The integral property of the spheroidal wave functions

TL;DR

This paper investigates the shape-invariance property of spheroidal wave functions using supersymmetric quantum mechanics and finds that they lack this property beyond the first order, indicating potential non-solvability.

Contribution

It applies the perturbation method in SUSYQM to analyze spheroidal equations and demonstrates the loss of shape-invariance at second order.

Findings

Superpotential loses shape-invariance after second term

Spheroidal equations may be non-solvable using SUSYQM

First-order approximation suggests initial shape-invariance

Abstract

The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study whether the spheroidal equations have the shape-invariance property. Expanding the super-potential term by term in the parameter alpha and solving it, we find that the superpotential loses its shape-invariance property upon to the second term. This first means that we could not solve the spheroidal problems by the SUSQM; further it is not unreasonable to say they are non-solvable in some sense.