Hermitian quasi-exactly solvable matrix Shroedinger operators

TL;DR

This paper constructs six families of Hermitian quasi-exactly solvable matrix Schrödinger operators using Lie algebra representations, providing new models with explicit solutions and square-integrable eigenfunctions.

Contribution

It introduces novel multi-parameter families of quasi-exactly solvable matrix Schrödinger operators based on o(2,2) Lie algebra representations.

Findings

Six new families of operators constructed

Examples with square-integrable eigenfunctions provided

Method relies on special Lie algebra representations

Abstract

We construct six multi-parameter families of Hermitian quasi-exactly solvable matrix Schroedinger operators in one variable. The method for finding these operators relies heavily upon a special representation of the Lie algebra o(2,2) whose representation space contains an invariant finite-dimensional subspace. Besides that we give several examples of quasi-exactly solvable matrix models that have square-integrable eigenfunctions. These examples are in direct analogy with the quasi-exactly solvable scalar Schroedinger operators obtained by Turbiner and Ushveridze.