### 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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References

1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
2. Atakishiyev M.N., Atakishiyev N.M., Klimyk A.U., On ${\rm su}_q(1,1)$-models of quantum oscillator, J. Math. Phys. 47 (2006), 093502, 21 pages.
3. Atakishiyev N.M., Pogosyan G.S., Vicent L.E., Wolf K.B., Finite two-dimensional oscillator. I. The Cartesian model, J. Phys. A: Math. Gen. 34 (2001), 9381-9398.
4. Atakishiyev N.M., Pogosyan G.S., Vicent L.E., Wolf K.B., Finite two-dimensional oscillator. II. The radial model, J. Phys. A: Math. Gen. 34 (2001), 9399-9415.
5. Atakishiyev N.M., Pogosyan G.S., Wolf K.B., Finite models of the oscillator, Phys. Part. Nuclei 36 (2005), 247-265.
6. Atakishiyev N.M., Suslov S.K., Difference analogues of the harmonic oscillator, Theoret. and Math. Phys. 85 (1990), 1055-1062.
7. Bailey W.N., Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, no. 32, Stechert-Hafner, Inc., New York, 1964.
8. Bargmann V., Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568-640.
9. Basu D., Wolf K.B., The unitary irreducible representations of ${\rm SL}(2,\,{\bf R})$ in all subgroup reductions, J. Math. Phys. 23 (1982), 189-205.
10. Berezans'ki Ju.M., Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968.
11. Biedenharn L.C., The quantum group ${\rm SU}_q(2)$ and a $q$-analogue of the boson operators, J. Phys. A: Math. Gen. 22 (1989), L873-L878.
12. Groenevelt W., Koelink E., Meixner functions and polynomials related to Lie algebra representations, J. Phys. A: Math. Gen. 35 (2002), 65-85, math.CA/0109201.
13. Horváthy P.A., Plyushchay M.S., Valenzuela M., Bosons, fermions and anyons in the plane, and supersymmetry, Ann. Physics 325 (2010), 1931-1975, arXiv:1001.0274.
14. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
15. Ismail M.E.H., Letessier J., Valent G., Quadratic birth and death processes and associated continuous dual Hahn polynomials, SIAM J. Math. Anal. 20 (1989), 727-737.
16. Jafarov E.I., Stoilova N.I., Van der Jeugt J., Finite oscillator models: the Hahn oscillator, J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310.
17. Jafarov E.I., Stoilova N.I., Van der Jeugt J., The $\mathfrak{su}(2)_\alpha$ Hahn oscillator and a discrete Fourier-Hahn transform, J. Phys. A: Math. Theor. 44 (2011), 355205, 18 pages, arXiv:1106.1083.
18. Klimyk A.U., On position and momentum operators in the $q$-oscillator, J. Phys. A: Math. Gen. 38 (2005), 4447-4458.
19. Klimyk A.U., The $su(1,1)$-models of quantum oscillator, Ukr. J. Phys. 51 (2006), 1019-1027.
20. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
21. Koelink H.T., Van Der Jeugt J., Convolutions for orthogonal polynomials from Lie and quantum algebra representations, SIAM J. Math. Anal. 29 (1998), 794-822, q-alg/9607010.
22. Koornwinder T.H., Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials, in Orthogonal Polynomials and their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, Springer, Berlin, 1988, 46-72.
23. Koornwinder T.H., Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal. 13 (1982), 1011-1023.
24. Macfarlane A.J., On $q$-analogues of the quantum harmonic oscillator and the quantum group ${\rm SU}(2)_q$, J. Phys. A: Math. Gen. 22 (1989), 4581-4588.
25. Ohnuki Y., Kamefuchi S., Quantum field theory and parastatistics, University of Tokyo Press, Tokyo, 1982.
26. Plyushchay M.S., Deformed Heisenberg algebra with reflection, Nuclear Phys. B 491 (1997), 619-634, hep-th/9701091.
27. Post S., Vinet L., Zhedanov A., Supersymmetric quantum mechanics with reflections, J. Phys. A: Math. Theor. 44 (2011), 435301, 15 pages, arXiv:1107.5844.
28. Regniers G., Van der Jeugt J., Wigner quantization of some one-dimensional Hamiltonians, J. Math. Phys. 51 (2010), 123515, 21 pages, arXiv:1011.2305.
29. Slater L.J., Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966.
30. Srinivasa Rao K., Van der Jeugt J., Raynal J., Jagannathan R., Rajeswari V., Group theoretical basis for the terminating ${}_3F_2(1)$ series, J. Phys. A: Math. Gen. 25 (1992), 861-876.
31. Sun C.P., Fu H.C., The $q$-deformed boson realisation of the quantum group ${\rm SU}(n)_q$ and its representations, J. Phys. A: Math. Gen., 22 (1989), L983-L986.
32. Tsujimoto S., Vinet L., Zhedanov A., From $sl_q(2)$ to a parabosonic Hopf algebra, SIGMA 7 (2011), 093, 13 pages, arXiv:1108.1603.
33. Tsujimoto S., Vinet L., Zhedanov A., Jordan algebras and orthogonal polynomials, J. Math. Phys. 52 (2011), 103512, 8 pages, arXiv:1108.3531.