Scattering from Singular Potentials in Quantum Mechanics

TL;DR

This paper extends analytic methods for scattering from singular potentials in quantum mechanics to multiple dimensions, providing new asymptotic solutions and an algorithm for solving complex equations with negative powers of r.

Contribution

It generalizes earlier two-dimensional results to arbitrary dimensions and introduces a Darboux-based algorithm for solving complex singular potential problems.

Findings

Derived asymptotic forms of wave functions at r→0 and r→∞

Extended Hill equation approach to multiple dimensions

Developed an algorithm for solving Schrödinger equations with complex singular potentials

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.