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 $B\Subset {\mathbb R}^{n}\backslash \overline \Omega$, the obstacle $\Omega$ can be reconstructed by the data \begin{gather*} F^{\Lambda }_{\lambda }f(x)\coloneq \int _{0}^{\infty }e^{-\sqrt \lambda t}\big (u^{\Lambda }_{f}(t,x)-u^{0}_{f}(t,x)\big ) dt,\\ x\in B ,\ f\in L^{2}({\mathbb R}^{n}) ,\ \mathrm {supp}(f)\subset B . \end{gather*} 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 $(-\Delta _{\Lambda }+\lambda )^{-1}-(-\Delta +\lambda )^{-1}$.