Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles
Vesselin Petkov
TL;DR
This work analyzes the dissipative wave equation outside a strictly convex obstacle under boundary damping $0<\gamma(x)<1$ and studies the spectrum of the generator $G$ of the associated contraction semigroup. It develops a semiclassical framework based on the exterior Dirichlet-to-Neumann map $N(\lambda)$, constructs a refined parametrix in the hyperbolic region, and derives a trace formula linking spectral data to the boundary operator $\mathcal C(\lambda)=N(\lambda)-\lambda\gamma$. The authors prove sharp localization of eigenvalues (for $\mathrm{Re}\lambda<0$) and incoming resonances (for $\mathrm{Re}\lambda>0$) within a region $\Lambda$, and establish a Weyl-type asymptotic for their counting in terms of the geometry and damping via $\int_{\Gamma}(1-\gamma^2(x))^{(d-1)/2} dS_x$, with a leading constant $\frac{2\omega_{d-1}}{(2\pi)^{d-1}}$. In the ball case with constant $\gamma$, there are no eigenvalues, so the Weyl formula concerns only incoming resonances; the resulting framework advances spectral and scattering theory for dissipative wave problems in exterior domains, with implications for inverse problems.
Abstract
We examine the wave equation in the exterior of a strictly convex bounded domain $K$ with dissipative boundary condition $\partial_ν u - γ(x) \partial_t u = 0$ on the boundary $Γ$ and $0 < γ(x) <1, \:\forall x \in Γ.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The poles $λ$ of the meromorphic incoming resolvent $(G - λ)^{-1}: \:{ \mathcal H}_{comp} \rightarrow {\mathcal D}_{loc}$ are eigenvalues of G if ${\rm Re}\: λ< 0$ and incoming resonances if ${\rm Re}\: λ> 0$. We obtain sharper results for the location of the eigenvalues of $G$ and incoming resonances in $Λ= \{λ\in \mathbb C:\: |{\rm Re}\: λ| \leq C_2(1 + |{\rm Im}\: λ|)^{-2},\: |{\rm Im}\: λ| \geq A_2 > 1\}$ and we prove a Weyl formula for their asymptotic. For $K = \{x \in {\mathbb R}^3:\:|x| \leq 1\}$ and $γ$ constant we show that $G$ has no eigenvalues so the Weyl formula concerns only the incoming resonances.