Body resonances for classical waves

We provide a detailed study of the spectral properties of the linear operator \(H(\varepsilon )=-(\varepsilon ^{2}\chi _{\Omega _{\varepsilon }}+\chi _{\Omega ^{c}_{\varepsilon }})\Delta \) $H(\epsilon )=-({\epsilon }^2{\chi }_{{\mathrm{\Omega }}_{\epsilon }}+{\chi }_{{\mathrm{\Omega }}_{\epsilon }^c})\mathrm{\Delta }$ modeling, through the wave equation \((\partial _{tt}+H(\varepsilon ))u=0\) $({\partial }_{\mathit{tt}}+H(\epsilon ))u=0$ , the dynamics of acoustic waves in the presence of a small inhomogeneity of size \(\varepsilon \) $\epsilon$ having high contrast \(\varepsilon ^{-2}\) ${\epsilon }^{-2}$ . In particular, we give precise results on the localization of the resonances of \(H(\varepsilon )\) $H(\epsilon )$ and their first-order \(\varepsilon \) $\epsilon$ -expansions; the latter are explicitly expressed in terms of the eigenvalues and eigenvectors of the Newton potential operator of the set \(\Omega \) $\mathrm{\Omega }$ whose rescaling of size \(\varepsilon \) $\epsilon$ defines \(\Omega _{\varepsilon }\) ${\mathrm{\Omega }}_{\epsilon }$ .