The Spin-weighted Spheroidal Wave functions in the Case of s=1/2

TL;DR

This paper applies supersymmetry quantum mechanics perturbation methods to thoroughly analyze spin-weighted spheroidal wave functions for s=1/2, deriving eigenfunctions and exploring shape-invariance properties.

Contribution

It introduces a systematic perturbation approach in supersymmetry quantum mechanics to solve spin-weighted spheroidal equations for s=1/2, including general superpotential terms and eigenfunctions.

Findings

Derived the first five terms of the super-potential series.

Proved the shape-invariance property in the series form.

Obtained ground and excited eigenfunctions from the superpotential.

Abstract

The spin-weighted spheroidal equations in the case s=1/2 is thoroughly studied in the paper by means of the perturbation method in supersymmetry quantum mechanics. The first-five terms of the super-potential in the series of the parameter beta are given. The general form of the nth term of the superpotential is also obtained, which could derived from the previous terms W_{k}, k<n. From the results, it is easy to give the ground eigenfunction of the equation. Furthermore, the shape-invariance property is investigated in the series form of the parameter beta and is proven kept in this series form for the equations. This nice property guarantee one could obtain the excited eigenfunctions in the series form from the ground eigenfunctions by the method in supersymmetry quantum mechanics. This shows the perturbation method method in supersymmetry quantum mechanics could solve the spin-weight spheroidal wave equations completely in the series form of the small parameter beta.