High-order asymptotics for the Spin-Weighted Spheroidal Equation at large real frequency

TL;DR

This paper develops high-order asymptotic expansions for spin-weighted spheroidal eigenvalues and eigenfunctions at large real frequencies, improving accuracy and extending previous results in black hole perturbation theory.

Contribution

It introduces a high-order asymptotic expansion for eigenvalues and eigenfunctions, correcting and extending prior literature, with validation through numerical calculations.

Findings

Asymptotic expansion accurately matches numerical results

Extension of existing asymptotic formulas to higher order

Improved understanding of spin-field perturbations in Kerr black holes

Abstract

The spin-weighted spheroidal eigenvalues and eigenfunctions arise in the separation by variables of spin-field perturbations of Kerr black holes. We derive a large, real-frequency asymptotic expansion of the spin-weighted spheroidal eigenvalues and eigenfunctions to high order. This expansion corrects and extends existing results in the literature and we validate it via a numerical calculation.