Inverse scattering. IV. Three dimensions: generalized Marchenko construction with bound states, and generalized Gel’fand–Levitan equations

Abstract

This paper represents the final installment in a series on the solution of the inverse scattering problem for the Schrödinger equation in three dimensions. The potential is constructed from a given scattering amplitude without assuming its existence, even in the presence of bound states. For exponentially decreasing potentials, properties of the Jost function and of the regular solution are derived that are sufficient to establish the triangularity of the kernel on which the generalized Gel’fand–Levitan (GL) equation is based. Other generalized GL equations, for nonzero reference potentials, and a nonlinear equation are derived, and for central potentials they are shown to reduce to the well-known radial equations. The contents of the series of papers is summarized.