Study of the Spin-weighted Spheroidal Equation in the Case of s=1

TL;DR

This paper applies supersymmetric quantum mechanics to analyze the spin-weighted spheroidal wave equation for s=1, deriving super-potentials, eigenvalues, and eigenfunctions relevant to astrophysical radiation near rotating black holes.

Contribution

It introduces a series solution method for the spin-weighted spheroidal equation using SUSYQM, providing explicit super-potential terms and eigenvalues for the first time.

Findings

Derived the first four terms of the super-potential.

Proved the general form of the n-th super-potential term by induction.

Computed ground eigenvalues and eigenfunctions for s=1.

Abstract

We present series study of using the method of super-symmetric quantum mechanics(SUSYQM) solving the spin-weighted spheroidal wave equation. In this paper, we obtain the first four terms of super-potential of the spin-weighted spheroidal wave equation in the case of s=1. These results may help summary the general form for the n-th term of the super-potential, which is proved correct by means of induction. We finally compute the ground eigenvalues and ground eigenfunction. All the results may be of significative for studies of electromagnetic radiation processes near rotating black holes and compute radiation reaction in curved space-time.