Resonance Eigenfunction Hypothesis for Chaotic Systems

Abstract

A hypothesis about the average phase-space distribution of resonance eigenfunctions in chaotic systems with escape through an opening is proposed. Eigenfunctions with decay rate γ are described by a classical measure that (i) is conditionally invariant with classical decay rate γ and (ii) is uniformly distributed on sets with the same temporal distance to the quantum resolved chaotic saddle. This explains the localization of fast-decaying resonance eigenfunctions classically. It is found to occur in the phase-space region having the largest distance to the chaotic saddle. We discuss the dependence on the decay rate γ and the semiclassical limit. The hypothesis is numerically demonstrated for the standard map.

Structure of resonance eigenfunctions for chaotic systems with partial escape

“…We observe that the distance grows with γ, seen already qualitatively in Ref. [28]. Note that the quantitative analysis confirms that these measures are in better agreement than the cmeasures of Ref.…”

“…Best understood are resonance eigenfunctions in open systems with full escape [21][22][23][24][25][26][27][28][29], in which particles are completely absorbed or escape to infinity. Examples are scattering systems (e.g., three-discs [30]) and systems in which a phase-space region Ω ⊂ Γ acts as a leak [19].…”

O Gato e a Borboleta: propriedades quânticas de sistemas caóticos

“…A morfologia no espaço de fase da função de onda associada a uma ressonância com taxa de decaimento Γ parece ser bem capturada semiclassicamente pelas seguintes hipóteses [119]: 1) Ela corresponde a uma medida condicionalmente invariante do sistema clássico com taxa de decaimento Γ e 2) elaé constante sobre pontos que estãoà mesma distância temporal da sela caótica, quando estaé resolvida com precisãoh. Essa prescrição leva a uma excelente concordância com dados numéricos, como podemos ver na Figura 9.…”

“…As colunas correspondem a taxas de decaimento dadas por Γ = γ = 0.21, Γ = 0.6, Γ = 1 e Γ = 2, sendo γ a taxa de decaimento clássica. Reproduzido de[119], com permissão.…”

Local random vector model for semiclassical fractal structure of chaotic resonance states

“…Here we focus on the semiclassical structure of resonance states in chaotic systems with escape. They are described by multifractal measures on phase space [37,38,39,40,41,42,43,44,45,46,47,48,49] with exponentially distributed intensity fluctuations [50]. These measures are conditionally invariant under the classical dynamics with escape [51,19] and strongly depend on the corresponding quantum decay rate γ [40,47,48].…”