Weyl laws for partially open quantum maps

TL;DR

This paper investigates the spectral properties of damped quantum maps on the torus, deriving Weyl law analogues in the semiclassical limit and providing detailed estimates for chaotic dynamics.

Contribution

It introduces a model of partially open quantum systems with damping, extending Weyl law concepts to these systems and analyzing their spectral behavior in chaotic regimes.

Findings

Derived Weyl law analogues for damped quantum maps.

Provided precise spectral estimates in chaotic cases.

Connected quantum damping effects with classical chaotic dynamics.

Abstract

We study a toy model for "partially open" wave-mechanical system, like for instance a dielectric micro-cavity, in the semiclassical limit where ray dynamics is applicable. Our model is a quantized map on the 2-dimensional torus, with an additional damping at each time step, resulting in a subunitary propagator, or "damped quantum map". We obtain analogues of Weyl's laws for such maps in the semiclassical limit, and draw some more precise estimates when the classical dynamic is chaotic.