From Lieb-Thirring inequalities to spectral enclosures for the damped wave equation

TL;DR

This paper connects spectral properties of the damped wave equation with non-self-adjoint Schrödinger operators, deriving bounds on eigenvalues and identifying conditions under which damping is spectrally undetectable.

Contribution

It introduces a novel spectral analysis method linking damped wave operators to Schrödinger operators, providing sharp bounds and conditions for spectral similarity.

Findings

Small damping with L^1 norm less than 2 makes the damped wave operator similar to the undamped one.

Established bounds on complex eigenvalues of the damped wave equation.

Identified conditions where damping does not affect the spectrum.

Abstract

Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schroedinger operators, we derive various bounds on complex eigenvalues of the former. In particular, we establish a sharp result that the one-dimensional damped wave operator is similar to the undamped one provided that the L^1 norm of the (possibly complex-valued) damping is less than 2. It follows that these small dampings are spectrally undetectable.