LIE ALGEBRAIZATION OF GENERALIZED LAM\'E EQUATION

Abstract

The paper presents a comprehensive Lie algebraic framework for the generalized Lamé equation (GLE), addressing its deformed \( sl(2,\mathbb{C}) \) Lie algebra \( \g_p \) where the GLE operator is rewritten as an element of the centre of the universal enveloping algebra $U(\g_p)$. We establish a correspondence between finite-dimensional irreducible representations of \( \g_p \) and the quasi-exactly solvable sectors of the generalized Lamé equation under specific conditions. The paper derives the Jacobi tri-diagonal matrix whose eigenvalues coincide with the quasi-exactly solvable eigenvalues of the generalized Lamé operator. Additionally, a gauge transformation is applied to obtain a normalized three-term recurrence for the eigenfunction expansion coefficients. This work not only leads to closed-form expressions for these coefficients but also confirms the existence of a rigorous deformed symmetry for the generalized Lamé equation. The results have implications for other quasi-exactly solvable systems, encouraging future applications within this framework.