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arXiv · 1506.04148

Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis

Shahar Hod
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Abstract

The spheroidal harmonics $S_{lm}(θ;c)$ 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 $\{A_{lm}(c)\}$ 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.

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Shahar Hod. 2015-06-12. Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis. https://doi.org/10.1016/j.physletb.2015.05.036

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