Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis

Abstract

The spheroidal harmonics Slm(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 {Alm(c)} of these functions have been determined by many authors. However, it should be emphasized that all the previous asymptotic analyzes were restricted either to the regime m→∞ with a fixed value of c, or to the complementary regime |c|→∞ 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→∞ and |c|→∞ with a fixed m/c ratio.