Quantum chaos and the thermodynamical formalism Stphane Nonnenmacher - PowerPoint PPT Presentation

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Quantum chaos and the thermodynamical formalism Stéphane Nonnenmacher (Orsay) Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism, ICERM, March 7-11, 2016

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Outline quantum chaos on a compact manifold: structure of the high-frequency eigenstates • quantum ergodicity • a lower bound on the metric entropy (with N.Anantharaman) open quantum chaos: quantum scattering • quantum resonances, in the semiclassical regime • hyperbolic trapped sets (Axiom A) • "gap" in the resonance spectrum, in terms of a topological pressure (with M.Zworski) In both problems, crucial role played by the hyperbolic dispersion of wavepackets.

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Structure of chaotic eigenmodes Quantum (unique?) ergodicity

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Spectral geometry: spatial structure of vibration modes Quantum particle propagating on ( X, g ) compact manifold, possibly with (piecewise smooth) boundary: def • Schrödinger equation ih∂ t ψ ( t, x ) = P h ψ ( t, x ) , with P h = − h 2 ∆ X . ⇒ relevant to consider the spectrum of the Laplacian: discrete Linear = spectrum (∆ X + k 2 n ) ψ n = 0 ( ⇐ ⇒ ( − h 2 n ∆ X − 1) ψ n = 0 ) What can we say about the spectrum { k n } and eigenmodes { ψ n } in the high-frequency limit k n → ∞ ? ( ⇐ ⇒ semiclassical limit h n → 0 ) Local Weyl’s law: for any test function f ∈ C ∞ ( X ) , � � f ( x ) | ψ n ( x ) | 2 dx = C d K d � f ( x ) dx + o ( K d ) , X X k n ≤ K On average, the eigenstates become equidistributed on X . How about individual eigenstates? Semiclassical analysis makes the connection with the underlying Hamiltonian dynamics: (broken) geodesic flow Φ t : S ∗ X → S ∗ X .

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Chaotic dynamics: Quantum Ergodicity Quantum Chaos : preferably consider ( X, g ) s.t. the geodesic flow Φ t has chaotic features. Theorem (Quant. Ergod. [S CHNIRELMAN , Z ELDITCH , C OLIN DE V ERDIÈRE . . . ] ) If Φ t is ergodic on S ∗ X w.r.t. the Liouville measure, almost all the eigenmodes ψ n become asymptotically equidistributed on X : 1 � j →∞ � ψ n j , fψ n j � L 2 → f ( x ) dx along subsequence of density 1 . Vol( X ) X Qu: Can there be exceptional modes, for instance localizing along certain periodic geodesics? [L INDENSTRAUSS ’06] : X arithmetic surface of const. negative curvature and ( ψ n ) “Hecke" eigenmodes: Quantum Unique Ergodicity . [H ASSELL ’10] : for X a generic stadium billiard, ∃ bouncing-ball modes

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Localization of high-frequency eigenstates: Semiclassical measures To connect with classical dynamics, lift the localization to phase space T ∗ X . c ( T ∗ X ) �→ F ( x, hD ) , pseudodiff. operator on X . F ( x, ξ ) ∈ C ∞ Allows to test the localization of ψ n ( x ) both in position space and in Fourier space at the scale h − 1 (microlocalization). Ex: the local plane wave ψ h ( x ) = a ( x ) e iξ 0 · x/h is microlocalized on the Lagrangian plane Λ ξ 0 = { ( x, ξ 0 ) , x ∈ supp a } . Adapt "Planck’s constant" h to ψ n : ( − h n 2 ∆ − 1) ψ n = 0 , so that ψ n is microlocalized on S ∗ X = { ( x, ξ ) : | ξ | = 1 } . j →∞ Extracting subsequences, � ψ n j , F ( x, h n j D ) ψ n j � � T ∗ X F dµ sc , → where µ sc is called a semiclassical measure. Each µ sc is a probability measure supported on S ∗ X , and is invariant through Φ t . It represents the asymptotic phase space distribution of the subsequence ( ψ n j ) . ⇒ JOB FOR DYN. SYS.: describe the possible invariant measures of Φ t . =

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Anosov flows: Entropy of semiclassical measures Choose ( X, g ) with Anosov geodesic flow, e.g. with negative sectional t ( ρ ) = | det( d Φ t ↾ E u curvature. Important quantity: unstable Jacobian J u ρ ) | * S X t=0 6 s 1 E ρ t=3 0 ρ E ρ t=1 E s E u Φ(ρ) t=2 ρ 3 4 2 Φ(ρ) t=4 5 E u Φ(ρ) Attempt to characterize the localization properties of eigenstates: study the metric entropy of the semiclassical measure µ sc . l S ∗ X = � J partition of unity on S ∗ X : 1 j =1 π j , π j = 1 l V j . Refined partitions: π α 0 ··· α n − 1 = π α n − 1 ◦ Φ n − 1 × · · · π α 1 ◦ Φ 1 × π α 0 . H KS ( µ ) = lim n →∞ 1 n H n ( µ ) , where H n ( µ ) = � | α | = n − µ ( π α ) log µ ( π α ) . Indicator of localization: µ very localized (e.g. µ = δ γ ) = ⇒ H ( µ ) small. If µ ( π α ) ≤ Ce − β | α | when | α | → ∞ , then H ( µ ) ≥ β . ⇒ can we show that µ sc ( π α ) ≤ Ce − β | α | ? =

