Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the eigenvalues of the spheroidal wave equation

Harald Schmid

TL;DR

This work reframes spheroidal eigenvalue problems via a 2×2 linear Hamiltonian system depending on parameters $(u_1,u_2,u_3)$, proving that eigenvalues $\Lambda$ depend analytically on the parameters and satisfy a first-order quasilinear PDE relating $\Lambda$ to $(u_1,u_2,u_3)$. Using the method of characteristics, it shows that zeros of a connection coefficient propagate along characteristic curves and that a two-parameter reduction is possible in the prolate case, yielding a simpler PDE for a transformed variable. This PDE-based viewpoint provides a practical, numerically stable route to compute spheroidal eigenvalues and clarifies the link between CSWE eigenvalues ($\Lambda=0$) and spheroidal eigenvalues. The results also point to broader applicability to the confluent Heun equation and other linear eigenvalue problems via deformation theory.

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.

On the eigenvalues of the spheroidal wave equation

TL;DR

This work reframes spheroidal eigenvalue problems via a 2×2 linear Hamiltonian system depending on parameters , proving that eigenvalues depend analytically on the parameters and satisfy a first-order quasilinear PDE relating to . Using the method of characteristics, it shows that zeros of a connection coefficient propagate along characteristic curves and that a two-parameter reduction is possible in the prolate case, yielding a simpler PDE for a transformed variable. This PDE-based viewpoint provides a practical, numerically stable route to compute spheroidal eigenvalues and clarifies the link between CSWE eigenvalues () and spheroidal eigenvalues. The results also point to broader applicability to the confluent Heun equation and other linear eigenvalue problems via deformation theory.

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.
Paper Structure (4 sections, 5 theorems, 64 equations, 2 figures)

This paper contains 4 sections, 5 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A linear Hamiltonian system associated to the CSWE
  3. Analysis of the PDE for the eigenvalues
  4. Conclusion

Key Result

Lemma 1

$\Lambda=0$ is an eigenvalue of the self-adjoint operator $T(u_1,u_2,u_3)$ associated to the linear Hamiltonian system GSWE if and only if $\lambda = u_3+\mu(\mu+1)$ is an eigenvalue of the Coulomb spheroidal wave equation CSWE for the parameters $\gamma^2 = u_1$ and $\beta = -u_3-2(\mu+1)u_2$.

Figures (2)

  • Figure 1: The function $\tilde{\Theta}(\Lambda)$ for $\mu=1$ and $a=5$ with a zero at $\Lambda_0=-0.841720\ldots$
  • Figure 2: The solution $\Lambda(t)$ of \ref{['RedChar']} for $\mu=1$ starting from $u_1(0)=5$, $u_2(0)=0$ and $\Lambda(0)=-0.841720\ldots$

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof