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

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 $(\widetildeΔ,Δ)$, where $Δ$ is the free Laplacian in $L^{2}({\mathbb R}^{3})$ and $\widetildeΔ$ is one of its singular perturbations, i.e., such that the set $\{u\in H^{2}({\mathbb R}^{3})\cap \text{dom}(\widetildeΔ)\, :\, Δu=\widetildeΔu\}$ is dense. Typically $\widetildeΔ$ 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 $Γ=\partialΩ$, where $Ω\subset{\mathbb R}^{3}$ is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on $Σ\subsetΓ$, a relatively open subset with a Lipschitz boundary. We show that either $Γ$ or $Σ$ are determined by the knowledge of the Scattering Matrix, equivalently of the Far Field Operator, at a single frequency.