The spectral concentration for damped waves on compact Anosov manifolds

TL;DR

This paper investigates the spectral distribution of damped waves on compact Anosov manifolds, revealing that most eigenvalues concentrate near the average damping and lie in regions approaching this average logarithmically as eigenvalues grow.

Contribution

It proves that eigenvalues of damped waves on Anosov manifolds are confined to regions approaching the average damping logarithmically, extending previous results on spectral concentration.

Findings

Eigenvalues concentrate near the average damping function.

Most eigenvalues lie in regions with imaginary parts approaching the average logarithmically.

Application to zeros of twisted Selberg zeta functions near the critical line.

Abstract

We study the spectral distribution of damped waves on compact Anosov manifolds. Sj\"ostrand \cite{SJ1} proved that the imaginary parts of the majority of the eigenvalues concentrate near the average of the damping function, see also Anantharaman \cite{AN2}. In this paper, we prove that the most of eigenvalues actually lie in certain regions with imaginary parts that approach the average logarithmically as the real parts tend to infinity. The proof relies on the moderate deviation principles for Anosov geodesic flows. As an application, we show the concentration of non-trivial zeros of twisted Selberg zeta functions in a logarithmic region asymptotically close to $\Re s=\frac{1}{2}$.