RECOVERING SINGULARITIES FROM BACKSCATTERING IN TWO DIMENSIONS

Abstract

We have shown that in two dimensions the leading singularities of the quantum mechanical scattering potential are determined by the backscattering data. We assume that the short range potential belongs to a suitable weighted Sobolev space, and by estimating the iterative terms in the Born-expansion we are able to show, that for example for Heaviside-type singularities across a smooth hypersurface, both the location and the size of the jump are recovered from backscattering. The main part of the proof consists in getting sharp enough estimates for the first non-linear Born-term. These estimates are proven using a recent characterization of W 1,p -functions due to P. Hajlasz, and a modification of the classical Triebel's Maximal Inequality.