Normally hyperbolic trapping: from quantum dispersion to classical - PowerPoint PPT Presentation

Normally hyperbolic trapping: from quantum dispersion to classical mixing S. Nonnenmacher (CEA-Saclay) + M. Zworski (Berkeley) Quantum chaos: fundamentals and applications, Luchon, March 2015 − E+ E ρ ρ E+ t Φ(ρ) Φ t (ρ) ρ K 0 E � c E E+c h �

Outline reminder on quantum (/wave) scattering, resonance spectrum. semiclassical distribution of long-living resonances near energy E → structure of set K E of classical trapped trajectories ← focus: K E normally hyperbolic symplectic submanifold Normal hyperbolicity = ⇒ explicit resonance gap Application to classical chaos: quantitative exponential mixing for Anosov geodesic flows

Classical and quantum scattering V(x,y) y x ( X , g ) of infinite volume, Euclidean outside of a bounded region → scattering by geometry / potential / obstacles Classical scattering: particles follow the geodesic / Hamiltonian flow (with reflection on obstacles). Quantum scattering: wave propagation. Two types of situations: Schrödinger equation: i � ∂ t ψ = H � ψ , with the Hamiltonian operator = − � 2 ∆ + V ( x ) , or H � = − � 2 ∆ Dir def H � 2 2 √ wave equation ( ∂ 2 t − ∆) ψ ( x , t ) = 0 ( ⇔ Schrödinger with H � = − � 2 ∆ ) High frequency régime: fix E > 0, take the semiclassical limit � → 0.

Quantum resonances C h 0 E 2θ Im(r) Γ θ Re(r) z j θ 0 X Euclidean near infinity = ⇒ Spec ( H � ) purely abs. continuous on R + . ⇒ the resolvent ( H � − z ) − 1 diverges when Im z → 0. = however, the Green’s function G ( y , x ; z ) = � y | ( H � − z ) − 1 | x � can be meromorphically continued from { Im z > 0 } to { Im z < 0 } . Poles (of finite multiplicity) = resonances { z j ( � ) } (indep. of x , y ) each z j ← � → lifetime τ j ( � ) = 2 | Im z j | Long-living resonance: | Im z j ( � ) | ≤ C � A way to uncover resonances: complex deformation of H � [A GUILAR -B ALSLEV -C OMBES ,S IMON ,H ELFFER -S JÖSTRAND ..] ⇒ H � ,θ on X , H � ,θ = − e − 2 i θ � 2 ∆ H � on Γ θ ⇐ 2 for | x | > R

Questions in the semiclassical régime h ≪ 1 Ch 0 E gh For E > 0 fixed, what is the semiclassical distribution of the long-living resonances z j ( � ) near E ? Resonance-free strip? bounds on G ( x , y ; z ) (or on the cutoff resolvent operator) for z in the resonance free strip? Gap + good resolvent bound = ⇒ fast decays as t → ∞ • Schrödinger "correlations" � e − itH � / � ψ 1 , ψ 2 � • wave eq.: local energy E Ω ( ψ ( t )) def = 1 Ω ( | ∂ t ψ ( t , x ) | 2 + |∇ ψ ( t , x ) | 2 ) dx � 2 • correlations for Anosov geodesic flow f ( x ) g ( ϕ t x ) dx − � � f ( x ) dx � g ( x ) dx

Semiclassical distribution of resonances - Trapped set Main idea : the distribution of long-living resonances near E is guided by the set of trapped classical trajectories for the Hamiltonian flow Φ t , K E = { ( x , p ) ∈ T ∗ X , H ( x , p ) = E , Φ t ( x , p ) �→ ∞ , t → ±∞} K E compact subset of { H ( x , p ) = E } , invariant through Φ t . ⇒ Im z j ≤ − C � log � − 1 . K E = ∅ = No long-living resonances [L AX -P HILLIPS ’69. . . M ARTINEZ ’02] . K E contains a stable periodic orbit. Resonances Im z j ( � ) = O ( � ∞ ) : � h very long lifetimes E 0 [P OPOV ,V ODEV ,T ANG - � c/h Im z~ e Z WORSKI ,S TEFANOV ]

