Inverse Scattering for the Laplace operator with boundary conditions on Lipschitz surfaces

Abstract

We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of self-adjoint operators $(\tilde{\Delta },\Delta )$, where $\Delta$ is the free Laplacian in $L^2({\mathbb{R}}^3)$ and $\tilde{\Delta }$ is one of its singular perturbations, i.e., such that the set $\{u\in H^2({\mathbb{R}}^3)\cap \text{dom}(\tilde{\Delta }):\Delta u=\tilde{\Delta }u\}$ is dense. Typically $\tilde{\Delta }$ corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular our results apply to standard, either separating or semi-transparent, boundary conditions at $\Gamma =\partial \Omega$, where $\Omega \subset {\mathbb{R}}^3$ is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on $\Sigma \subset \Gamma$, a relatively open subset with a Lipschitz boundary. We show that either $\Gamma$ or $\Sigma$ are determined by the knowledge of the Scattering Matrix, equivalently of the Far Field Operator, at a single frequency.

Final version, to appear in Inverse Problems

References in corpus (4)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Boundary triples and Weyl functions for singular perturbations of self-adjoint operators
• Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces
• Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

Cited by in corpus (2)

• Inverse wave scattering in the time domain for point scatterers
• Inverse wave scattering in the Laplace domain: a factorization method approach