Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles

TL;DR

This paper studies the spectral properties of a dissipative acoustic operator outside a convex obstacle, providing sharper eigenvalue and resonance location results, and establishing a Weyl asymptotic formula.

Contribution

It offers new bounds on eigenvalues and resonances for the dissipative wave operator and proves a Weyl law for their asymptotic distribution in convex obstacle settings.

Findings

Sharper eigenvalue and resonance location results.

Weyl asymptotic formula for eigenvalues and resonances.

No eigenvalues for constant damping on the unit ball.

Abstract

We examine the wave equation in the exterior of a strictly convex bounded domain $K$ with dissipative boundary condition $\partial_{\nu} u - \gamma(x) \partial_t u = 0$ on the boundary $\Gamma$ and $0 < \gamma(x) <1, \:\forall x \in \Gamma.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The poles $\lambda$ of the meromorphic incoming resolvent $(G - \lambda)^{-1}: \:{ \mathcal H}_{comp} \rightarrow {\mathcal D}_{loc}$ are eigenvalues of G if ${\rm Re}\: \lambda < 0$ and incoming resonances if ${\rm Re}\: \lambda > 0$. We obtain sharper results for the location of the eigenvalues of $G$ and incoming resonances in $\Lambda = \{\lambda \in \mathbb C:\: |{\rm Re}\: \lambda| \leq C_2(1 + |{\rm Im}\: \lambda|)^{-2},\: |{\rm Im}\: \lambda| \geq A_2 > 1\}$ and we prove a Weyl formula for their asymptotic. For $K = \{x \in {\mathbb R}^3:\:|x| \leq 1\}$ and $\gamma$ constant we show that $G$ has no eigenvalues so the Weyl formula concerns only the incoming resonances.