Study of the Spin-weighted Spheroidal Wave Equation in the Case of s=3/2

TL;DR

This paper applies supersymmetric quantum mechanics to analyze the spin-weighted spheroidal wave equation for s=3/2, deriving super-potentials, eigenfunctions, eigenvalues, and exploring shape invariance for excited states.

Contribution

It introduces a supersymmetric approach to solve the spin-weighted spheroidal wave equation for s=3/2, providing explicit super-potentials and eigenvalues, which is a novel application.

Findings

Derived first five terms of super-potential

Obtained ground eigen-function and eigenvalue

Computed excited eigenvalues and eigen-functions

Abstract

In this paper, we use the means of super-symmetric quantum mechanics to study of the Spin-weighted Spheroidal Wave in the case of s=3/2. We obtain some interesting results: the first-five terms of the super-potential, the general form of the super-potential. The ground eigen-function and eigenvalue of the equation are also given. According these results, we make use of the shape invariance property to compute the exited eigenvalues and eigen-functions. These results help us to understand the Spin-weighted Spheroidal Wave and show that it is integral.