A finite oscillator model with equidistant position spectrum based on an extension of $\mathfrak{su}(2)$

TL;DR

This paper introduces a finite oscillator model based on an extended $rak{su}(2)$ algebra with a parity operator, resulting in an equidistant position spectrum similar to the standard $rak{su}(2)$ oscillator, but with new wavefunction properties.

Contribution

It extends $rak{su}(2)$ algebra with a parity operator and classifies its finite-dimensional representations, leading to a novel finite oscillator model with equidistant spectrum.

Findings

Position spectrum is equidistant and matches the $rak{su}(2)$ oscillator spectrum.

Discrete wavefunctions are expressed via dual Hahn polynomials and depend on a new parameter.

Spectrum remains independent of the extension parameter $c$.

Abstract

We consider an extension of the real Lie algebra $\mathfrak{su}(2)$ by introducing a parity operator $P$ and a parameter $c$. This extended algebra is isomorphic to the Bannai-Ito algebra with two parameters equal to zero. For this algebra we classify all unitary finite-dimensional representations and show their relation with known representations of $\mathfrak{su}(2)$. Moreover, we present a model for a one-dimensional finite oscillator based on the odd-dimensional representations of this algebra. For this model, the spectrum of the position operator is equidistant and coincides with the spectrum of the known $\mathfrak{su}(2)$ oscillator. In particular the spectrum is independent of the parameter $c$ while the discrete position wavefunctions, which are given in terms of certain dual Hahn polynomials, do depend on this parameter.