Eigenvalue asymptotics for the damped wave equation on metric graphs

TL;DR

This paper studies the spectral properties of the damped wave equation on finite metric graphs, revealing how eigenvalues behave asymptotically, especially in equilateral cases with standard coupling conditions.

Contribution

It provides a detailed analysis of eigenvalue asymptotics for the damped wave equation on metric graphs, highlighting the influence of graph structure and damping averages.

Findings

Finite number of high-frequency abscissas in equilateral graphs

Eigenvalue locations depend on damping averages

Behavior varies with edge length commensurability

Abstract

We consider the linear damped wave equation on finite metric graphs and analyse its spectral properties with an emphasis on the asymptotic behaviour of eigenvalues. In the case of equilateral graphs and standard coupling conditions we show that there is only a finite number of high-frequency abscissas, whose location is solely determined by the averages of the damping terms on each edge. We further describe some of the possible behaviour when the edge lengths are no longer necessarily equal but remain commensurate.