Computation of the eigenvalues for the angular and Coulomb spheroidal wave equation

TL;DR

This paper presents a new computational method for accurately and efficiently calculating eigenvalues and eigenfunctions of the angular and Coulomb spheroidal wave equations using an entire function derived from Floquet solutions.

Contribution

It introduces a recurrence-based approach to compute the entire function whose zeros are the eigenvalues, improving accuracy and reducing computational cost.

Findings

Recurrence formula enables precise eigenvalue calculation.

Method provides easy computation of eigenfunctions.

Efficient algorithm with low computational cost.

Abstract

In this paper we study the eigenvalues of the angular spheroidal wave equation and its generalization, the Coulomb spheroidal wave equation. An associated differential system and a formula for the connection coefficients between the various Floquet solutions give rise to an entire function whose zeros are exactly the eigenvalues of the Coulomb spheroidal wave equation. This entire function can be calculated by means of a recurrence formula with arbitrary accuracy and low computational cost. Finally, one obtains an easy-to-use method for computing spheroidal eigenvalues and the corresponding eigenfunctions.