Asymptotic spectrum of the oblate spin-weighted spheroidal harmonics: a WKB analysis

TL;DR

This paper derives the asymptotic eigenvalues of spin-weighted spheroidal harmonics using a WKB analysis by transforming the angular equation into a Schrödinger-like form, aiding understanding of their spectral properties.

Contribution

It introduces a novel, simplified method to obtain asymptotic eigenvalues of spin-weighted spheroidal harmonics through a Schrödinger-like transformation and WKB analysis.

Findings

Derived asymptotic eigenvalues for spin-weighted spheroidal harmonics

Provided a new analytical approach for spectral analysis

Enhanced understanding of physical phenomena involving these harmonics

Abstract

Spin-weighted spheroidal harmonics play a central role in the mathematical description of diverse physical phenomena, including black-hole perturbation theory and wave scattering. We present a novel and compact derivation of the asymptotic eigenvalues of these important functions. Our analysis is based on a simple trick which transforms the corresponding spin-weighted spheroidal angular equation into a Schr\"odinger-like wave equation which is amenable to a standard WKB analysis.