Real Lie Algebras of Differential Operators and Quasi-Exactly Solvable Potentials

TL;DR

This paper classifies finite-dimensional real Lie algebras of first-order differential operators in two dimensions, explores their quasi-exact solvability, and constructs new solvable Schrödinger operators using these algebras.

Contribution

It provides a complete classification of real Lie algebras of differential operators in R^2 and introduces new quasi-exactly solvable Schrödinger operators based on these algebras.

Findings

Classified all finite-dimensional real Lie algebras in R^2.

Identified all quasi-exactly solvable algebras and modules.

Constructed new quasi-exactly solvable Schrödinger operators.

Abstract

We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators in $R^2$. Furthermore, we find all algebras which are quasi-exactly solvable, along with the associated finite-dimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasi-exactly solvable Schroedinger operators on $R^2$.