University of Washington. Quantum Chaos in Scattering Theory

Transcription

1 University of Washington Quantum Chaos in Scattering Theory Maciej Zworski UC Berkeley 10 April 2007

2 In this talk I want to relate objects of classical chaotic dynamics such as trapped sets and topological pressure of hyperbolic flows to the distribution of quantum resonances. The plan is to

3 In this talk I want to relate objects of classical chaotic dynamics such as trapped sets and topological pressure of hyperbolic flows to the distribution of quantum resonances. The plan is to introduce quantum resonances using simple one-dimensional models

4 In this talk I want to relate objects of classical chaotic dynamics such as trapped sets and topological pressure of hyperbolic flows to the distribution of quantum resonances. The plan is to introduce quantum resonances using simple one-dimensional models show how to compute them in that setting and how their distribution is related to dynamics

5 In this talk I want to relate objects of classical chaotic dynamics such as trapped sets and topological pressure of hyperbolic flows to the distribution of quantum resonances. The plan is to introduce quantum resonances using simple one-dimensional models show how to compute them in that setting and how their distribution is related to dynamics describe simple models of chaotic scattering in dimension two

6 In this talk I want to relate objects of classical chaotic dynamics such as trapped sets and topological pressure of hyperbolic flows to the distribution of quantum resonances. The plan is to introduce quantum resonances using simple one-dimensional models show how to compute them in that setting and how their distribution is related to dynamics describe simple models of chaotic scattering in dimension two relate the dimension of the trapped set to the density of resonances

7 In this talk I want to relate objects of classical chaotic dynamics such as trapped sets and topological pressure of hyperbolic flows to the distribution of quantum resonances. The plan is to introduce quantum resonances using simple one-dimensional models show how to compute them in that setting and how their distribution is related to dynamics describe simple models of chaotic scattering in dimension two relate the dimension of the trapped set to the density of resonances relate the pressure to the quantum decay rate.

8 Simplest model: V (x) R,

9 Simplest model: V (x) R, V (x) C,

10 Simplest model: V (x) R, V (x) C, V (x) = 0

11 Simplest model: V (x) R, V (x) C, V (x) = 0 for

12 Simplest model: V (x) R, V (x) C, V (x) = 0 for x > L,

13 Simplest model: V (x) R, V (x) C, V (x) = 0 for x > L, and the corresponding Schrödinger operator,

14 Simplest model: V (x) R, V (x) C, V (x) = 0 for x > L, and the corresponding Schrödinger operator, H V

15 Simplest model: V (x) R, V (x) C, V (x) = 0 for x > L, and the corresponding Schrödinger operator, H V =

16 Simplest model: V (x) R, V (x) C, V (x) = 0 for x > L, and the corresponding Schrödinger operator, H V = 2 x + V (x).

17 Simplest model: V (x) R, V (x) C, V (x) = 0 for x > L, and the corresponding Schrödinger operator, H V = 2 x + V (x). Quantum resonances are the poles of the resolvent of the meromorphic continuation of Green s function of

18 Simplest model: V (x) R, V (x) C, V (x) = 0 for x > L, and the corresponding Schrödinger operator, H V = 2 x + V (x). Quantum resonances are the poles of the resolvent of the meromorphic continuation of Green s function of H V λ 2.

19 This definition although very elegant is not very intuitive. Resonances manifest themselves very concretely in wave expansions, peaks of the scattering cross sections, and phase shift transitions.

20 This definition although very elegant is not very intuitive. Resonances manifest themselves very concretely in wave expansions, peaks of the scattering cross sections, and phase shift transitions. But first let us compute resonances in real time using a MATLAB code (Bindel 2006).

21 This definition although very elegant is not very intuitive. Resonances manifest themselves very concretely in wave expansions, peaks of the scattering cross sections, and phase shift transitions. But first let us compute resonances in real time using a MATLAB code (Bindel 2006). dbindel/resonant1d

22 The next movie shows the change of resonances of h 2 + V (x) as we make h 0 where

23 The next movie shows the change of resonances of h 2 + V (x) as we make h 0 where

24 The next movie shows the change of resonances of h 2 + V (x) as we make h 0 where We went from h = 1 to h = 1/7 in 49 steps.

