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

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.