Inverse wave scattering in the time domain for point scatterers

Abstract

Let $Δ_{α,Y}$ be the bounded from above self-adjoint realization in $L^{2}({\mathbb R}^{3})$ of the Laplacian with $n$ point scatterers placed at $Y=\{y_{1},\dots,y_{n}\}\subset{\mathbb R}^{3}$, the parameters $(α_{1},\dotsα_{n})\equivα\in {\mathbb R}^{n}$ being related to the scattering properties of the obstacles. Let $u^{α,Y}_{f_ε}$ and $u^{\varnothing}_{f_ε}$ denote the solutions of the wave equations corresponding to $Δ_{α,Y}$ and to the free Laplacian $Δ$ respectively, with a source term given by the pulse $f_ε(x)=\sum_{k=1}^{N}f_{k}\,φ_ε(x-x_{k}) $ supported in $ε$-neighborhoods of the points in $X_{N}=\{x_{1},\dots, x_{N}\}$, $X_{N}\cap Y=\varnothing$. We show that, for any fixed $λ>\supσ(Δ_{α,Y})$, there exits $N_{\circ}\ge 1$ such that the locations of the points in $Y$ can be determined by the knowledge of the finite-dimensional scattering data operator $F^{N}_λ:{\mathbb R}^{N}\to{\mathbb R}^{N}$, $N\ge N_{\circ}$, $$ (F^{N}_λf)_{k}:=\lim_{ε\searrow 0}\int_{0}^{\infty}e^{-\sqrtλ\,t}\big(u^{α,Y}_{f_ε}(t,x_{k})-u^{\varnothing}_{f_ε}(t,x_{k})\big)\,dt\,. $$ We exploit the factorized form of the resolvent difference $(-Δ_{α,Y}+λ)^{-1}-(-Δ+λ)^{-1}$ and a variation on the finite-dimensional factorization in the MUSIC algorithm; multiple scattering effects are not neglected.