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 , where is the free Laplacian in and is one of its singular perturbations, i.e. such that the set is dense. Typically 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 , where is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on , a relatively open subset with a Lipschitz boundary. We show that either the obstacle or the screen are determined by the knowledge of the scattering matrix, equivalently of the far field operator, at a single frequency.