Quasi-exactly solvable Lie superalgebras of differential operators

Abstract

In this paper, we study Lie superalgebras of 2 x 2 matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional superalgebras whose odd subspace is non-trivial, we find those admitting a finite-dimensional invariant module of smooth vector-valued functions, and classify all the resulting finite-dimensional modules. The latter Lie superalgebras and their modules are the building blocks in the construction of quasi-exactly solvable quantum mechanical models for spin-1/2 particles in one dimension.