Normally hyperbolic trapped set Focus on the case where K = ∪ | E ′ − E |≤ δ K E ′ is a (smooth) 2 d � -dimensional symplectic submanifold of T ∗ X , and such that the transverse dynamics is hyperbolic. Normally Hyperbolic Invariant Manifold [W IGGINS ’94...] ( T ρ K ) ⊥ = E − T ρ ( T ∗ X ) = T ρ K ⊕ ( T ρ K ) ⊥ , ρ ⊕ E + dim E ± forall ρ ∈ K , ρ , ρ = d − d � E − ρ , E + ρ are the transverse stable and unstable subspaces: � d Φ t ↾ E − � d Φ − t ↾ E + ∀ t > 0 , ρ � ≤ C e − λ t , ρ � ≤ C e − λ t ∀ ρ ∈ K , The subspaces { E ∓ − ρ , ρ ∈ K } E+ E ρ ρ are Φ t -invariant, and assumed E+ t continuous w.r.t. ρ . Φ(ρ) Φ t (ρ) ρ E ∓ ρ tangent to the stable/unstable manifolds Γ ∓ . K

1st example: trapped set = 1 hyperbolic orbit • K E = single hyperbolic periodic orbit ( d � = 1) [I KAWA ’85,G ÉRARD -S JÖSTRAND ’87,G ÉRARD ’88. . . ] Construct a Quantum Normal Form for H � near the orbit Ex. (d=2): NF variables ( x 1 , x 2 ) ∈ R × S 1 , K E = { x 1 = p 1 = p 2 = 0 , x 2 ∈ S 1 } = E + λ E x 1 p 1 + p 2 NF: H ( x 1 , p 1 , x 2 , p 2 ) + . . . T E ≡ E + λ E � + � ∂ x 2 + . . . on L 2 ( R × S 1 ) U ∗ QNF: � H � U � 2 i ( x 1 ∂ x 1 + ∂ x 1 x 1 ) iT E � �� � dilation op . ❀ explicit resonances near z = E : Ch E 0 deformed half-lattice Im z< � h /2 � z ℓ, k ( � ) = E ( � ) − i � λ E ( 1 / 2 + ℓ ) + � k + O ( � 2 ) , ℓ ∈ N , k ∈ Z T E Hyperbolicity = ⇒ resonance gap: hyperbolic dispersion

Another example from quantum chemistry Chemical reaction dynamics [G OUSSEV -S CHUBERT -W AALKENS -W IGGINS ’10] : Neighbourhood of a saddle-center-center fixed point ( d � = d − 1) p p p 1 2 d s t n a ω t ω d c 2 a e R x . . . x x x 1 x x 2 d s t c u d o r P "Bath" coordinates "Reaction" coordinates Quadratic approximation near the fixed point: d ω k � 2 ( x 2 k + ξ 2 K = { x 1 = p 1 = 0 } H ( x , p ) = E + λ x 1 p 1 + k ) + . . . , k = 2 d ω k H � = E + λ � � � − � 2 ∂ 2 x k + x 2 � 2 i ( x 1 ∂ x 1 + ∂ x 1 x 1 ) + + . . . k 2 k = 2 Nonresonance condition on the ω 2 , . . . , ω d = ⇒ QNF Explicit resonances : z ℓ, n ≈ E − i � λ ( 1 / 2 + ℓ ) + � d k = 2 � ω k ( n k + 1 / 2 ) .

Our main result: Normal hyperbolicity implies a resonance gap � E+ E � � E− δ δ + 0 E E+ J ( ) � t h Λ/2 � t ( � ) � K E If the dynamics on K is not integrable, NO normal forms, NO expression for resonances. Still, one can prove a resonance gap. Normal hyperbolicity → | det d Φ t ↾ E + ρ | ∼ e Λ( ρ ) t for t ≫ 1 ❀ minimal transverse expanding rate Λ def = inf ρ ∈ K Λ( ρ ) Theorem ( N-Z WORSKI ’14 ) Assume the trapped set K is a normally hyperbolic symplectic manifold. Then, for δ, ǫ > 0 and � > 0 small enough, the strip {| E − Re z | ≤ δ, 0 ≥ Im z ≥ − � Λ / 2 + ǫ } is free of resonances. (+ polynomial bound for the resolvent in the strip) Intuition: wavepackets localized on K disperse exponentially fast along Γ + , due to transverse hyperbolicity. Consequences: exponential decay for wave dynamics

