Scarring on invariant manifolds for quantum maps on the torus DUBI - PDF document

Scarring on invariant manifolds for quantum maps on the torus DUBI KELMER Banff workshop Quantum chaos: Routes to RMT and beyond February 2008

Overview • Quantum ergodicity and scarring • Quantum maps on the torus • Scarring on periodic orbits • Scarring on invariant manifolds 1

Quantization • Classical flow: Φ t : X → X • Unitary flow U t on a Hilbert space H h • Quantum states ψ ∈ H h interpreted as dis- tribution W ψ on X . • Semi-classical limit h → 0 retrieve classical behavior ( W U t ψ ∼ W ψ ◦ φ t ) • If ψ is an eigenfunction then in semiclassi- cal limit W ψ becomes Φ t invariant. • What are the possible limiting (invariant) measures? 2

Quantum Ergodicity • Chaotic dynamics: Small changes in initial condition result in drastic changes in out- come • Lose all information as t → ∞ • Quantum interpretation: for “small” h , U t ψ become evenly distributed as t → ∞ • Eigenfunctions become evenly distributed as h → 0 Quantum Ergodicity Theorem: When the classical dynamics is ergodic, in the semiclas- sical limit W ψ → vol for “almost all” eigenfunc- tions 3

QUE and scarring • Scarring: There are eigenfunction localiz- ing on an invariant set h → 0 W ψ ( P ) � = 0 lim • Quantum Unique Ergodicity (QUE): the volume measure is the only limiting mea- sure • QUE is conjectured to hold for negatively curved surfaces. (Proved for “arithmetic” surfaces [Lindenstrauss]) Theorem (Anantharaman,Koch,Nonnenmacher) . For Anosov flows, any limiting measure has positive entropy. Conjecture. Entropy bounded below by half of maximal entropy 4

Quantum mechanics on the torus • Phase space T 2 d = R 2 d / Z 2 d � � p coordinates x = q • Hilbert space H h = L 2 [( h Z / Z ) d ] (where h = 1 /N ). • Weyl quantization: f ∈ C ∞ ( T 2 d ) � Op h ( f ) • Wigner distribution ψ ∈ H h � W ψ ( f ) = � Op h ( f ) ψ, ψ � • If f = f ( q ) (function of position) f ( Q N ) | ψ ( Q W ψ ( f ) = h d � N ) | 2 5

Quantization of Hamiltonian flows • H ∈ C ∞ ( T 2 d ) real valued Hamiltonian H : T 2 d → T 2 d • Hamiltonian flow Φ t d dt ( f ◦ Φ t H ) = { f, H } ◦ Φ t H • Quantization: unitary flow H ) = exp( it U ( φ t � Op h ( H )) • Egorov Theorem: U ∗ Op h ( f ) U = Op h ( f ◦ Φ H ) + O ( h ) implying: W Uψ ( f ) = W ψ ( f ◦ Φ H ) + O ( h ) 6

Quantization of linear maps • A ∈ Sp(2 d, Z ) acts on T 2 d ( x �→ Ax (mod 1)) • A hyperbolic implies map is Anosov • Quantization [Hannay-Berry]: There is a unique unitary operator satisfying that U h ( A ) ∗ Op h ( f ) U h ( A ) = Op h ( f ◦ A ) [ Remark: A �→ U ( A ) is the Weil represen- tation of Sp(2 d, Z /N Z )] • For perturbation: Φ = Φ H ◦ A quantization: U (Φ) = U (Φ h ) U ( A ). 7

Scarring on periodic orbits Theorem (Faure, Nonnenmacher, De-Bievre) . For linear maps on T 2 , there are e.f. satisfying W ψ ( f ) → 1 2 f (0) + 1 � fdx 2 • These scars occur only when U ( A ) has large spectral degeneracies • Arithmetic symmetries remove degenera- cies With arithmetic symmetries linear maps on T 2 are QUE [Kurlberg, Rudnick] • Perturbation remove degeneracies Open question: Is a generic perturbation QUE? 8

Invariant manifolds for linear maps on T 2 d • A : T 2 d → T 2 d by x �→ Ax (mod 1) Dual action: A : Z 2 d → Z 2 d by n �→ nA • Correspondence: Λ ⊂ Z 2 d invariant lattice of rank d 0 ⇒ � x ∈ T 2 d | e n ( x ) = 1 , ∀ n ∈ Λ � X Λ = invariant manifold of co-dimension d 0 . • Invariant manifolds only exist for d > 1 • We say Λ is isotropic (or X Λ co-isotropic) if the symplectic form ω ( n, m ) = n 1 · m 2 − n 2 · m 1 vanishes on Λ × Λ. 9

Scarring on Invariant manifolds Theorem (K.) . Let A ∈ Sp(2 d, Z ) . Assume that Λ ⊆ Z 2 d is invariant and isotropic: • There are eigenfunctions of U ( A ) localizing on X Λ � W ψ ( f ) → fdx X Λ • This also holds after taking arithmetic sym- metries into account • This also holds for perturbation Φ H ◦ A by any Hamiltonian flow preserving X Λ • If A has a fixed point ξ ∈ T 2 d there are also eigenfunctions localizing on X Λ + ξ . 10

The simplest example B t � � 0 • Take A = for B ∈ GL( d, Z ) B − 1 0 �� p � ∈ T 2 d � • Invariant manifold X = 0 • Perturbation: (by Hamiltonian H = H ( q )) � p � B t p + ∇ H ( B − 1 q ) � Φ( � ) = q B − 1 q • Quantization: U (Φ) ψ ( Q N ) = exp( i � H ( BQ N )) ψ ( BQ N ) • The state ψ = δ 0 is an eigenfunction of U localized on X 11

Sketch of proof • Consider the family of operators A = { Op h ( e n ) | n ∈ Λ } • Λ isotropic ⇒ A ∼ = ( Z /N Z ) d 0 is commuta- tive. • Decomposition into joint eigenspaces, � H h = H λ , (Op h ( e n ) ψ = λ ( n ) ψ ) . Each of dimension N d − d 0 . • For any ψ ∈ H 1 and n ∈ Λ, W ψ ( e n ) = � Op h ( e n ) ψ, ψ � = 1. • States from H 1 are concentrated on X Λ . 12

• Exact Egorov ⇒ U ( A ) permutes eigenspaces U ( A ) : H λ → H λ ◦ A • Hamiltonian flow preserves X Λ ⇒ U (Φ H ) preserves all eigenspaces • The trivial eigenspace H 1 , is preserved by purerbed map: U (Φ H ◦ A ) H 1 = H 1 . • There is a basis for H 1 composed of eigen- functions • These are the localized eigenfunctions 13

Concluding remarks • Quantum states localize on a co-isotropic manifold (uncertainty principle does not ap- ply) • The entropy of the scarred states is always bounded below by half the maximal entropy and it is equal if and only if dim X Λ = d • If A has an invariant lattice then U ( A ) has large spectral degeneracies. However, perturbation (generically) removes all degeneracies • Scarring on invariant manifolds does not imply spectral degeneracies 14

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