Body resonances for classical waves

We provide a detailed study of the spectral properties of the linear operator H(ε)=-(ε²χ_Ωε+χ_Ωᶜε)Δ modeling, through the wave equation (∂ₜₜ+H(ε))u=0, the dynamics of acoustic waves in the presence of a small inhomogeneity of size ε having high contrast ε⁻². In particular, we give precise results on the localization of the resonances of H(ε) and their first-order ε-expansions; the latter are explicitly expressed in terms of the eigenvalues and eigenvectors of the Newton potential operator of the set Ω whose rescaling of size ε defines Ωε.