Resonance eigenfunction hypothesis for chaotic systems

TL;DR

This paper proposes a hypothesis describing the phase-space distribution of resonance eigenfunctions in chaotic systems with escape, linking quantum decay rates to classical measures and explaining eigenfunction localization.

Contribution

It introduces a new hypothesis connecting resonance eigenfunctions to classical measures, supported by numerical evidence in the standard map.

Findings

Eigenfunctions with decay rate γ are described by a conditionally invariant measure.

Fast-decaying eigenfunctions localize in phase-space regions far from the chaotic saddle.

The hypothesis is numerically validated for the standard map.

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 $\gamma$ are described by a classical measure that $(i)$ is conditionally invariant with classical decay rate $\gamma$ 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 $\gamma$ and the semiclassical limit. The hypothesis is numerically demonstrated for the standard map.