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

## TL;DR

This paper develops a unified approach combining Mathematical Scattering Theory and the Factorization Method to solve inverse scattering problems involving the Laplace operator with boundary conditions on Lipschitz surfaces, determining boundaries from scattering data.

## Contribution

It introduces a general scheme for inverse scattering with singular perturbations of the Laplacian, applicable to boundary conditions on Lipschitz surfaces, using a combined theoretical framework.

## Key findings

- Boundary surfaces are uniquely determined by single-frequency scattering data.
- The method applies to standard boundary conditions on Lipschitz domains.
- Results extend to boundary conditions on subsets of 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 $(\widetilde\Delta,\Delta)$, where $\Delta$ is the free Laplacian in $L^{2}({\mathbb R}^{3})$ and $\widetilde\Delta$ is one of its singular perturbations, i.e., such that the set $\{u\in H^{2}({\mathbb R}^{3})\cap \text{dom}(\widetilde\Delta)\, :\, \Delta u=\widetilde\Delta u\}$ is dense. Typically $\widetilde\Delta$ 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.