Spectral asymptotics for the vectorial damped wave equation

TL;DR

This paper extends spectral asymptotic results from scalar to vectorial damped wave equations, showing eigenfrequencies are confined within a band determined by Lyapunov exponents, generalizing previous scalar results.

Contribution

It generalizes spectral asymptotics to vectorial damped wave equations, replacing Birkhoff limits with Lyapunov exponents for the damping term.

Findings

Eigenfrequencies lie in a band parallel to the real axis.

Eigenfrequency confinement is governed by Lyapunov exponents.

Results extend scalar damping spectral asymptotics to vectorial cases.

Abstract

The eigenfrequencies associated to a scalar damped wave equation are known to belong to a band parallel to the real axis. In [Sj{\"o}00] J. Sj{\"o}strand showed that up to a set of density 0, the eigenfrequencies are confined in a thinner band determined by the Birkhoff limits of the damping term. In this article we show that this result is still true for a vectorial damped wave equation. In this setting the Lyapunov exponents of the cocycle given by the damping term play the role of the Birkhoff limits of the scalar setting.