Course - Wavefunctions in chaotic quantum systems

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II Classical billiards – Mathematical description Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 6 Two–dimensional billiards are conservative systems. Thus the motion is restricted to a hypersurface in R 4 of dimension 3. I.e., the trajectories lie on surfaces of constant energy E { } Σ E := (p, q) ∈ R 2 × Ω | p 2 = E . (4) Scaling property Σ E = E 1 2 Σ1 := {(E 1 2 p, q) | (p, q) ∈ Σ1 } . Thus we can restrict ourselves to |p| = 1 and identify the equi-energy surface Σ E as follows Σ 1 = {(p, q) | q ∈ Ω, p ∈ R 2 , |p| = 1} ≃ Ω × S 1 , (5) where S 1 is the unit circle.
II Classical billiards – Mathematical description Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 7 So we have • T ⋆ Ω: phase space • {Φ t }: billiard flow The invariant measure for the flow is the Liouville measure 1 dν = vol(Σ E ) δ(E − H(p, q)) d2 p d 2 q . (6) Remark: A measure ν is called invariant if ν(A) = ν(φ t A) for all measurable A ⊂ T ⋆ Ω.
Page 1 and 2: Wavefunctions in chaotic quantum sy
Page 3 and 4: II Classical billiards Arnd Bäcker
Page 5: II Classical billiards - Mathematic
Page 9 and 10: II Classical billiards - ergodicity
Page 11 and 12: II Classical billiards - mixing Def
Page 13 and 14: II Classical billiards — periodic
Page 15 and 16: III Quantum billiards — some eige
Page 17 and 18: III Quantum billiards - integrable
Page 19 and 20: III Quantum billiards — spectral
Page 21 and 22: III Quantum billiards — mean beha
Page 23 and 24: IV SoE - Statistics of eigenfunctio
Page 25 and 26: IV SoE - Random wave model - chaoti
Page 27 and 28: IV SoE - Random wave model - conseq
Page 29 and 30: IV SoE - Maximum norms - (some) kno
Page 31 and 32: Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 31
Page 33 and 34: V Quantum ergodicity Arnd Bäcker
Page 35 and 36: Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 35
Page 37 and 38: V QE — observables and operators
Page 39 and 40: V Quantum ergodicity theorem Arnd B
Page 41 and 42: V QET - Example - observable in pos
Page 43 and 44: V Quantum ergodicity theorem —
Page 45 and 46: V Quantum ergodicity theorem —
Page 47 and 48: V QET — quantum limits Arnd Bäck
Page 49 and 50: V QET - quantum limits (sketch of i
Page 51 and 52: V QET - eigenfunctions stadium Arnd
Page 53 and 54: V QET - BBMs - Counting function Qu
Page 55 and 56: V BBMs - Counting function Counting
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V BBMs - Counting function Arnd Bä
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V BBMs - Counting function Arnd Bä
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V QET - quasimodes (Arnold ’72) A
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V QET — Scars Arnd Bäcker ⇓
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V QET — Scars Arnd Bäcker ⇓
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V QET — Scars Some results on thi
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V QET and random wave model Amplitu
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V Sketch: proof of the QET Theorem
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V Sketch: proof of the QET Arnd Bä
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V QET - Summary Arnd Bäcker ⇓
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VI Autocorrelation function and rat
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VII More recent results - Poincaré
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VII Poincaré Husimi functions - ex
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VII Poincaré Husimi representation
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VII Poincaré Husimi representation
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VII Relation between Husimi functio
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VII Poincaré Husimi functions —
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VIII Time evolution Arnd Bäcker
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VIII Time evolution Arnd Bäcker
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VIII Time evolution Plot of (x(t),
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Arnd Bäcker ⇓ ⇐ ⇒ Σ ⊕ 10
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Course - Wavefunctions in chaotic quantum systems

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