Body resonances for classical waves

Abstract

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