New Investigation on the Spheroidal Wave Equations

TL;DR

This paper transforms spheroidal wave equations into a Schrödinger-like form, deriving super-potentials expanded in a parameter series, which simplifies obtaining ground eigenfunctions but loses shape-invariance, limiting recurrence relation extensions.

Contribution

The paper introduces a new Schrödinger form of spheroidal equations with super-potentials expanded in a parameter series, facilitating eigenfunction calculation.

Findings

Super-potentials expanded in parameter series are derived.

Ground eigenfunctions can be obtained more easily.

Shape-invariance property is not preserved.

Abstract

Changing the spheroidal wave equations into new Schro$dinger's form, the super-potential expanded in the series form of the parameter $\alpha$are obtained in the paper. This general form of the super-potential makes it easy to get the ground eigenfunctions of the spheroidal equations. But the shape-invariance property is not retained and the corresponding recurrence relations of the form (4) could not be extended from the associated Legendre functions to the case of the spheroidal functions.