Resonances in open quantum maps

TL;DR

This paper reviews recent research on the resonance spectra of open quantum maps with chaotic classical dynamics, exploring fractal structures, spectral gaps, and statistical models in the semiclassical limit.

Contribution

It provides a comprehensive overview of the spectral properties of open quantum maps, including fractal Weyl laws, spectral gaps, and effects of partial transparency, integrating classical and quantum perspectives.

Findings

Fractal Weyl law describes resonance distribution

Spectral gap formation in chaotic maps

Eigenstate morphology linked to classical structures

Abstract

We review recent studies about the resonance spectrum of quantum scattering systems, in the semiclassical limit and assuming chaotic classical dynamics. Stationary quantum properties are related to fractal structures in the classical phase space. We focus attention on a particular class of problems that are chaotic maps in the torus with holes. Among the topics considered are the fractal Weyl law, the formation of a spectral gap and the morphology of eigenstates. We also discuss the situation when the holes are only partially transparent and the use of random matrices for a statistical description.