Italo Guarneri

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$Lectures on fractal geometry and dynamics$

Multi-Oscillator case: Wave operators (Evans, Solomyak, 2005) The wave operators: Ω ± = lim t→±∞ eiHt e −iĤ(b) t Above: Ĥ (b) . Below: Ĥ. exist and are complete.
Collapse If α > ω then, for any ψ in the absolutely continuous subspace of H, the probability distribution of the particle converges weakly as t → ±∞ to a superposition of δ functions supported in the interaction points: ∫ ∫ · · · t→±∞ R N dq 1 . . . dq N |ψ(x, q 1 , . . . , q N , t)| 2 −→ N∑ γ ± j δ(x − x j ) , j=1 where: γ ± j = ‖P j Ω ± (H (b) , H)ψ‖ 2 , (6) and P j denotes projection onto H j = L 2 (B j ) ⊗ L 2 (R N ), (B j : j-th box).
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 16 and 17: Application to the inverted oscilla
Page 18: Formal Eigenfunctions Sequences {u
Page 21 and 22: BA&WL again ’Exceptional’ energ
Page 23: Spectral Expansion Theorem For Ψ(E
Page 27 and 28: Band formalism Born-Oppenheimer-lik
Page 29 and 30: ”Phase transition” α < ω α >

Italo Guarneri

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