Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis

Shahar Hod

doi:10.1016/j.physletb.2015.05.036

paper

Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis

arXiv:1506.04148 · doi:10.1016/j.physletb.2015.05.036

Abstract

The spheroidal harmonics $S_{l m} (θ; c)$ 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 ${A_{l m} (c)}$ 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.

5 pages

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Analytic Study of Rotating Black-Hole Quasinormal Modes

Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis

Abstract

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