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Quantizing the partition. Hyperbolic dispersion estimate Smoothen and quantize π j into Π j = π j ( x, hD ) , to form a quantum partition of unity: Id = � J j =1 Π j . Π j = microlocal quasiprojector on the phase space region V j . Refine the quantum partition using Schrödinger evolution U t = e − itP h /h : def = U − n +1 Π α n − 1 · · · U 1 Π α 1 U 1 Π α 0 Π α evolution of observables: U − t a ( x, hD ) U t = a ◦ Φ t ( x, hD ) + O t ( h ) (Egorov theorem) product of observables: a ( x, hD ) b ( x, hD ) = ( ab )( x, hD ) + O ( h ) = ⇒ Π α = π α ( x, hD ) + O n ( h ) . ⊖ correspondence breaks down when V α becomes "quantum", that is for n > T E = log 1 /h λ max the Ehrenfest time. ⊕ beyond T E , exponential decay, governed by the unstable Jacobian along α -trajectories: 1 , Ch − ( d − 1) / 2 J u ( α ) − 1 / 2 � � Hyperbolic dispersion estimate . � Π α � L 2 → L 2 ≤ min

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Lower bounds on the entropy Formally, the weight of ψ h inside V α is � Π α ψ h � 2 , which decays exponentially when n > T E : � Π α ψ h � 2 ≤ h − ( d − 1) e − n Λ min ⊕ lower bound on quantum entropy H n ( ψ h ) ≥ n Λ min − ( d − 1) | log h − 1 | . ⊖ for times n ≫ T E , impossible to relate H n ( µ sc ) with H n ( ψ h ) . We obtain a nontrivial bound by taking n = 2 T E : Theorem ( [A NANTHARAMAN ’06,A NANTHARAMAN -N’07] ) If Φ t is Anosov, any semiclassical measure µ sc satisfies � log J u ( ρ ) dµ sc ( ρ ) − ( d − 1) λ max H ( µ sc ) ≥ . 2 S ∗ X If X is 2-dim. with nonpositive curv., H ( µ sc ) ≥ 1 � S ∗ X log J u ( ρ ) dµ sc ( ρ ) 2 [R IVIÈRE ’10] • (Ruelle: H ( µ ) ≤ S ∗ X log J u ( ρ ) dµ ( ρ ) , with equality iff µ = µ Liouv ). � • ∃ toy Anosov models (quantum maps) for which this lower bound is reached, µ sc = 1 2 δ γ + 1 2 µ Liouv [F AURE -N-D E B IÈVRE ’03] .

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Semiclassical propagation of Lagrangian states ξ Φ t ρ t Λ ϕ 0 Λ ϕ t ρ 0 t U x t x h x 0 A Lagrangian state ψ h ( x ) = a ( x ) e i ϕ ( x ) is microlocalized on the Lagrangian h leaf Λ ϕ = { ( x, dϕ ( x )) , x ∈ supp a } ⊂ T ∗ X . Ex: local plane wave a ( x ) e i η · x microlocalized on Λ η = { ( x, η ) , x ∈ supp a } . h Lagrangian states enjoy a simple semiclassical evolution: U t ( a e iϕ/h ) = a t e iϕ t /h + O ( h ) , with Λ ϕ t = Φ t (Λ ϕ ) . the amplitude a t is transported like a half-density: a t ( x t ) = a ( x 0 ) | det( ∂x t /∂x 0 ) | − 1 / 2 , ( x t , dϕ t ( x t )) = Φ t ( x 0 , dϕ ( x 0 )) where applying a pseudodiff F ( x, hD ) only modifies the symbol: [ F ( x, hD ) ae iϕ/h ]( x ) = F ( x, dϕ ( x )) a ( x ) e iϕ ( x ) /h + O ( h )

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Proof of Hyperbolic dispersive estimate 1 N−1 Φ Λ Φ η V a V a N 1 Λ ϕ N 1 Λ η Φ( )= Λ ϕ 1 ρ W u ρ W s V a ρ 0 1 Φ( ) V a 0 We want to show: � Π α n − 1 · · · U 1 Π α 1 U 1 Π α 0 ψ � L 2 � h − d − 1 J u ( α ) � ψ � L 2 2 Any state Π α 0 ψ can be "Fourier" expanded into Π α 0 ψ ( x ) = h − d − 1 I dη a ( x ) e i η · x ˜ � ψ ( η ) 2 h propagate individual Lagrangian states: U 1 ( a e iη · x/h ) = a 1 e iϕ 1 /h , with Λ ϕ 1 = Φ 1 (Λ η ) . the quasiprojector Π 1 cuts off the amplitude (norm reduction) propagate a 1 e iϕ 1 /h into a 2 e iϕ 2 /h , then truncate, etc. Hyperbolicity = ⇒ Λ ϕ N aligns along W u , and a N ∼ a 1 J u ( α 1 · · · α N ) − 1 / 2 . d − 1 linearity = 2 J u ( α 1 · · · α N ) − 1 / 2 � ψ � . ⇒ � Π α ψ � � h

Quantum ergodicity Hyperbolic dispersion Chaotic scattering Chaotic trapped set Resonance gap Quantum ergodicity Open quantum chaos: Chaotic scattering systems