On the eigenvalues of the spheroidal wave equation

Abstract

This paper presents some new results on the eigenvalues of the spheroidal wave equation. We study the angular and Coulomb spheroidal wave equation as a special case of a more general linear Hamiltonian system depending on three parameters. We prove that the eigenvalues of this system satisfy a first-order quasilinear partial differential equation with respect to the parameters. This relation offers a new insight on how the eigenvalues of the spheroidal wave equation depend on the spheroidal parameter. Apart from analytical considerations, the PDE we obtain can also be used for a numerical computation of spheroidal eigenvalues.