Delocalization of slowly damped eigenmodes on Anosov manifolds

TL;DR

This paper investigates high-frequency eigenmodes of the damped wave equation on Anosov manifolds, showing they cannot be fully localized on certain subsets and establishing a spectral gap related to the manifold's dynamics.

Contribution

It demonstrates that eigenmodes near the real axis cannot be localized on sets with negative topological pressure, and identifies a logarithmic spectral gap under dynamical conditions.

Findings

Eigenmodes cannot be fully localized on subsets with negative topological pressure.

Existence of a logarithmic spectral gap below the real axis.

Eigenmodes with spectral parameters close to the real axis are constrained by the manifold's dynamics.

Abstract

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with spectral parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a "strip" of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.