A non-quantum application: exponential mixing for Anosov flows ( Y , g ) compact Riemannian manifold of negative curvature. X = S ∗ Y (unit cotangent bundle) carries the geodesic flow ϕ t , generated by v ( x ) ∈ T x X ⇒ the flow ϕ t is Anosov (uniformly hyperbolic): Negative curvature = � d ϕ ∓ t ↾ ˜ T x X = R v ( x ) ⊕ ˜ E + x ⊕ ˜ E − x � ≤ C e − ν t , t > 0 . x , E ± ~ E − X=S*Y x v(x) ~ E + t ϕ( ) x Y ~ J ( ) + x t ⇒ ϕ t ergodic and mixing w.r.t. Liouville measure: decay of correlations = � � � def t →∞ f ( x ) g ( ϕ t ( x )) dx − → 0 C fg ( t ) = f ( x ) dx g ( x ) dx − − − [D OGOPYAT ’98,L IVERANI ’04] : the mixing is exponential : | C fg ( t ) | ≤ e − γ t The decay is controlled by Ruelle–Pollicott resonances { Z j } (Im Z j < 0). Question: how are the R-P resonances distributed?

Anosov flow ≡ scattering problem with K Normal. Hyp. Original idea [F AURE -S JÖSTRAND ’10] : analyze ϕ t : X → X as a quantum scattering propagator Fact: the transfer operator L t f = f ◦ ϕ − t is identical to ξ t Φ(ξ) the quantum propagator L t = e − itH � / � , for the Hamiltonian H � = � * * i v ( x ) · ∂ x T X T X x t ϕ x ❀ resonances of H � ≡ R-P resonances : z j ( � ) = � Z j t The corresponding classical Hamiltonian x ϕ x X H ( x , p ) = v ( x ) · p on T ∗ X generates the Hamiltonian flow Φ t : T ∗ X → T ∗ X , lift of ϕ t : X → X . ∀ E , the energy shell { H ( x , p ) = E } is unbounded * T X in the momentum direction ( ≃ scattering system) E+ ϕ t preserves the Liouville 1-form α on X K t ρ Φ(ρ) ⇒ trapped set K E = { ( x , p = E α x ) , x ∈ X } . = K = ∪ E K E normally hyperb. smooth submanifold, ρ = lift of ˜ E ± E ± x , ~ + E Λ = ˜ Λ minimal expanding rate along ˜ E + t x ϕ x X

Applying our gap result to the Ruelle-Pollicott resonances Theorem Consider the geodesic flow on ( Y , g ) compact of negative sectional curvature. Then there can be at most finitely many Ruelle-Pollicott resonances Z j in the strip { 0 ≥ Im Z j ≥ − ˜ Λ / 2 + ǫ } . As a consequence, the correlations C fg ( t ) decay as � e − iZ j t M j ( f , g ) + O ( e − t ˜ Λ / 2 ) = ⇒ C fg ( t ) = Im Z j > − ˜ Λ / 2 t log | det d ϕ t ↾ E + ( x ) | ) ( ˜ Λ = inf x ∈ X lim inf t →∞ 1 Same result by [T SUJII ’10,’12] , by studying the action of L t on anisotropic Sobolev spaces adapted to the dynamics.

Beyond this resonance gap: resonances in strips [D YATLOV ’13] [F AURE -T SUJII ’13] wave propagation on Kerr(-de Sitter) metrics. Assuming pinching condition Λ max < 2 Λ min , resonances in isolated strip ≤ Im z / � ≤ − ν min 2 } . Counting satisfies a Weyl’s law. [D YATLOV ’13] {− ν max 2 Anosov flow: same type of result for Ruelle-Pollicott resonances [F AURE -T SUJII ’13] .