Scattering from singular potentials in quantum mechanics

Abstract

In non-relativistic quantum mechanics, singular potentials in problems with spherical symmetry lead to a Schrodinger equation for stationary states with non-Fuchsian singularities both as r tends to zero and as r tends to infinity. In the sixties, an analytic approach was developed for the investigation of scattering from such potentials, with emphasis on the polydromy of the wave function in the r variable. The present paper extends those early results to an arbitrary number of spatial dimensions. The Hill-type equation which leads, in principle, to the evaluation of the polydromy parameter, is obtained from the Hill equation for a two-dimensional problem by means of a simple change of variables. The asymptotic forms of the wave function as r tends to zero and as r tends to infinity are also derived. The Darboux technique of intertwining operators is then applied to obtain an algorithm that makes it possible to solve the Schrodinger equation with a singular potential containing many negative powers of r, if the exact solution with even just one term is already known.