Deformed su(1,1) Algebra as a Model for Quantum Oscillators

TL;DR

This paper introduces a deformed su(1,1) algebra that extends quantum oscillator models, with wave functions expressed via continuous dual Hahn polynomials, and explores their properties and special cases.

Contribution

It develops a new deformed algebra framework for quantum oscillators, expanding the mathematical tools for modeling quantum systems with special functions.

Findings

Wave functions are expressed in terms of continuous dual Hahn polynomials.

The deformed algebra extends the positive discrete series representations of su(1,1).

Various limits and special cases of the oscillator models are discussed.

Abstract

The Lie algebra $\mathfrak{su}(1,1)$ can be deformed by a reflection operator, in such a way that the positive discrete series representations of $\mathfrak{su}(1,1)$ can be extended to representations of this deformed algebra $\mathfrak{su}(1,1)_\gamma$. Just as the positive discrete series representations of $\mathfrak{su}(1,1)$ can be used to model a quantum oscillator with Meixner-Pollaczek polynomials as wave functions, the corresponding representations of $\mathfrak{su}(1,1)_\gamma$ can be utilized to construct models of a quantum oscillator. In this case, the wave functions are expressed in terms of continuous dual Hahn polynomials. We study some properties of these wave functions, and illustrate some features in plots. We also discuss some interesting limits and special cases of the obtained oscillator models.