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 $u_f^{\Lambda }$ and $u_f^0$ 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 $$ F^Λ_λf(x):=\int_{0}^{\infty}e^{-\sqrtλ\,t}\big(u^Λ_{f}(t,x)-u^{0}_{f}(t,x)\big)\,dt\,,\qquad x\in B\,,\ f\in L^{2}({\mathbb R}^{n})\,,\ \mbox{supp}(f)\subset B\,. $$ 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 $(-{\Delta }_{\Lambda }+\lambda {)}^{-1}-(-\Delta +\lambda {)}^{-1}$.

Final version, to appear in Proceedings of the American Mathematical Society

References in corpus (3)

• Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces
• Asymptotic Completeness and S-Matrix for Singular Perturbations
• Inverse Scattering for the Laplace operator with boundary conditions on Lipschitz surfaces

Cited by in corpus (1)

• Inverse wave scattering in the time domain for point scatterers