Inversion of Discontinuities for the Schrödinger Equation in Three Dimensions

Abstract

This work deals with the inverse scattering problem for the Schrodinger operator in three dimensions. The following problem is studied: If the inverse scattering problem is solved approximately by using the linearizing Born approximation, what information about the true potential is obtained if the scatterer is not necessarily weak? It is shown that in a certain sense, the leading singularities of the potential are recovered exactly. More accurately, under certain a priori smoothness assumptions of the scattering potential, the difference of the potential and the one obtained by using the Born approximation is in a smoother class of functions. Especially, for bounded potentials the approximation agrees with the true potential up to a Lipschitz continuous function. For the a priori scale of function spaces we have chosen the Zygmund classes equipped with a suitable weight at infinity.