Associated Lam\'{E} Equation, Periodic Potentials and sl(2,R)

TL;DR

This paper introduces a new algebraic approach using sl(2,R) to analyze the Associated Lamé equation, deriving explicit solutions for periodic potentials and their spectra, including cases with integer and half-integer parameters.

Contribution

It presents a novel algebraization method for the Associated Lamé equation within sl(2,R), providing explicit solutions for various parameter cases.

Findings

Explicit band edge eigenfunctions and spectra for integer m, ℓ

Solutions for half-integer m and integer or half-integer ℓ

New algebraic framework for analyzing periodic potentials

Abstract

We propose a new approach based on the algebraization of the Associated Lam\'{e} equation \[-\psi''(x) + [ m(m+1)k^{2}sn^{2}x + \ell(\ell+1)k^{2}(cn^{2}x/dn^{2}x)]\psi(x) = E\psi(x)\] within sl(2,R) to derive the corresponding periodic potentials. The band edge eigenfunctions and energy spectra are explicitly obtained for integers m,$\ell$. We also obtain the explicit expressions of the solutions for half-integer m and integer or half-integer $\ell$.