Resolvent, spectrum and resonances for the acoustic operator with piecewise constant coefficients

TL;DR

This paper analyzes the spectral properties and resonances of a piecewise constant acoustic operator, providing resolvent formulas, a limiting absorption principle, and asymptotic expansions for small domain sizes and parameter limits.

Contribution

It introduces explicit resolvent difference formulas, characterizes the spectrum as purely absolutely continuous, and derives asymptotic resonance expansions for small domain sizes and converging material parameters.

Findings

Spectrum is purely absolutely continuous.

Resonance set characterized for smooth boundary components.

Analytic expansions of resonances for small domain sizes.

Abstract

We study the acoustic operator $A_{v,\rho }:=v^{2}\rho\nabla\!\cdot\rho^{-1}\nabla$ with transmission conditions at the boundary of $\Omega=\Omega_{1}\cup\dots\cup\Omega_{n}$, where the $\Omega_{\ell}$'s are connected disjoint open bounded Lipschitz domains, the positive functions $v$ and $\rho$ are constant on each connected component of $\Omega$ and $v=\rho=1$ on ${\mathbb R}^{3}\backslash\overline\Omega$. Through a formula for the resolvents difference $(-A_{v,\rho }+z)^{-1}-(-\Delta+z)^{-1}$, we provide a Limiting Absorption Principle, determine the spectrum, which turns out to be purely absolutely continuous, and, in the case the connected components of $\Omega$ are of class ${\mathcal C}^{1,\alpha}$, characterize the resonance set. The second part of the paper is devoted to the case where $\Omega=\Omega(\varepsilon)$ is connected with a small size $\varepsilon$ and the $\varepsilon$-analytic functions $v=v(\varepsilon)$ and/or $\rho=\rho(\varepsilon)$ converge to $0_{+}$ inside $\Omega(\varepsilon)$ as $\varepsilon\downarrow 0$; there, we provide the analytic $\varepsilon$-expansions of the resonances of $A_{v,\rho }$ according to different choices of the rate of convergence towards zero of the material parameters.