25 The next movie shows the change of resonances of h 2 + V (x) as we make h 0 where We went from h = 1 to h = 1/7 in 49 steps. Please note that as h decreases the density of resonances goes up.

26 Before explaining how resonances are related to the long time behaviour of scattered waves we discuss the more familiar case of eigenvalues and eigenfuctions.

27 Before explaining how resonances are related to the long time behaviour of scattered waves we discuss the more familiar case of eigenvalues and eigenfuctions. A vibrating string, is described using eigenvalues and eigefunctions: j=0 cos(tλ j )c j u j (x) + j=0 λ 1 j sin(tλ j )d j u j (x) (for simplicity we assumed there are no negative eigenvalues which holds, say, for V 0)

28 Before explaining how resonances are related to the long time behaviour of scattered waves we discuss the more familiar case of eigenvalues and eigenfuctions. A vibrating string, is described using eigenvalues and eigefunctions: j=0 cos(tλ j )c j u j (x) + j=0 λ 1 j sin(tλ j )d j u j (x) (for simplicity we assumed there are no negative eigenvalues which holds, say, for V 0) This decomposition in the basis of harmonic analysis, signal processing, and many other things.

29 On the whole line resonances and resonant states replace eigenvalues and eigenfunctions:

30 On the whole line resonances and resonant states replace eigenvalues and eigenfunctions: A<Im λ j 0 e itλ j c j u j (x) + r A (t, x), t +, (for simplicity we assumed there are no negative eigenvalues which holds, say, for V 0, and more seriously, that resonances are simple)

31 On the whole line resonances and resonant states replace eigenvalues and eigenfunctions: A<Im λ j 0 e itλ j c j u j (x) + r A (t, x), t +, (for simplicity we assumed there are no negative eigenvalues which holds, say, for V 0, and more seriously, that resonances are simple) The error satisfies the following estimate for any K > 0: r A (t, ) H 1 ([ K,K]) Ce At ( w 0 H 1 + w 1 L 2).

32 On the whole line resonances and resonant states replace eigenvalues and eigenfunctions: A<Im λ j 0 e itλ j c j u j (x) + r A (t, x), t +, (for simplicity we assumed there are no negative eigenvalues which holds, say, for V 0, and more seriously, that resonances are simple) The error satisfies the following estimate for any K > 0: r A (t, ) H 1 ([ K,K]) Ce At ( w 0 H 1 + w 1 L 2). where w 0 and w 1 are the initial conditions of our wave.

33 The waves resonate with frequencies given by Re λ j and decay rates given by Im λ j.

34 The waves resonate with frequencies given by Re λ j and decay rates given by Im λ j. This interpretation of resonances is quite common in popular culture, especially in the context of bells sounding its last dying notes

35 The waves resonate with frequencies given by Re λ j and decay rates given by Im λ j. This interpretation of resonances is quite common in popular culture, especially in the context of bells sounding its last dying notes An example from a poem by Li Bai: A thousand valleys rustling pines resound. My heart was cleansed, as if in flowing water. In bells of frost I heard the resonance die.

36 The waves resonate with frequencies given by Re λ j and decay rates given by Im λ j. This interpretation of resonances is quite common in popular culture, especially in the context of bells sounding its last dying notes An example from a poem by Li Bai: A thousand valleys rustling pines resound. My heart was cleansed, as if in flowing water. In bells of frost I heard the resonance die. The word appearing in the original led to this modern translation and can be interpreted as an early (8th century) mention of resonances in literature.

37 Here is an example from scattering of a soliton by a double barrier constructed by two delta functions:

38 Here is an example from scattering of a soliton by a double barrier constructed by two delta functions: nonresonant v = 1

39 Here is an example from scattering of a soliton by a double barrier constructed by two delta functions: nonresonant v = 1 resonant v = 2.4

40 Here is an example from scattering of a soliton by a double barrier constructed by two delta functions: nonresonant v = 1 resonant v = 2.4 nonresonant v = 3.4

41 Here is an example from scattering of a soliton by a double barrier constructed by two delta functions: nonresonant v = 1 resonant v = 2.4 nonresonant v = 3.4 Note the fat, resonant tail in the second figure.

