Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues for the one-dimensional damped wave operator, establishing uniqueness of damping determination from spectra and deriving a trace formula.

Contribution

It provides the general eigenvalue asymptotics, proves the unique determination of damping from spectral data, and derives a trace formula for the damped wave equation.

Findings

Eigenvalue asymptotics for the damped wave operator are established.

Spectral data uniquely determines the damping term.

A trace formula for the damped wave problem is derived.

Abstract

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.