The point scatterer approximation for wave dynamics

Abstract

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

final version, some typos fixed