On algebraic classification of quasi-exactly solvable matrix models

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.

LaTeX-file, 16 pages, submitted to J.Phys.A: Math.Gen

References in corpus (3)

• Quasi Exactly Solvable 2$\times$2 Matrix Equations
• On quasi-exactly solvable matrix models
• Conditional symmetry and spectrum of the one-dimensional Schrödinger equation

Cited by in corpus (2)

• Algebraic treatments of the problems of the spin-1/2 particles in the one and two-dimensional geometry: a systematic study
• Quasi-exactly solvable quartic Bose Hamiltonians