42 60 Spline potential generated by spline([ 2, 1,0,1,2],40*[0,0,1, 2,1,0,0]) Level sets of the corresponding classical Hamiltonian A double barrier potential and the phase portrait for the classical Hamiltonian ξ 2 + V (x).

43 Energy levels corresponding to bound states Resonances and bound states

44 Energy levels corresponding to bound states Resonances and bound states The bound states and resonances for the same potential. The colour coding gives an approximate classical/quantum correspondence between the bound states and energy level satisfying Bohr-Sommerfeld quantization conditions.

45 Energy levels for shape resonances Shape resonances with imaginary parts (blue) and (red)

46 Energy levels for shape resonances Shape resonances with imaginary parts (blue) and (red) Resonances close to the real axis. The real part is related to the bounded components of the energy levels and the imaginary part is related to tunneling between the bounded and unbounded energy levels.

47

48 Well in an island potential and the plot of (Re λ 2 j, log( Im λ2 j )) comparing the logarithms of resonance width (imaginary parts) to Agmon/tunneling distances (Helffer-Sjöstrand 1985):

49 Well in an island potential and the plot of (Re λ 2 j, log( Im λ2 j )) comparing the logarithms of resonance width (imaginary parts) to Agmon/tunneling distances (Helffer-Sjöstrand 1985): S 0 (E) = (V (x) E) dx, log( Im λ 2 j) S 0(Re λ 2 j ). h R

50 Unstable equilibrim points, the separatrix, and the energy levels of real parts of barrier top resonances Barrier top resonances

51 Unstable equilibrim points, the separatrix, and the energy levels of real parts of barrier top resonances Barrier top resonances Barrier top resonances corresponding to two unstable equilibria. The actual real parts are corrected due to quantum effects.

52 Energy levels at real parts of Regge resonances Regge resonances asymptotically lying on a logarithmic curve

53 Energy levels at real parts of Regge resonances Regge resonances asymptotically lying on a logarithmic curve Regge resonances generated by reflections by the singularities at the end points of the support of the potential and lying on logarithmic curves.

54 These resonances dominate large energy asymptotics:

55 These resonances dominate large energy asymptotics: #{λ j : λ j r} = 2(b a) r(1 + o(1)), r, π

56 These resonances dominate large energy asymptotics: #{λ j : λ j r} = 2(b a) r(1 + o(1)), r, π as was proved

57 These resonances dominate large energy asymptotics: #{λ j : λ j r} = 2(b a) r(1 + o(1)), r, π as was proved by Zworski 1987,

58 These resonances dominate large energy asymptotics: #{λ j : λ j r} = 2(b a) r(1 + o(1)), r, π as was proved by Zworski 1987, Froese 1997,

59 These resonances dominate large energy asymptotics: #{λ j : λ j r} = 2(b a) r(1 + o(1)), r, π as was proved by Zworski 1987, Froese 1997, and Simon 2000.

60 Now to dimension two...

61 Now to dimension two... or in fact any dimension...

62 Now to dimension two... or in fact any dimension...

63 Now to dimension two... or in fact any dimension... classically we consider

64 Now to dimension two... or in fact any dimension... classically we consider H = ξ 2 + V (x),

65 Now to dimension two... or in fact any dimension... classically we consider and on the quantum level, H = ξ 2 + V (x),

66 Now to dimension two... or in fact any dimension... classically we consider H = ξ 2 + V (x), and on the quantum level, Ĥ = h 2 + V (x).

