Gel’fand–Levitan equations with comparison measures and comparison potentials

Abstract

Using an abstract form of the Gel’fand–Levitan equation, it is shown how a solution of the equation corresponding to a given weight operator can be found in terms of a solution for the equation with a different weight operator. The resulting Gel’fand–Levitan equation is a generalization of the original one. To achieve our result, an analog of a canonical transformation for direct scattering is used. The effect of the use of the transformation is to include part of the scattering potential (the comparison potential) in the unperturbed Hamiltonian. The generalized Gel’fand–Levitan equation has the advantage that if the weight operator for a given Gel’fand–Levitan equation is close to that for an already solved Gel’fand–Levitan equation, the solution of the first can be obtained from the second by using the solution of the second as a first approximation in an iteration procedure or as a trial function in a variational procedure. The method is illustrated by considering the inverse problem for the one-dimensional Schrödinger equation, a generalized radial Schrödinger equation, and the Marchenko equation for the zero angular momentum radial Schrödinger equation. Though the use of a comparison weight function for some of the cases above has been given by others, the work of the present paper represents a systematic approach to the problem. The role of a variational principle will also be discussed.