Quasi-doubly periodic solutions to a generalized Lame equation

TL;DR

This paper investigates a generalized Lame equation with five parameters, transforming it into a Schrödinger equation with a quasi-doubly periodic potential, and identifies conditions for solutions expressible via generalized Jacobi functions.

Contribution

It introduces a generalized Jacobi function approach to solve the algebraic form of the equation and links it to the generalized Ince equation, revealing finite parameter sets for special solutions.

Findings

Existence of quasi-doubly periodic eigenfunctions for specific parameter values.

Transformation of the generalized Lame equation into a Schrödinger equation with periodic potential.

Connection established between the generalized Lame and Ince equations.

Abstract

We consider the algebraic form of a generalized Lame equation with five free parameters. By introducing a generalization of Jacobi's elliptic functions we transform this equation to a 1-dim time-independent Schroedinger equation with (quasi-doubly) periodic potential. We show that only for a finite set of integral values for the five parameters quasi-doubly periodic eigenfunctions expressible in terms of generalized Jacobi functions exist. For this purpose we also establish a relation to the generalized Ince equation.