The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials

TL;DR

This paper develops an algebraic quantum oscillator model based on the Lie superalgebra sh(2|2), linking its wavefunctions to Charlier polynomials and exploring spectral properties with a free parameter.

Contribution

It introduces a novel oscillator model using sh(2|2) superalgebra, connecting wavefunctions to Charlier polynomials and analyzing spectral characteristics with a new free parameter.

Findings

Spectrum of the position operator determined using Jacobi matrices.

Wavefunctions related to Charlier polynomials with parameter gamma^2.

Discussion of properties of the oscillator model and its wavefunctions.

Abstract

We investigate an algebraic model for the quantum oscillator based upon the Lie superalgebra sh(2|2), known as the Heisenberg-Weyl superalgebra or "the algebra of supersymmetric quantum mechanics", and its Fock representation. The model offers some freedom in the choice of a position and a momentum operator, leading to a free model parameter gamma. Using the technique of Jacobi matrices, we determine the spectrum of the position operator, and show that its wavefunctions are related to Charlier polynomials C_n with parameter gamma^2. Some properties of these wavefunctions are discussed, as well as some other properties of the current oscillator model.