zbMATH — the first resource for mathematics

Difference analogs of the harmonic oscillator. (English. Russian original) Zbl 1189.81099
Theor. Math. Phys. 85, No. 1, 1055-1062 (1990); translation from Teor. Mat. Fiz. 85, No. 1, 64-73 (1990).
From the text: The wave functions of the harmonic oscillator are often used as a basis to describe the properties of various quantum-mechanical systems. It is also of interest to construct difference analogs of these wave functions, which could be used to study physical systems on lattices. In this paper, we therefore consider two models of a difference oscillator; their wave functions admit representation in terms of Kravchuk (Krawtchouk) polynomials and Hermite \(q\)-polynomials, respectively. Factorization of the difference Hamiltonian makes it possible to determine the generators of the dynamical symmetry group for both models. The connections with representations of the classical and quantum \(\text{SU}(2)\) groups are discussed.

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81R12 Groups and algebras in quantum theory and relations with integrable systems
Full Text: DOI
[1] A. Érdelyi et al. (eds), Higher Transcendental Functions (California Institute of Technology H. Bateman M. S. Project) Vol. 2, McGraw Hill, New York (1953).
[2] A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable [in Russian], Nauka, Moscow (1985). · Zbl 0642.33020
[3] N. M. Atakishiev and S. K. Suslov, in: Modern Group Analysis: Methods and Applications [in Russian], Élm, Baku (1989), pp. 17-20.
[4] I. A. Malkin and V. I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979).
[5] N. M. Atakishiyev, Lecture Notes in Phys., Vol. 180, Springer-Verlag, Berlin (1983). pp. 393-396; N. M. Atakishiev, Teor. Mat. Fiz.,56, 154 (1983).
[6] I. M. Gel’fand, R. A. Minlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications, Pergamon Press, Oxford (1963).
[7] E. Inönü and E. P. Wigner, Proc. Nat. Acad. Sci. USA,39, 510 (1953). · Zbl 0050.02601 · doi:10.1073/pnas.39.6.510
[8] W. Miller (Jr.), Symmetry Groups and their Applications, Academic Press, New York (1972). · Zbl 0306.22001
[9] L. D. Landau and E. M. Lifshitz, Quantum Mechanics; Non-Relativistic Theory, 3rd ed., Pergamon Press, Oxford (1977). · Zbl 0178.57901
[10] A. M. Perelomov, Generalized Coherent States and their Applications [in Russian], Nauka, Moscow (1987). · Zbl 0672.22019
[11] A. J. Macfarlane, J. Phys. A,22, 4581 (1989). · Zbl 0722.17009 · doi:10.1088/0305-4470/22/21/020
[12] L. C. Biedenharn, J. Phys. A,22, L873 (1989). · Zbl 0708.17015 · doi:10.1088/0305-4470/22/18/004
[13] L. J. Rogers, Proc. London Math. Soc. (2),25, 318 (1894). · doi:10.1112/plms/s1-25.1.318
[14] L. J. Rogers, Proc. London Math. Soc. (2),16, 315 (1917).
[15] G. Szegö, Sitz. Preuss. Akad. Wiss. Phys,-Math. Kl.19, 242 (1926).
[16] R. Askey and M. E. H. Ismail, Studies in Pure Mathematics (P. Erdös, ed.), Birkhauser, Boston, Mass. (1983), pp. 55-78.
[17] R. Askey and J. A. Wilson, Memoirs Am. Math. Soc., No. 319 (1985).
[18] N. M. Atakishiyev and S. K. Suslov, ?On the Askey-Wilson polynomials,?, Preprint KMUNTZ 89-04, Leipzig (1989).
[19] M. E. H. Ismail, D. Stanton, and G. Viennot, Europ. J. Combinatorics,8, 379 (1987). · Zbl 0642.33006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.