# Structure of resonance eigenfunctions for chaotic systems with partial escape

## TL;DR

This paper introduces classical measures to explain the properties of resonance eigenfunctions in chaotic systems with partial escape, linking classical dynamics with quantum eigenfunction features.

## Contribution

It develops a family of conditionally-invariant measures that interpolate between natural forward and backward measures, accurately describing quantum eigenfunction properties.

## Key findings

- Classical measures match quantum eigenfunction phase space distributions.
- The measures describe the product structure along stable/unstable directions.
- Distance between classical and quantum measures vanishes in the semiclassical limit for certain eigenfunctions.

## 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.