The point scatterer approximation for wave dynamics

Given an open, bounded and connected set $$\Omega \subset \mathbb {R}^{3}$$ Ω ⊂ R 3 and its rescaling $$\Omega _{\varepsilon }$$ Ω ε of size $$\varepsilon \ll 1$$ ε ≪ 1 , we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$\begin{aligned} (\varepsilon ^{-2}\chi _{\Omega _{\varepsilon }}+\chi _{\mathbb {R}^{3}\backslash \Omega _{\varepsilon }})\partial _{tt}u=\Delta u+f \end{aligned}$$ ( ε - 2 χ Ω ε + χ R 3 \ Ω ε ) ∂ tt u = Δ u + f with initial data and source supported outside $$\Omega _{\varepsilon }$$ Ω ε ; here, $$\chi _{S}$$ χ S denotes the characteristic function of a set S. We provide the first-order $$\varepsilon $$ ε -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 ∞ ( ( 0 , 1 / ε τ ) , L 2 ( 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 ( Ω ) and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.