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

TL;DR

This paper constructs discrete series representations of the Lie superalgebra sl(2|1) to model a quantum oscillator, revealing connections to Meixner polynomials and paraboson oscillators, and generalizing the canonical oscillator.

Contribution

It introduces a new class of discrete series representations of sl(2|1) for quantum oscillator models, linking wavefunctions to Meixner polynomials and exploring parameter-dependent spectra.

Findings

Spectrum of the Hamiltonian matches the canonical oscillator.

Wavefunctions are expressed via Meixner polynomials in the discrete case.

Special parameter choices recover the canonical and paraboson oscillators.

Abstract

We explore a model for the one-dimensional quantum oscillator based upon 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 beta>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 gamma. In each representation, the spectrum of the Hamiltonian is the same as that of the canonical oscillator. The spectrum of the momentum operator can be continuous or infinite discrete, depending on the value of gamma. We determine the position wavefunctions both in the continuous and 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 sl(2|1), it can be seen why the case gamma=1 corresponds to the paraboson oscillator. Consequently, taking the values (beta,gamma)=(1/2,1) in the sl(2|1) model yields the canonical oscillator.