67

68 We assume that the flow is hyperbolic on the trapped set: K E

69 We assume that the flow is hyperbolic on the trapped set: K E =

70 We assume that the flow is hyperbolic on the trapped set: K E = Γ + E Γ E

71 We assume that the flow is hyperbolic on the trapped set: where K E = Γ + E Γ E

72 We assume that the flow is hyperbolic on the trapped set: K E = Γ + E Γ E where Γ ± E

73 We assume that the flow is hyperbolic on the trapped set: where Γ ± E = K E = Γ + E Γ E

74 We assume that the flow is hyperbolic on the trapped set: where Γ ± E = {(x, ξ) : ξ 2 + V (x) = E, K E = Γ + E Γ E

75 We assume that the flow is hyperbolic on the trapped set: where Γ ± E = K E = Γ + E Γ E {(x, ξ) : ξ 2 + V (x) = E, (x(t), ξ(t)),

76 We assume that the flow is hyperbolic on the trapped set: where Γ ± E = K E = Γ + E Γ E {(x, ξ) : ξ 2 + V (x) = E, (x(t), ξ(t)), t },

77 We assume that the flow is hyperbolic on the trapped set: K E = Γ + E Γ E where Γ ± E = {(x, ξ) : ξ 2 + V (x) = E, (x(t), ξ(t)), t }, and the flow is defined by

78 We assume that the flow is hyperbolic on the trapped set: K E = Γ + E Γ E where Γ ± E = {(x, ξ) : ξ 2 + V (x) = E, (x(t), ξ(t)), t }, and the flow is defined by Newton 1687

79 We assume that the flow is hyperbolic on the trapped set: K E = Γ + E Γ E where Γ ± E = {(x, ξ) : ξ 2 + V (x) = E, (x(t), ξ(t)), t }, and the flow is defined by Newton 1687 x (t) = 2ξ(t),

80 We assume that the flow is hyperbolic on the trapped set: K E = Γ + E Γ E where Γ ± E = {(x, ξ) : ξ 2 + V (x) = E, (x(t), ξ(t)), t }, and the flow is defined by Newton 1687 x (t) = 2ξ(t), ξ (t) = V (x(t)),

81 We assume that the flow is hyperbolic on the trapped set: K E = Γ + E Γ E where Γ ± E = {(x, ξ) : ξ 2 + V (x) = E, (x(t), ξ(t)), t }, and the flow is defined by Newton 1687 x (t) = 2ξ(t), ξ (t) = V (x(t)), x(0) = x,

82 We assume that the flow is hyperbolic on the trapped set: K E = Γ + E Γ E where Γ ± E = {(x, ξ) : ξ 2 + V (x) = E, (x(t), ξ(t)), t }, and the flow is defined by Newton 1687 x (t) = 2ξ(t), ξ (t) = V (x(t)), x(0) = x, ξ(0) = ξ.

83 Where do the resonant state live?

84 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.,

85 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006),

86 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ).,

87 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ). with Re z(h k ) = E + o(1) and Im z(h) Ch.,

88 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ). with Re z(h k ) = E + o(1) and Im z(h) Ch. Let µ be a semiclassical measure associated to u(h k ):,

89 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ). with Re z(h k ) = E + o(1) and Im z(h) Ch. Let µ be a semiclassical measure associated to u(h k ): Then,

90 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ). with Re z(h k ) = E + o(1) and Im z(h) Ch. Let µ be a semiclassical measure associated to u(h k ): Then suppµ Γ + E,,

91 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ). with Re z(h k ) = E + o(1) and Im z(h) Ch. Let µ be a semiclassical measure associated to u(h k ): Then suppµ Γ + E, λ > 0,,

92 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ). with Re z(h k ) = E + o(1) and Im z(h) Ch. Let µ be a semiclassical measure associated to u(h k ): Then suppµ Γ + E, λ > 0, lim Im z(h k )/h k = λ/2, k,

93 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ). with Re z(h k ) = E + o(1) and Im z(h) Ch. Let µ be a semiclassical measure associated to u(h k ): Then suppµ Γ + E, λ > 0, lim Im z(h k )/h k = λ/2, k Lµ = λµ,

94 Where do the resonant state live? In phase space they live on Γ + E. Theorem 1.(Nonnenmacher-Rubin 2006) Let u(h k ) be resonant states corresponding to z(h k ). with Re z(h k ) = E + o(1) and Im z(h) Ch. Let µ be a semiclassical measure associated to u(h k ): Then suppµ Γ + E, λ > 0, lim Im z(h k )/h k = λ/2, k Lµ = λµ, where L is the Lie derivative along the flow.

