Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis

TL;DR

This paper provides a uniform asymptotic analysis of the eigenvalue spectrum of spheroidal harmonics in the double limit where both the order and the parameter grow large with a fixed ratio, filling a gap in previous asymptotic studies.

Contribution

It introduces a uniform asymptotic analysis of spheroidal harmonic eigenvalues for simultaneous large m and c with fixed m/c ratio, extending prior work limited to single-parameter limits.

Findings

Derived asymptotic eigenvalue spectrum in the double limit

Unified understanding of eigenvalues for large m and c

Bridged gap between previous asymptotic regimes

Abstract

The spheroidal harmonics $S_{lm}(\theta;c)$ have attracted the attention of both physicists and mathematicians over the years. These special functions play a central role in the mathematical description of diverse physical phenomena, including black-hole perturbation theory and wave scattering by nonspherical objects. The asymptotic eigenvalues $\{A_{lm}(c)\}$ of these functions have been determined by many authors. However, it should be emphasized that all previous asymptotic analyzes were restricted either to the regime $m\to\infty$ with a fixed value of $c$, or to the complementary regime $|c|\to\infty$ with a fixed value of $m$. A fuller understanding of the asymptotic behavior of the eigenvalue spectrum requires an analysis which is asymptotically uniform in both $m$ and $c$. In this paper we analyze the asymptotic eigenvalue spectrum of these important functions in the double limit $m\to\infty$ and $|c|\to\infty$ with a fixed $m/c$ ratio.