Inverse wave scattering in the Laplace domain: a factorization method approach

Abstract

Let $\Delta_{\Lambda}\le \lambda_{\Lambda}$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle $\Omega$. Let $u^{\Lambda}_{f}$ and $u^{0}_{f}$ denote the solutions of the wave equations corresponding to $\Delta_{\Lambda}$ and to the free Laplacian $\Delta$ respectively, with a source term $f$ concentrated at time $t=0$ (a pulse). We show that for any fixed $\lambda>\lambda_{\Lambda}\ge 0$ and any fixed $B\subset\subset{\mathbb R}^{n}\backslash\bar\Omega$, the obstacle $\Omega$ can be reconstructed by the data $$ F^{\Lambda}_{\lambda}f(x):=\int_{0}^{\infty}e^{-\sqrt\lambda\,t}\big(u^{\Lambda}_{f}(t,x)-u^{0}_{f}(t,x)\big)\,dt\,,\qquad x\in B\,,\ f\in L^{2}({\mathbb R}^{n})\,,\ \mbox{supp}(f)\subset B\,. $$ A similar result holds in the case of screens reconstruction, when the boundary conditions are assigned only on a part of the boundary. Our method exploits the factorized form of the resolvent difference $(-\Delta_{\Lambda}+\lambda)^{-1}-(-\Delta+\lambda)^{-1}$.