Discrete series representations for sl(2|1), Meixner polynomials and oscillator models

We explore a model for a one-dimensional quantum oscillator based on the Lie superalgebra sl(2|1). For this purpose, a class of discrete series representations of sl(2|1) is constructed, each representation characterized by a real number β > 0. In this model, the position and momentum operators of the oscillator are odd elements of sl(2|1) and their expressions involve an arbitrary parameter γ. In each representation, the spectrum of the Hamiltonian is the same as that of a canonical oscillator. The spectrum of a position operator can be continuous or infinite discrete, depending on the value of γ. We determine the position wavefunctions both in the continuous and the discrete case and discuss their properties. In the discrete case, these wavefunctions are given in terms of Meixner polynomials. From the embedding osp(1|2) subset of sl(2|1), it can be seen why the case γ = 1 corresponds to a paraboson oscillator. Consequently, taking the values (β, γ) = (1/2, 1) in the sl(2|1) model yields a canonical oscillator. (paper)