Classical and Quantum Liquids Induced by Quantum Fluctuatons
Resonances in classical and quantum dynamics
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Resonances in classical and quantum dynamics
Semyon Dyatlov
March 30, 2015
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Overview
What are resonances?
Resonances: complex characteristic frequenciesassociated to open or dissipative systems
real part = rate of oscillation, imaginary part = rate of decay
For an observable u(t), the resonance expansion is
u(t) =∑
ωj resonanceImωj≥−ν
e−itωjuj +O(e−νt), t → +∞
which is analogous to eigenvalue expansions for closed systems
−ν
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Classical (Pollicott–Ruelle) resonances
Motivation: statistics for billiards
One billiard ball in a Sinai billiard with finite horizon
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Classical (Pollicott–Ruelle) resonances
10000 billiard balls in a Sinai billiard with finite horizon#(balls in the box) → volume of the box
velocity angles distribution → uniform measure
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Classical (Pollicott–Ruelle) resonances
10000 billiard balls in a three-disk system#(balls in the box) → 0 exponentially
velocity angles distribution → some fractal measure
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Classical (Pollicott–Ruelle) resonances
Dynamical systems
U phase space of the dynamical systemϕt : U → U flow of the system
Correlations: f , g ∈ C∞(U)
ρf ,g (t) =
∫U
(f ◦ ϕ−t)g dxdv
yv
Examples
Billiard ball flow on U = {(y , v) | y ∈ M, |v | = 1}, M ⊂ R2
Geodesic flow on U = {(y , v) | y ∈ M, |v |g = 1},(M, g) a negatively curved Riemannian manifold
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Classical (Pollicott–Ruelle) resonances
Pollicott–Ruelle resonances
ρf ,g (t) =
∫U
(f ◦ ϕ−t)g dxdv
Pollicott–Ruelle resonances would appear in resonance expansions of ρf ,gfor smooth hyperbolic systems and are independent of f , g :
ρf ,g (t) =∑
ωj PR resonanceImωj≥−ν
e−itωj cj(f , g) +O(e−νt), t → +∞
−ν
They are defined as poles of meromorphic continuations of
ρf ,g (ω) =
∫ ∞0
e itωρf ,g (t) dt
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Classical (Pollicott–Ruelle) resonances
Pollicott–Ruelle resonances
ρf ,g (t) =
∫U
(f ◦ ϕ−t)g dxdv
Pollicott–Ruelle resonances would appear in resonance expansions of ρf ,gfor smooth hyperbolic systems and are independent of f , g :
ρf ,g (t) =∑
ωj PR resonanceImωj≥−ν
e−itωj cj(f , g) +O(e−νt), t → +∞
Closed system: ρf ,g (t) = c( ∫U f dxdv
)( ∫U g dxdv
)+ o(1)
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Classical (Pollicott–Ruelle) resonances
Pollicott–Ruelle resonances
ρf ,g (t) =
∫U
(f ◦ ϕ−t)g dxdv
Pollicott–Ruelle resonances would appear in resonance expansions of ρf ,gfor smooth hyperbolic systems and are independent of f , g :
ρf ,g (t) =∑
ωj PR resonanceImωj≥−ν
e−itωj cj(f , g) +O(e−νt), t → +∞
Open system: ρf ,g (t) = e−δt( ∫U f dµ−
)( ∫U g dµ+
)+ o(e−δt)
−δ
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Classical (Pollicott–Ruelle) resonances
Pollicott–Ruelle resonances
ρf ,g (t) =
∫U
(f ◦ ϕ−t)g dxdv
Pollicott–Ruelle resonances would appear in resonance expansions of ρf ,gfor smooth hyperbolic systems and are independent of f , g :
ρf ,g (t) =∑
ωj PR resonanceImωj≥−ν
e−itωj cj(f , g) +O(e−νt), t → +∞
Ruelle ’76,’86,’87, Pollicott ’85,’86, Parry–Pollicott ’90, Rugh ’92,Fried ’95, Kitaev ’99, Blank–Keller–Liverani ’02, Liverani ’04,’05,Gouëzel–Liverani ’06, Baladi–Tsujii ’07, Butterley–Liverani ’07,Faure–Roy–Sjöstrand ’08, Faure–Sjöstrand ’11, D–Guillarmou ’14
Climate models: Chekroun–Neelin–Kondrashov–McWilliams–Ghil ’14
Inverse problems: Guillarmou ’14
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Classical (Pollicott–Ruelle) resonances
Ruelle zeta function
ζR(ω) =∏γ
(1− e iωTγ ), Imω � 1
where Tγ are periods of primitive closed trajectories γ
Theorem [Giulietti–Liverani–Pollicott ’12,D–Zworski ’13,D–Guillarmou ’14]
For a hyperbolic dynamical system (open or closed)∗, the Ruelle zetafunction continues meromorphically to ω ∈ C.
