Italo Guarneri

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$Lectures on fractal geometry and dynamics$

Application to the inverted oscillator φ(2n, E) ∼ 1 √ n cos ( E log(n)/2ω + n π 4 + β(E)) consistent with q → +∞ asymptotics : G.Barton, Ann. Phys. 166 (1986) ⇒ ac spectrum. ac spectrum + n −1/2 eigfs decay ⇒ exponential growth of energy.
Spectral Analysis over threshold. Expansion in normalized Hermite functions (HO eigenfunctions): ψ(x, q) = ∞∑ ψ n(x) h n(q) , n=0 identifies the Hilbert space of the system as H = l 2 (N 0) ⊗ L 2 +(S) of sequences {ψ n(x)} n∈N0 , ψ n(x) ∈ L 2 (S), ψ even, ∑ ‖ψ n‖ 2 < +∞. n Hamiltonian {ψ n (x)} ↦→ {L n ψ n (x)}, L n = − 1 2 + (n + 1 dx 2 2 )ω with periodic bdary conditions and matching conditions at x = 0: ψ ′ n(0+) − ψ ′ n(0−) = 2α(2ω) −1(√ n + 1ψ n+1 (0) + √ nψ n−1 (0) ) d 2
Page 1 and 2: Inverted oscillators & collapse of
Page 3 and 4: Table of contents Smilansky’s mod
Page 5 and 6: Smilansky’s ”irreversible” mo
Page 7 and 8: Boxing the particle Let the particl
Page 9 and 10: Dynamics of Smilansky’s model II
Page 12 and 13: Band formalism, II (ψ, Hψ) = (ψ,
Page 14 and 15: BA&WL Theory Birkhoff (1911), Adams
Page 18: Formal Eigenfunctions Sequences {u
Page 21 and 22: BA&WL again ’Exceptional’ energ
Page 23 and 24: Spectral Expansion Theorem For Ψ(E
Page 25 and 26: Collapse If α > ω then, for any
Page 27 and 28: Band formalism Born-Oppenheimer-lik
Page 29 and 30: ”Phase transition” α < ω α >

Italo Guarneri

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