Real Lie algebras of differential operators and quasi-exactly solvable potentials

Abstract

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