The point scatterer approximation for wave dynamics

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