A finite oscillator model related to sl(2|1)

TL;DR

This paper introduces a new finite quantum oscillator model based on the Lie superalgebra sl(2|1), with discrete spectra for position and momentum, and explicit wave functions expressed via Krawtchouk polynomials.

Contribution

It presents a novel finite oscillator model using sl(2|1), defining position and momentum as odd elements, with explicit wave functions and a discrete Fourier transform.

Findings

Position spectrum is discrete, of the form ±√k.

Wave functions are expressed in terms of Krawtchouk polynomials.

Explicit discrete Fourier transform is constructed.

Abstract

We investigate a new model for the finite one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1). In this setting, it is natural to present the position and momentum operators of the oscillator as odd elements of the Lie superalgebra. The model involves a parameter p (0<p<1) and an integer representation label j. In the (2j+1)-dimensional representations W_j of sl(2|1), the Hamiltonian has the usual equidistant spectrum. The spectrum of the position operator is discrete and turns out to be of the form $\pm\sqrt{k}$, where k=0,1,...,j. We construct the discrete position wave functions, which are given in terms of certain Krawtchouk polynomials. These wave functions have appealing properties, as can already be seen from their plots. The model is sufficiently simple, in the sense that the corresponding discrete Fourier transform (relating position wave functions to momentum wave functions) can be constructed explicitly.