Discrete Gel'fand-Levitan and Marchenko matrix equations and layer stripping algorithms for the discrete two-dimensional Schrödinger equation inverse scattering problem with a nonlocal potential

Abstract

We develop discrete counterparts to the Gel'fand-Levitan and Marchenko integral equations for the two-dimensional (2D) discrete inverse scattering problem in polar coordinates with a nonlocal potential. We also develop fast layer stripping algorithms that solve these systems of equations exactly. The significance of these results is: (1) they are the first numerical implementation of Newton's multidimensional inverse scattering theory; (2) they show that the result will almost always be a nonlocal potential, unless the data are `miraculous'; (3) they show that layer stripping algorithms implement fast `split' signal processing fast algorithms; (4) they link 2D discrete inverse scattering with 2D discrete random field linear least-squares estimation; and (5) they formulate and solve 2D discrete Schrödinger equation inverse scattering problems in polar coordinates.