The point scatterer approximation for wave dynamics

Given an open, bounded and connected set $\Omega\subset\mathbb{R}^{3}$ and its rescaling $\Omega_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$(\varepsilon^{-2}\chi_{\Omega_{\varepsilon}}+\chi_{\mathbb{R}^{3}\backslash\Omega_{\varepsilon}})\partial_{tt}u=\Delta u+f$$ with initial data and source supported outside $\Omega_{\varepsilon}$; here, $\chi_{S}$ denotes the characteristic function of a set $S$. We provide the first-order $\varepsilon$-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the $L^{\infty}((0,1/\varepsilon^{\tau}),L^{2}(\mathbb{R}^{3}))$-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in $L^{2}(\Omega)$ and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.