists a subset S ⊂ N of density 1, such that � � k ∈S a ( x ) | ψ k ( x ) | 2 d Vol ( x ) − − − − − → a ( x ) d Vol ( x ) . k − → + ∞ M M

QE on manifolds QE on discrete graphs Perspectives on random matrices Equivalently, there exists a subset S ⊂ N of density 1, such that � � k ∈S a ( x ) | ψ k ( x ) | 2 d Vol ( x ) − − − − − → a ( x ) d Vol ( x ) . k − → + ∞ M M Equivalently, k ∈S | ψ k ( x ) | 2 d Vol ( x ) − − − → + ∞ d Vol ( x ) − − → k − in the weak topology.

QE on manifolds QE on discrete graphs Perspectives on random matrices 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 ), ( D x = 1 a ( x , D x ) 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 ).

QE on manifolds QE on discrete graphs Perspectives on random matrices − ∆ ψ k = λ k ψ k , λ k ≤ λ k +1 . For a ∈ S 0 ( T ∗ M ), we consider � ψ k , a ( x , D x ) ψ k � L 2 ( M ) .

QE on manifolds QE on discrete graphs Perspectives on random matrices − ∆ ψ k = λ k ψ k , λ k ≤ λ k +1 . For a ∈ S 0 ( T ∗ M ), we consider � ψ k , a ( x , D x ) ψ k � L 2 ( M ) . M a ( x ) | ψ k ( x ) | 2 d Vol ( x ) if a = a ( x ). � This amounts to

QE on manifolds QE on discrete graphs Perspectives on random matrices Let ( ψ k ) k ∈ N be an orthonormal basis of L 2 ( M ), with − ∆ ψ k = λ k ψ k , λ k ≤ λ k +1 . QE theorem : Theorem (Shnirelman, Zelditch, Colin de Verdi` ere) Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a ( x , ξ ) ∈ S 0 ( T ∗ M ) . Then 2 � � 1 � � � � � � ψ k , a ( x , D x ) ψ k � L 2 ( M ) − a ( x , ξ ) dxd ξ − → 0 . � � N ( λ ) � � | ξ | =1 � λ k ≤ λ

QE on manifolds QE on discrete graphs Perspectives on random matrices Recall : the geodesic flow φ t : SM − → SM ( t ∈ R ) is the flow ξ generated by the vector field | 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 | · ∂ x a = 0 are the constant functions.

QE on manifolds QE on discrete graphs Perspectives on random matrices Idea of proof. For any bounded operator K on L 2 ( M ), define the quantum variance 1 � 2 . � � � Var λ ( K ) = � � ψ k , K ψ k � L 2 ( M ) N ( λ ) λ k ≤ λ

QE on manifolds QE on discrete graphs Perspectives on random matrices Idea of proof. For any bounded operator K on L 2 ( M ), define the quantum variance 1 � 2 . � � � Var λ ( K ) = � � ψ k , K ψ k � L 2 ( M ) N ( λ ) λ k ≤ λ The proof start from the trivial observation that Var λ ([ f (∆) , K ]) = 0 for any K , for any function f . [ f (∆) , K ] = f (∆) K − Kf (∆) .

QE on manifolds QE on discrete graphs Perspectives on random matrices The proof start from the trivial observation that √ Var λ ([ − ∆ , K ]) = 0 for any K . In addition, if K = a ( x , D x ) is a pseudodifferential operator with a ∈ S 0 ( T ∗ M ), then √ [ − ∆ , a ( x , D x )] = b ( x , D x ) where � ξ � b ( x , ξ ) = | ξ | · ∂ x a ( x , ξ ) + r ( x , ξ ) ξ where r is − 1-homogeneous in ξ and | ξ | · ∂ x is the derivative along the geodesic flow.

QE on manifolds QE on discrete graphs Perspectives on random matrices This implies that � ( ξ � Var λ | ξ | · ∂ x a )( x , D x ) − → + ∞ 0 . → λ −

QE on manifolds QE on discrete graphs Perspectives on random matrices This implies that � ( ξ � Var λ | ξ | · ∂ x a )( x , D x ) − → + ∞ 0 . → λ − In addition, for any a , � | a ( x , ξ ) | 2 dxd ξ. Var λ ( a ( x , D x )) ≤ C | ξ | =1

QE on manifolds QE on discrete graphs Perspectives on random matrices This implies that � ( ξ � Var λ | ξ | · ∂ x a )( x , D x ) − → + ∞ 0 . → λ − In addition, for any a , � | a ( x , ξ ) | 2 dxd ξ. Var λ ( a ( x , D x )) ≤ C | ξ | =1 If the geodesic flow is ergodic, this implies Var λ ( a ( x , D x )) − → + ∞ 0 → λ − if a has zero mean.

QE on manifolds QE on discrete graphs Perspectives on random matrices [Arnd B¨ acker]

QE on manifolds QE on discrete graphs Perspectives on random matrices

QE on manifolds QE on discrete graphs Perspectives on random matrices 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 ) dxd ξ.

QE on manifolds QE on discrete graphs Perspectives on random matrices 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 ) dxd ξ. Proven by E. Lindenstrauss in the special case of arithmetic congruence surfaces, for joint eigenfunctions of the Laplacian and the Hecke operators.

QE on manifolds QE on discrete graphs Perspectives on random matrices A-Nonnenmacher (06) proved a weaker statement valid in greater generality. Let M have negative curvature. Assume � � ψ k , a ( x , D x ) ψ k � L 2 ( M ) − → a ( x , ξ ) d µ ( x , ξ ) ( x ,ξ ) ∈ SM Then µ must have positive Kolmogorov-Sinai entropy. For constant negative curvature, our result implies that the support of µ has dimension ≥ d = dim M .

QE on manifolds QE on discrete graphs Perspectives on random matrices Other geometries The sphere

QE on manifolds QE on discrete graphs Perspectives on random matrices The flat torus, R d / 2 π Z d (Jakobson-Bourgain 97, Jaffard 90, A-Fermanian-Maci` a, A-Maci` a 2012) It’s not possible for a sequence of eigenfunctions to concentrate on a closed geodesic. � c k e ik . x . φ λ ( x ) = k ∈ Z d , | k | 2 = λ 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 (independent on the sequence φ λ ) such that ν (Ω) ≥ c (Ω) .

QE on manifolds QE on discrete graphs Perspectives on random matrices QE on discrete graphs Since the 90s there has been the idea of using graphs as a testing ground/toy model for quantum chaos. Smilansky, Kottos, Elon,... Keating, Berkolaiko, Winn, Piotet, Gnutzmann Marklof...

QE on manifolds QE on discrete graphs Perspectives on random matrices Here we focus on the case of large regular (discrete) graphs. Let G = ( V , E ) be a ( q + 1)-regular graph. Discrete laplacian : f : V − → C , � � ∆ f ( x ) = ( f ( y ) − f ( x )) = f ( y ) − ( q + 1) f ( x ) . y ∼ x y ∼ x ∆ = A − ( q + 1) I

QE on manifolds QE on discrete graphs Perspectives on random matrices Sp ( A ) ⊂ [ − ( q + 1) , q + 1] Let | V | = N . We look at the limit N − → + ∞ .

QE on manifolds QE on discrete graphs Perspectives on random matrices Sp ( A ) ⊂ [ − ( q + 1) , q + 1] Let | V | = N . We look at the limit N − → + ∞ . We assume that G N has “few” short loops (= converges to a tree in the sense of Benjamini-Schramm).