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

TL;DR

This work analyzes the acoustic operator $A_{v,ρ}=v^{2}ρ\nabla\cdot ρ^{-1}\nabla$ with piecewise-constant coefficients and transmission conditions across a collection of Lipschitz domains. It develops a resolvent-difference framework relative to the free Laplacian, establishes a Limiting Absorption Principle, and proves that the spectrum is purely absolutely continuous on $(-\infty,0]$, with resonances characterized via a meromorphic acoustic $Q$-function when the boundary is $\mathcal{C}^{1,α}$. The second part studies small-inhomogeneity regimes $Ω(ε)$ and provides analytic $ε$-expansions for resonances across four scaling cases, including leading-order corrections from volume, surface, and Neumann/Laplacian spectral data, as well as zero-energy resonances. Together, these results connect micro-resonator geometry and material contrast to the spectral and resonant structure of the acoustic system, with implications for wave localization and scattering in highly contrasted media.

Abstract

We study the acoustic operator $A_{v,ρ}:=v^{2}ρ\nabla\!\cdotρ^{-1}\nabla$ with transmission conditions at the boundary of $Ω=Ω_{1}\cup\dots\cupΩ_{n}$, where the $Ω_{\ell}$'s are connected disjoint open bounded Lipschitz domains, the positive functions $v$ and $ρ$ are constant on each connected component of $Ω$ and $v=ρ=1$ on ${\mathbb R}^{3}\backslash\overlineΩ$. Through a formula for the resolvents difference $(-A_{v,ρ}+z)^{-1}-(-Δ+z)^{-1}$, 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 $Ω$ are of class ${\mathcal C}^{1,α}$, characterize the resonance set. The second part of the paper is devoted to the case where $Ω=Ω(\varepsilon)$ is connected with a small size $\varepsilon$ and the $\varepsilon$-analytic functions $v=v(\varepsilon)$ and/or $ρ=ρ(\varepsilon)$ converge to $0_{+}$ inside $Ω(\varepsilon)$ as $\varepsilon\downarrow 0$; there, we provide the analytic $\varepsilon$-expansions of the resonances of $A_{v,ρ}$ according to different choices of the rate of convergence towards zero of the material parameters.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.1
• proof
• Remark 3.2
• Remark 4.1
• Theorem 4.2
• proof
• Theorem 4.3
• proof