Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 025, 15 pages      arXiv:1202.3541      https://doi.org/10.3842/SIGMA.2012.025

Deformed $\mathfrak{su}(1,1)$ Algebra as a Model for Quantum Oscillators

Elchin I. Jafarov a, b, Neli I. Stoilova c and Joris Van der Jeugt a
a) Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium
b) Institute of Physics, Azerbaijan National Academy of Sciences, Javid Av. 33, AZ-1143 Baku, Azerbaijan
c) Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria

Received February 17, 2012, in final form May 08, 2012; Published online May 11, 2012

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.

Key words: oscillator model; deformed algebra $\mathfrak{su}(1,1)$; Meixner-Pollaczek polynomial; continuous dual Hahn polynomial.

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