Differential and integral methods for three-dimensional inverse scattering problems with a non-local potential

Abstract

The inverse scattering problem for the three-dimensional Schrodinger equation with a non-local potential diagonal in the radius variable is considered. Generalised Gel'fand-Levitan and Marchenko integral equations are derived using Gram-Schmidt orthogonalisations of free-space wave solutions. A generalised miracle equation is then used to recover the non-local potential. This is the first unified presentation of both the generalised Gel'fand-Levitan and Marchenko equations for this problem. New differential fast algorithms for solving these integral equations and recovering the potential are presented. An application to the problem of linear least-squares estimation of a random field is noted.