Prime orbit theorem (POT): #{γ | Tγ ≤ T} = ehtopT
htopT(1 + o(1))
htop
Margulis, Parry–Pollicott ’90
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Classical (Pollicott–Ruelle) resonances
Ruelle zeta function
ζR(ω) =∏γ
(1− e iωTγ ), Imω � 1
where Tγ are periods of primitive closed trajectories γ
Theorem [Giulietti–Liverani–Pollicott ’12,D–Zworski ’13,D–Guillarmou ’14]
For a hyperbolic dynamical system (open or closed)∗, the Ruelle zetafunction continues meromorphically to ω ∈ C.
Prime orbit theorem (POT): #{γ | Tγ ≤ T} = ehtopT
htopT(1 + o(1))
htop
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Classical (Pollicott–Ruelle) resonances
Spectral gaps
Essential spectral gap of size β > 0:there are finitely many resonances in {Imω ≥ Imω0 − β},
where ω0 is the top resonance
ω0
β
ω0
gap no gap
Spectral gap∗ =⇒ resonance expansion:
ρf ,g (t) =∑
ωj PR resonanceImωj≥−ν
e−itωj cj(f , g) +O(e−νt), ν := Imω0 − β
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Classical (Pollicott–Ruelle) resonances
Spectral gaps
Essential spectral gap of size β > 0:there are finitely many resonances in {Imω ≥ Imω0 − β},
where ω0 is the top resonance
ω0
β
ω0
gap no gap
Spectral gap for ζR =⇒ exponential remainder in POT:
#{γ | Tγ ≤ T} =ehtopT
htopT(1 +O(e−βT ))
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Classical (Pollicott–Ruelle) resonances
Spectral gaps
Essential spectral gap of size β > 0:there are finitely many resonances in {Imω ≥ Imω0 − β},
where ω0 is the top resonance
ω0
β
ω0
gap no gap
Gaps known for geodesic flows on compact negatively curved manifolds:Dolgopyat ’98, Liverani ’04, Tsujii ’12, Giulietti–Liverani–Pollicott ’12,Nonnenmacher–Zworski ’13, Faure–Tsujii ’13and some special noncompact cases: Naud ’05, Petkov–Stoyanov ’10,Stoyanov ’11,’13
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Quantum resonances
We now switch to a different case of quantum resonances, featured inexpansions of solutions to wave equations rather than classical correlations
Examples
Potential scattering (Schrödinger operators)Obstacle scatteringBlack hole ringdown
QuestionsCan resonances be defined?Is there a spectral gap?How fast does the number of resonances grow?
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Quantum resonances
We now switch to a different case of quantum resonances, featured inexpansions of solutions to wave equations rather than classical correlations
Examples
Potential scattering (Schrödinger operators)Obstacle scatteringBlack hole ringdown
QuestionsCan resonances be defined?Is there a spectral gap?How fast does the number of resonances grow?
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Quantum resonances
Example: scattering on the line
Wave equation:
{(∂2
t − ∂2x )u = f ∈ C∞0 ((0,∞)t × Rx)
u|t<0 = 0
Question: how does u(t, x) behave for t →∞ and |x | ≤ R?
