Eigenvalue Problems for Lam\'e's Differential Equation

TL;DR

This paper explores eigenvalue problems related to Lamé's differential equation, including Floquet and Wangerin types, providing comparison theorems, analytic continuation, and connections to algebraic functions and polynomials.

Contribution

It introduces generalized eigenvalue problems for Lamé's equation and analyzes their properties, including zeros, limiting cases, and special functions, expanding understanding of Lamé-related eigenfunctions.

Findings

Comparison theorems for eigenvalues

Analytic continuation of eigenfunctions

Identification of algebraic Lamé functions and polynomials

Abstract

The Floquet eigenvalue problem and a generalized form of the Wangerin eigenvalue problem for Lam\'e's differential equation are discussed. Results include comparison theorems for eigenvalues and analytic continuation, zeros and limiting cases of (generalized) Lam\'e-Wangerin eigenfunctions. Algebraic Lam\'e functions and Lam\'e polynomials appear as special cases of Lam\'e-Wangerin functions.