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 SIGMA, 2012, Volume 8, 025, 15 pp. (Mi sigma702)

Deformed $\mathfrak{su}(1,1)$ algebra as a model for quantum oscillators

Elchin I. Jafarovab, Neli I. Stoilovac, Joris Van der Jeugtb

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

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.

Keywords: oscillator model, deformed algebra $\mathfrak{su}(1,1)$, Meixner–Pollaczek polynomial, continuous dual Hahn polynomial.

DOI: https://doi.org/10.3842/SIGMA.2012.025

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ArXiv: 1202.3541
MSC: 81R05, 81Q65, 33C45
Received: February 17, 2012; in final form May 8, 2012; Published online May 11, 2012
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Citation: Elchin I. Jafarov, Neli I. Stoilova, Joris Van der Jeugt, “Deformed $\mathfrak{su}(1,1)$ algebra as a model for quantum oscillators”, SIGMA, 8 (2012), 025, 15 pp.

Citation in format AMSBIB
\Bibitem{JafStoVan12} \by Elchin I. Jafarov, Neli I. Stoilova, Joris Van der Jeugt \paper Deformed $\mathfrak{su}(1,1)$ algebra as a model for quantum oscillators \jour SIGMA \yr 2012 \vol 8 \papernumber 025 \totalpages 15 \mathnet{http://mi.mathnet.ru/sigma702} \crossref{https://doi.org/10.3842/SIGMA.2012.025} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2942814} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000303998000001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84882364826} 

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This publication is cited in the following articles:
1. Jafarov E.I., Van der Jeugt J., “Discrete Series Representations for Sic (2 Vertical Bar 1), Meixner Polynomials and Oscillator Models”, J. Phys. A-Math. Theor., 45:48 (2012), 485201
2. Genest V.X. Ismail M.E.H. Vinet L. Zhedanov A., “The Dunkl Oscillator in the Plane: I. Superintegrability, Separated Wavefunctions and Overlap Coefficients”, J. Phys. A-Math. Theor., 46:14 (2013), 145201
3. Genest V.X., Vinet L., Zhedanov A., “The Algebra of Dual-1 Hahn Polynomials and the Clebsch–Gordan Problem of Sl(-1)(2)”, J. Math. Phys., 54:2 (2013), 023506
4. Roychoudhury R., Roy B., Dube P.P., “Non-Hermitian Oscillator and R-Deformed Heisenberg Algebra”, J. Math. Phys., 54:1 (2013), 012104
5. Jafarov E.I., Van der Jeugt J., “The Oscillator Model for the Lie Superalgebra Sh(2 Vertical Bar 2) and Charlier Polynomials”, J. Math. Phys., 54:10 (2013), 103506
6. Genest V.X. Vinet L. Zhedanov A., “The Singular and the 2:1 Anisotropic Dunkl Oscillators in the Plane”, J. Phys. A-Math. Theor., 46:32 (2013), 325201
7. Tierz M., Phys. Rev. D, 93:12 (2016), 126003
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