Spectral asymptotics for the vectorial damped wave equation

Abstract

The eigenfrequencies associated to a scalar damped wave equation are known to belong to a band parallel to the real axis. In [Sj{ö}00] J. Sj{ö}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.