Eigenvalues of the Laplace-Beltrami Operator on a Prolate Spheroid

Abstract

The Laplace-Beltrami operator on a prolate spheroidadmits a sequence of eigenvalues. These eigenvalues are determined by a singular Sturm-Liouville problem.Properties of the eigenvalues are obtained using the minimum-maximum principle and the Pr\"ufer angle.In particular, eigenvalues are approximated by those of generalized matrix eigenvalue problems,and their behavior is studied when the eccentricity of the spheroid approaches $0$ or $1$.

Keywords: Laplace-Beltrami operator, prolate spheroid, singular Sturm-Liouville problem, generalized matrix eigenvalue problem