Structure of resonance eigenfunctions for chaotic systems with partial escape

Abstract

Physical systems are often neither completely closed nor completely open, but instead they are best described by dynamical systems with partial escape or absorption. In this paper we introduce classical measures that explain the main properties of resonance eigenfunctions of chaotic quantum systems with partial escape. We construct a family of conditionally-invariant measures with varying decay rates by interpolating between the natural measures of the forward and backward dynamics. Numerical simulations in a representative system show that our classical measures correctly describe the main features of the quantum eigenfunctions: their multi-fractal phase space distribution, their product structure along stable/unstable directions, and their dependence on the decay rate. The (Jensen-Shannon) distance between classical and quantum measures goes to zero in the semiclassical limit for long-and short-lived eigenfunctions, while it remains finite for intermediate cases.

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

“…(iv) The classical measure µ p , q γ for the special case γ nat (γ inv ) is consistent with the general observation that at the (inverse) natural decay rate resonance states are uniform along the unstable (stable) direction [19,48].…”

“…The natural measure is uniform along the unstable q-direction such that the natural decay rate is given by γ nat = − ln( 1 n n−1 k=0 r k ) [66]. The natural measure of the backward dynamics is uniform along the stable p-direction, such that the natural growth rate of the inverse map is [65,48]. Another important classical decay rate is the average decay of a typical ergodic orbit [34], the so-called typical decay rate γ typ = − 1 n n−1 k=0 ln r k .…”

“…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].…”

“…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]. Such questions are also of interest in mixed systems and in non-Hermitian PT-symmetric systems [52,53,54,55].…”

“…Such questions are also of interest in mixed systems and in non-Hermitian PT-symmetric systems [52,53,54,55]. For fully chaotic systems there exist approximative descriptions of resonance states with arbitrary decay rates γ for full [47] and partial escape [48], however, the precise semiclassical limit is not known.…”