Spectral deviations for the damped wave equation

TL;DR

This paper establishes a fractal upper bound on the spectrum of the damped wave equation on negatively curved manifolds, linking spectral deviations to ergodic entropy and providing specific results for arithmetic surfaces.

Contribution

It introduces a Weyl-type fractal upper bound for eigenvalue distribution and explores spectral deviations related to ergodic entropy, with new results for arithmetic surfaces.

Findings

Proves a fractal upper bound for the spectrum on negatively curved manifolds.

Counts eigenvalues deviating from typical damping behavior using entropy.

Provides a trace formula-based result for arithmetic surfaces with strong damping.

Abstract

We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the `entropy' that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction.