Raising and Lowering operators of spin-weighted spheroidal harmonics

TL;DR

This paper extends the mathematical framework of spin-raising and lowering operators from spin-weighted spherical harmonics to spheroidal harmonics, incorporating an additional parameter and enabling higher-order generalizations.

Contribution

It generalizes the operators to spin-weighted spheroidal harmonics with a new parameter, providing explicit calculations of related raising and lowering operators.

Findings

Derived generalized operators for spheroidal harmonics.

Explicitly calculated the raising and lowering operators for spherical harmonics.

Established a foundation for higher-order operator generalizations.

Abstract

In this paper we generalize the spin-raising and lowering operators of spin-weighted spherical harmonics to linear-in-$\gamma$ spin-weighted spheroidal harmonics where $\gamma$ is an additional parameter present in the second order ordinary differential equation governing these harmonics. One can then generalize these operators to higher powers in $\gamma$. Constructing these operators required calculating the $\ell$-, $s$- and $m$-raising and lowering operators (and various combinations of them) of spin-weighted spherical harmonics which have been calculated and shown explicitly in this paper.