x
t
−R R
f
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Quantum resonances
Example: scattering on the line
Wave equation:
{(∂2
t − ∂2x )u = f ∈ C∞0 ((0,∞)t × Rx)
u|t<0 = 0
Fourier–Laplace transform in time:
u(ω)(x) :=
∫ ∞0
e itωu(t, x) dt ∈ L2(R), Imω > 0
(−∂2x − ω2)u(ω) = f (ω), Imω > 0
Resolvent: u(ω) = R(ω)f (ω), where
R(ω) := (−∂2x − ω2)−1 : L2(R)→ L2(R), Imω > 0
Fourier inversion formula:
u(t) =12π
∫Imω=1
e−itωR(ω)f (ω) dω
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Quantum resonances
Example: scattering on the line
Wave equation:
{(∂2
t − ∂2x )u = f ∈ C∞0 ((0,∞)t × Rx)
u|t<0 = 0
R(ω) := (−∂2x − ω2)−1 : L2(R)→ L2(R), Imω > 0
u(t) =12π
∫Imω=1
e−itωR(ω)f (ω) dω
Meromorphically continue R(ω) : L2comp(R)→ L2
loc(R)
R(ω)g(x) =i
2ω
∫Re iω|x−y |g(y) dy , ω ∈ C
and deform the contour, with the integral being O(e−νt) in L2(−R,R):
u(t) = cf +12π
∫Imω=−ν
e−itωR(ω)f (ω) dω
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Quantum resonances
Potential scattering on the line
Introduce a potential V ∈ L∞(R)
R(ω) = (−∂2x +V − ω2)−1 : L2(R)→ L2(R), Imω > 0
continues meromorphically to a family of operators
R(ω) : L2comp(R)→ L2
loc(R), ω ∈ C
The poles of R(ω), called resonances, are featured in resonance expansionsfor the wave equation (∂2
t − ∂2x + V )u = f , and sound like this:
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Quantum resonances
Potential scattering on the line
Introduce a potential V ∈ L∞(R)
R(ω) = (−∂2x +V − ω2)−1 : L2(R)→ L2(R), Imω > 0
continues meromorphically to a family of operators
R(ω) : L2comp(R)→ L2
loc(R), ω ∈ C
!3 !2 !1 0 1 2 3 4 5
0
50
100
Potential
!20 !15 !10 !5 0 5 10 15 20!0.8
!0.6
!0.4
!0.2
0Pole locations
Computed using codes byDavid Bindel
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Quantum resonances
Obstacle scattering
∆E : the Laplacian on E = R3 \ O with Dirichlet boundary conditions,where O ⊂ R3 is an obstacle
R(ω) = (−∆E − ω2)−1 : L2(R3)→ L2(R3), Imω > 0
continues meromorphically to a family of operators
R(ω) : L2comp(R3)→ L2
loc(R3), ω ∈ C
and the poles of R(ω) are called resonances
A rich mathematical theory dating back to Lax–Phillips ’69, Vainberg ’73,Melrose, Sjöstrand
D–Zworski, Mathematical theory of scattering resonances, available online
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Quantum resonances
A real experimental example
Microwave experiments:
Potzuweit–Weich–Barkhofen–Kuhl–Stöckmann–Zworski ’12
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Quantum resonances
Essential spectral gap for obstacles
Essential spectral gap: R(ω) has finitely many poles in {Imω > −β}
Implies∗ exponential decay of local energy of waves modulo a finitedimensional space
Is there a gap? Depends on the structure of trapped billiard ball trajectories
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Quantum resonances
Essential spectral gap for obstacles
Is there a gap? Depends on the structure of trapped billiard ball trajectories
One convex obstacle:
No trapping =⇒ gap of any size
Lax–Phillips ’69, Morawetz–Ralston–Strauss ’77, Vainberg ’89,Melrose–Sjöstrand ’82, Sjöstrand–Zworski ’91. . .
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Quantum resonances
Essential spectral gap for obstacles
Is there a gap? Depends on the structure of trapped billiard ball trajectories
Two convex obstacles:
One trapped trajectory =⇒ a lattice of resonances and gap of fixed size
Ikawa ’82, Gérard–Sjöstrand ’87, Christianson ’06Related case of black holes: Wunsch–Zworski ’10,Nonnenmacher–Zworski ’13, Dyatlov ’13,’14
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Quantum resonances
Essential spectral gap for obstacles
Is there a gap? Depends on the structure of trapped billiard ball trajectories
Three convex obstacles:
Fractal set of trapped trajectories =⇒ gap under a pressure condition
Ikawa ’88, Gaspard–Rice ’89, Naud ’04, Nonnenmacher–Zworski ’09,Petkov–Stoyanov ’10. . .
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Quantum resonances
Experimental observation of the gap
Three-disk system:
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Quantum resonances
Fractal Weyl laws
Weyl law for −∆uj = λ2j uj on a compact manifold M of dimension n:
#{λj ≤ R} = cn Vol(M)Rn(1 + o(1)), R →∞
On a noncompact manifold with a hyperbolic trapped set, for each ν > 0
#{ωj ∈ Res : |Reωj | ≤ R, Imωj ≥ −ν} ≤ CR1+δ,
where 2δ + 2 is the upper Minkowski dimension of the trapped setMelrose ’83, Sjöstrand ’90, Zworski ’99, Wunsch–Zworski ’00,Guillopé–Lin–Zworski ’04, Sjöstrand–Zworski ’07,Nonnenmacher–Sjöstrand–Zworski ’11, Datchev–Dyatlov ’12,Datchev–D–Zworski ’12Weyl laws and band structure for some cases with smooth trapped sets:
Black holes (Kerr–de Sitter): Dyatlov ’13Closed hyperbolic systems (contact Anosov): Faure–Tsujii ’11,’13Semyon Dyatlov Resonances, classical and quantum March 30, 2015 17 / 18
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Thank you for your attention!
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