On algebraic classification of quasi-exactly solvable matrix models

TL;DR

This paper extends the Lie algebraic approach to classify and construct matrix models of quasi-exactly solvable Schrödinger equations, providing new classes of such models with matrix potentials.

Contribution

It generalizes the algebraic method to matrix models, classifies low-dimensional Lie algebra representations, and constructs new quasi-exactly solvable matrix Schrödinger equations.

Findings

Classified inequivalent Lie algebra representations up to dimension three.

Described invariant subspaces for certain Lie algebras including sl(2,R).

Constructed two classes of quasi-exactly solvable matrix Schrödinger equations.

Abstract

We suggest a generalization of the Lie algebraic approach for constructing quasi-exactly solvable one-dimensional Schroedinger equations which is due to Shifman and Turbiner in order to include into consideration matrix models. This generalization is based on representations of Lie algebras by first-order matrix differential operators. We have classified inequivalent representations of the Lie algebras of the dimension up to three by first-order matrix differential operators in one variable. Next we describe invariant finite-dimensional subspaces of the representation spaces of the one-, two-dimensional Lie algebras and of the algebra sl(2,R). These results enable constructing multi-parameter families of first- and second-order quasi-exactly solvable models. In particular, we have obtained two classes of quasi-exactly solvable matrix Schroedinger equations.