Confluent Heun functions and separation of variables in spheroidal coordinates

Abstract

The Helmholtz equation and Schrödinger's equation for $H_2 ^ +$H2+ are separable in (respectively) oblate and prolate spheroidal coordinates. They share the same form of the angular equation. In both cases the radial and angular equations have solutions in terms of confluent Heun functions. We show that the zeros of the Wronskian of a pair of solutions to the angular equation give the allowed values of the separation of variables parameter. Since the Heun functions and their derivatives are implemented in Maple, this provides a new method of calculating the physical values of the separation of variables parameter, without programming. We also derive the asymptotic forms of the radial solutions of the Helmholtz equation, and obtain integral relations between the radial and angular solutions.