New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential

Abstract

We prove that in dimension three and higher potential scattering the leading order singularities (and in some special cases - all singularities) of unknown potential are obtained exactly by the Born approximation. The proof is based on new estimates for the continuous spectrum of the Laplacian in weighted L p-spaces. These estimates allow us to consider the potentials with stronger singularities than in previous publications.