Spectral theory of damped quantum chaotic systems

TL;DR

This paper studies the spectral properties of damped wave equations on negatively curved manifolds, linking decay rates of wave energy to dynamical features of geodesic flow and damping functions.

Contribution

It introduces new conditions for spectral gaps based on minimally damped trajectories and reviews high-frequency spectral estimates related to dynamical quantities.

Findings

Conditions for exponential energy decay derived from geodesic flow and damping.

New spectral gap criteria depending on minimally damped trajectories.

Estimates of high-frequency spectrum in terms of dynamical quantities.

Abstract

We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on $X$ and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condition for a spectral gap, depending on the set of minimally damped trajectories.