Oscillators in the Framework of Unified (q, α, β, γ, ν)-Deformation and Their Oscillator Algebras
The aim of this paper is to review our results on the description of multiparameter deformed oscillators and their oscillator algebras. We define generalized (q; α, β, γ, ν)-deformed oscillator algebras and study their irreducible representations. The Arik–Coon oscillator with the main relation aa+ – qa+a = 1, where q >1, is embedded in this framework. We have found the connection of this oscillator with the Askey q–1-Hermite polynomials. We construct a family of generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of the appropriate classical Stieltjes moment problem, we prove the property of (over)completeness of these states.
M. Arik and D.D. Coon, J. Math. Phys. 17, 524 (1976).
L.C. Biedenharn, J. Phys. A: Math. Gen. 22, L873 (1989).
A.J. Macfarlane, J. Phys. A: Math. Gen. 22, 4581 (1989).
R. Chakrabarty and R.J. Jagannathan, J. Phys. A: Math. Gen. 24, L711 (1991).
S. Jing, Nuovo Cim. A 105, 1267 (1992).
I.M. Burban and A.U. Klimyk, Integr. Transform. Spec. Funct. 2, 15 (1994).
M. Arik, E. Demircan, T. Turgut, and L. Ekinci, and M. Mungan, Z. Phys. C 55, 89 (1992).
J. Ben Geloun, and M.N. Hounkonnou, J. Phys. A: Math. Theor. 40, F817 (2007).
Chung, K.-S. Chung, S.-T. Nam, and C.-I. Um, Phys. Lett. A 183, 363 (1993).
V.V. Borzov, E.V. Damaskinsky, and S.V. Yegorov, Some remarks on the representations of the generalized deformed algebra, q-alg/9509022.
I.M. Burban, Phys. Lett. A 366, 308 (2007).
S.S. Mizrahi, J.P. Lima, and V.V. Dodonov, J. Phys. A: Math. Gen. 37, 3707 (2004).
I.M. Burban, Mod. Phys. Lett. A 25, 1239 (2010).
I.M. Burban, J. Phys. A: Math. Gen. 43, 305204 (2010).
N. Reshetikhin, Lett. Math. Phys. 20, 321 (1990).
CZ T.L. Curtright and C.K. Zachos, Phys. Lett. B 243, 237 (1990).
A.P. Polychronacos, Mod. Phys. Lett. A 5, 2325 (1990).
D.F. Walls and G.J. Milburn, Quantum Optics (Springer, Berlin, 1994).
M.A. Vasiliev, Int. J. Mod. Phys. A 6, 1115 (1991).
L. Brink, T.H. Hanson, and M.A. Vasiliev, Phys. Lett. 286, 109 (1992).
C. Quesne and N. Vanstineekiste, C_λ-extended oscillator algebras and some of their deformation, math-ph/0003025.
J. Engquist, Anyons, deformed oscillator algebras and projectors hep-th/0809.3226 v 1.
T. Brzezinski, I.L. Egusquiza, and A. Macfarlane, Phys. Lett. B 276, 10 (1992).
K. Odaka, T. Kishi, and S.J. Kamefuchi, J. Phys. A: Math. Gen. 24, L591 (1991).
C.J. Daskaloyannis, J. Phys. A 24, L789 (1991).
S. Meljanac, M. Milekovic, and S. Pallua, Phys. Lett. 11, 3081 (1996).
F.J. Calogero, Math. Phys. 10, 2191 (1969).
I.M. Burban, J. Phys. A: Math. Gen. 42, 065201 (2009).
G. Rideau, Lett. Math. Phys. 24, 147 (1992).
H. Chaichian and P.J. Grosse, J. Phys. A: Math. Gen. 27, 2045 (1994).
C. Quesne and N. Vanstineekiste, Representation theory of generalized oscillator algebras, q-alg/9701031.
P. Kosinski, M. Majevski, and P. Maslianka, Representation of generalized oscillator algebra, q-alg/9501012.
V.V. Borzov, and E.V. Damaskinsky, Zap. Nauchn. Sem. ROMI 308, 48 (2004).
I.M. Burban and A.U. Klimyk, Lett. Math. Phys. 29, 13 (1993).
R.A. Askey, in q-Series and Partitions, edited by D. Stanton (Springer, New York, 1989), p. 151.
Yu.M. Berezansky, Expansions in Eigenfunctions of Self-Adjoint Operators (Amer. Math. Soc., Providence, RI, 1969).
M.E.H. Ismail and D.R. Masson, Trans. Amer. Math. Soc. 346, 43 (1994).
R. Koekoek and R.F. Swarttouw, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-analogue (Delft University of Technology, Report 98-17, 1998).
G. Gasper and M. Rahman, Basic Hypergeometric Series (Cambridge Univ. Press, Cambridge, 1990).
J. Ben Geloun and M.N. Hounkonnou, J. Math. Phys. 48, 093505 (2007).
E.H. El Kinani and A. Ouhadon, Int. J. Algebra. 2, 751 (2008).
A. Aleixo and A.B. Balatekin, J. Math. Phys. 50, 112103 (2009).
M.N. Hounkonnou and E.B.N. Nkouankam, J. Math. Phys. 51, 103517 (2010).
A.M. Gavrililk, I.I. Kachurik, and A.P. Rebesh, J. Phys. A: Math. Theor. 43, 245294 (2010).
A.M. Gavrililk and A.P. Rebesh, J. Phys. A: Math. Theor. 43, 095203 (2010).
How to Cite
License to Publish the Paper
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.