Investigation of the recurrence relations for the spheroidal wave functions

TL;DR

This paper uses supersymmetric quantum mechanics to analyze recurrence relations of spheroidal wave functions, revealing shape-invariance properties and deriving approximate eigenfunctions in closed form, aiding physical problem investigations.

Contribution

It introduces a novel application of SUSYQM to derive recurrence relations and closed-form eigenfunctions for spheroidal wave functions, highlighting shape-invariance properties.

Findings

First-order shape-invariance property confirmed.

Closed-form eigenfunctions obtained.

Approximate recurrence relations derived.

Abstract

The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study the spheroidal wave functions' recurrence relations, which are revealed by the shape-invariance property of the super-potential. The super-potential is expanded by the parameter alpha and could be gotten by approximation method. Up to the first order, it has the shape-invariance property and the excited spheroidal wave functions are gotten. Also, all the first term eigenfunctions obtained are in closed form. They are advantageous to investigating for involved physical problems of spheroidal wave function.