95

96 A potential with a simple trapped set.

97

98 The first resonant function for h = 1/16.

99

100

101 FBI transform thanks to Laurent Demanet laurent

102 The trapped set K E lives in the three dimensional energy surface and looks similar to

103 The trapped set K E lives in the three dimensional energy surface and looks similar to

104 The trapped set K E lives in the three dimensional energy surface and looks similar to This is actually a picture of a Julia set but the similarity is more than formal and similar ideas apply to zeros of Ruelle zeta functions.

105 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions:

106 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions:

107 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E))

108 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) =

109 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ))

110 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ

111 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ

112 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ

113 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ )

114 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) =

115 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ)

116 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ)

117 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ) λ > 0,

118 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ) λ > 0, dφ t ρ(v)

119 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ) λ > 0, dφ t ρ(v)

120 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ) λ > 0, dφ t ρ(v) Ce λ t v

121 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ) λ > 0, dφ t ρ(v) Ce λ t v for all v E ρ,

122 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ) λ > 0, dφ t ρ(v) Ce λ t v for all v E ρ, and ±t 0.

123 We say that the flow Φ t (x, ξ) = (x(t), ξ(t)) is hyperbolic on K E, if any ρ K E, the tangent space to H 1 (E) at ρ splits into the flow, unstable and stable directions: T ρ (H 1 (E)) = R 2ξ(ρ), V (x(ρ)) E + ρ E ρ dφ t ρ(e ± ρ ) = E ± Φ t (ρ) λ > 0, dφ t ρ(v) Ce λ t v for all v E ρ, and ±t 0. Verification of this is not easy but in our setting it is available thanks to the work of Sinai, Ikawa, Sjöstrand, and Morita.

124 The Poincaré section is given by a surface in H 1 (E) transversal to the flow:

125 The Poincaré section is given by a surface in H 1 (E) transversal to the flow:

126 The Poincaré section is given by a surface in H 1 (E) transversal to the flow:

127 The Poincaré section is given by a surface in H 1 (E) transversal to the flow: We write the Hausdorff dimension of K E as

128 The Poincaré section is given by a surface in H 1 (E) transversal to the flow: We write the Hausdorff dimension of K E as dim K E = 2d E + 1.

129 The Poincaré section is given by a surface in H 1 (E) transversal to the flow: We write the Hausdorff dimension of K E as Pesin-Sadovskaya 2001 dim K E = 2d E + 1.

130 Theorem 2.

131 Theorem 2.(Sjöstrand-Zworski 2005)

132 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of

133 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x).

134 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x). Under the assumptions of hyperbolicity near energy E,

135 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x). Under the assumptions of hyperbolicity near energy E, R(h)

136 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x). Under the assumptions of hyperbolicity near energy E, R(h)

137 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x). Under the assumptions of hyperbolicity near energy E, R(h) [E h, E + h]

138 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x). Under the assumptions of hyperbolicity near energy E, R(h) [E h, E + h] i[0, Mh]

139 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x). Under the assumptions of hyperbolicity near energy E, R(h) [E h, E + h] i[0, Mh] =

140 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x). Under the assumptions of hyperbolicity near energy E, R(h) [E h, E + h] i[0, Mh] = O(h d E ).

141 Theorem 2.(Sjöstrand-Zworski 2005) Let R(h) denote the set of resonances of Ĥ = h 2 + V (x). Under the assumptions of hyperbolicity near energy E, R(h) [E h, E + h] i[0, Mh] = O(h d E ). This is the analogue of the counting law for eigenvalues of a closed system. Classically everything is trapped in a closed system, so dim K E = 3, d E = 1 and the number of eigenvalues is asymptotic to C E h 1.

142 If u is a resonant state for

143 If u is a resonant state for z

144 If u is a resonant state for z =

145 If u is a resonant state for z = E

146 If u is a resonant state for z = E i Γ

147 If u is a resonant state for then z = E i Γ

148 If u is a resonant state for z = E i Γ then exp( itĥ/h)u

149 If u is a resonant state for z = E i Γ then exp( itĥ/h)u =

150 If u is a resonant state for z = E i Γ then exp( itĥ/h)u = e ite/h tγ/h u.

151 If u is a resonant state for z = E i Γ then exp( itĥ/h)u = e ite/h tγ/h u. Hence states with Γ h decay too fast to be visible.

