Quasi Exactly Solvable 2$\times$2 Matrix Equations

TL;DR

This paper explores the algebraic structures of quasi exactly solvable 2x2 matrix differential equations, providing conditions for solvability and analyzing a specific example related to the Lamé equation.

Contribution

It introduces new conditions for quasi exact solvability in 2x2 matrix differential systems and presents explicit algebraic structures and an illustrative example.

Findings

Identified algebraic conditions for quasi exact solvability

Described explicit algebraic structures in these systems

Provided a specific example related to Lamé equation

Abstract

We investigate the conditions under which systems of two differential eigenvalue equations are quasi exactly solvable. These systems reveal a rich set of algebraic structures. Some of them are explicitely described. An exemple of quasi exactly system is studied which provides a direct counterpart of the Lam\'e equation.