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

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.