Jost solutions and Green’s functions for the three-dimensional Schrödinger equation

Abstract

In an earlier paper, in which the minimal scattering data needed to reconstruct the scattering potential was found for the three-dimensional inverse problem, a Green’s function appeared very naturally in this context. We use this Green’s function to construct Jost solutions for the three-dimensional problem which are closely analogous to those for the one-dimensional problem. The Green’s function and the Jost solutions differ from those given by Faddeev. The completeness relations of the new Jost solutions are given simply in terms of the scattering amplitude. They are the same as those given in an earlier attempt to obtain an algorithm of the Gelfand–Levitan type for the three-dimensional problem. However, it is not yet clear that the Jost solutions of the present paper are the same as those of the earlier paper, since different methods are used to define the two sets of Jost solutions. The one-dimensional problem is discussed in some detail to motivate our definition of the Jost solutions for the three-dimensional problem. The present paper is the first of three papers which report on research arising from the three-dimensional inverse problem.