152 If u is a resonant state for z = E i Γ then exp( itĥ/h)u = e ite/h tγ/h u. Hence states with Γ h decay too fast to be visible. Interpretation of the imaginary part as decay rate brings us to the next theorem.

153 Question: What properties of the flow Φ t, or of K E alone, imply the existence of a gap γ > 0 such that, for h > 0 sufficiently small, z R(h), Re z E = Im z < γh? In other words, what dynamical conditions guarantee a lower bound on the quantum decay rate?

154 Question: What properties of the flow Φ t, or of K E alone, imply the existence of a gap γ > 0 such that, for h > 0 sufficiently small, z R(h), Re z E = Im z < γh? In other words, what dynamical conditions guarantee a lower bound on the quantum decay rate? 0 hbar= gamma E

155 Theorem 3.

156 Theorem 3.(Nonnenmacher-Zworski 2006)

157 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension

158 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E

159 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies

160 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2.

161 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0

162 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0 such that

163 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0 such that R(h)

164 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0 such that R(h)

165 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0 such that R(h) ([E 0 δ, E 0 + δ] i[0, hγ])

166 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0 such that R(h) ([E 0 δ, E 0 + δ] i[0, hγ]) =

167 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0 such that R(h) ([E 0 δ, E 0 + δ] i[0, hγ]) =.

168 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0 such that R(h) ([E 0 δ, E 0 + δ] i[0, hγ]) =. What is γ?

169 Theorem 3.(Nonnenmacher-Zworski 2006) Suppose that the dimension d E satisfies d E < 1 2. Then there exists δ, γ > 0 such that R(h) ([E 0 δ, E 0 + δ] i[0, hγ]) =. What is γ? It can be described using the topological pressure of the flow on K E.

170 We can take any γ satisfying 0 < γ < min E 0 E δ ( P E(1/2)), P E (s) = pressure of the flow on K E.

171 We can take any γ satisfying 0 < γ < min E 0 E δ ( P E(1/2)), P E (s) = pressure of the flow on K E. The existence of a resonance gap depends on the sign of the pressure at s = 1/2, P E (1/2).

172 We can take any γ satisfying 0 < γ < min E 0 E δ ( P E(1/2)), P E (s) = pressure of the flow on K E. The existence of a resonance gap depends on the sign of the pressure at s = 1/2, P E (1/2). The connection between the pressure and the quantum decay rate first appeared in the physics/chemistry literature in the work of Gaspard-Rice 1989.

173 Numerical results (Lin 2002):

174 Numerical results (Lin 2002): Quantum resonances for the three bumps potential.

175 Numerical results (Lin 2002): Quantum resonances for the three bumps potential.

176 Numerical results (Lin 2002): Quantum resonances for the three bumps potential.

177 Numerical results (Lin 2002): Quantum resonances for the three bumps potential.

178 Numerical results (Lin 2002): Quantum resonances for the three bumps potential. This and also some quantum map rigorous models of Nonnenmacher-Zworski 2005 suggest that Theorem 2 is optimal.

179 Preliminary experimental results (Kuhl-Stöckmann 2006):

180 Preliminary experimental results (Kuhl-Stöckmann 2006):

181 Preliminary experimental results (Kuhl-Stöckmann 2006):

182 Preliminary experimental results (Kuhl-Stöckmann 2006):

183 The optimality of Theorem 3 is not clear even on the heuristic or numerical grounds.

184 The optimality of Theorem 3 is not clear even on the heuristic or numerical grounds. In the analogous case of scattering on convex co-compact hyperbolic surfaces the results of Dolgopyat, Naud, and Stoyanov show that the resonance free strip is larger at high energies than the strip predicted by the pressure.

185 The optimality of Theorem 3 is not clear even on the heuristic or numerical grounds. In the analogous case of scattering on convex co-compact hyperbolic surfaces the results of Dolgopyat, Naud, and Stoyanov show that the resonance free strip is larger at high energies than the strip predicted by the pressure. That relies on delicate zeta function analysis following the work of Dolgopyat: at zero energy there exists a Patterson-Sullivan resonance with the imaginary part (width) given by the pressure but all other resonances have more negative imaginary parts.