Geodesic
Generalized coherent states for oscillators connected with Meixner and Meixner–Pollachek polynomials
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 18, Tome 317 (2004), pp. 66-93 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We are continuing here the study of generalized coherent states for the oscillator-like systems connected with the given set of orthogonal polynomials. In this work we concider the case of Meixner and Meixner–Pollachek polynomials. We construct the oscillator-like systems associated with these polynomials and define their coherent states. By the solution of the related classical moment problem we prove the overcompleteness of constructed families of coherent states. We compare the obtained results with the results of other works. In particular, we shaw that the Hamiltonian of the relativistic model of linear harmonic oscillator can be concidered as special linearization of the quadratic Hamiltonin naturally appearing in our approach. 
@article{ZNSL_2004_317_a4,
     author = {V. V. Borzov and E. V. Damaskinsky},
     title = {Generalized coherent states for oscillators connected with {Meixner} and {Meixner{\textendash}Pollachek} polynomials},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {66--93},
     year = {2004},
     volume = {317},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a4/}
}
TY  - JOUR
AU  - V. V. Borzov
AU  - E. V. Damaskinsky
TI  - Generalized coherent states for oscillators connected with Meixner and Meixner–Pollachek polynomials
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2004
SP  - 66
EP  - 93
VL  - 317
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a4/
LA  - ru
ID  - ZNSL_2004_317_a4
ER  - 
%0 Journal Article
%A V. V. Borzov
%A E. V. Damaskinsky
%T Generalized coherent states for oscillators connected with Meixner and Meixner–Pollachek polynomials
%J Zapiski Nauchnykh Seminarov POMI
%D 2004
%P 66-93
%V 317
%U http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a4/
%G ru
%F ZNSL_2004_317_a4
V. V. Borzov; E. V. Damaskinsky. Generalized coherent states for oscillators connected with Meixner and Meixner–Pollachek polynomials. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 18, Tome 317 (2004), pp. 66-93. http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a4/

[1] R. Floreanini, L. Vinet, “$q$-orthogonal polynomials and the oscillator quantum group”, Lett. Math. Phys., 22:1 (1991), 45–54 | DOI | MR | Zbl

[2] R. Floreanini, L. Vinet, Representations of quantum algebras and $q$-special functions, arXiv: /hep-th/9111023 | MR

[3] E. V. Damaskinskii, P. P. Kulish, “$q$-polinomy Ermita i $q$-ostsillyatory”, Zap. nauchn. semin. LOMI, 199, 1992, 81–90 | MR | Zbl

[4] Z. Chang, H.-Y. Guo, H. Yan, “The $q$-Hermite polynomial and the representations of Heisenberg and quantum Heisenberg algebra”, J. Phys. A, 25:6 (1992), 1517–1526 | DOI | MR

[5] R. Floreanini, J. Le Tourneaux, L. Vinet, “More on the $q$-oscillator algebra and $q$-orthogonal polynomials”, J. Phys. A, 28:10 (1994), L287–L294 ; arXiv: /math.CA/9504218 | DOI | MR

[6] R. Floreanini, J. Le Tourneaux, L. Vinet, “An algebraic interpretation of the continuous big $q$-Hermite polynomials”, J. Math. Phys., 36:9 (1995), 5091–5097 | DOI | MR | Zbl

[7] E. V. Damaskinsky, P. P. Kulish, “Irreducible representations of deformed oscillator algebra and $q$-special functions”, Intern. J. Mod. Phys. A, 12:1 (1996), 153–158 ; arXiv: /q-alg/9610002 | DOI | MR

[8] R. Hinterding, J. Wess, “$q$-deformed Hermite polinomials in $q$-quantum mechanics”, Z. Phys. C, 71 (1996), 533–537 ; arXiv: /math.QA/9803050 | DOI | MR

[9] H.-C. Fu, R. Sasaki, “Exponential and Laguerre squeezed states for $su(1,1)$ algebra and the Calogero–Sutherland model”, Phys. Rev. A, 53:6 (1996), 3836–3844 | DOI

[10] N. M. Atakishiyev, E. I. Jafarov, Sh. M. Nagiyev, K. B. Wolf, “Meixner oscillators”, Revista Mexicana de Fisica, 44:3 (1998), 235–244 ; arXiv: /math-ph/9807035 | MR

[11] A. Odzijewicz, “Quantum algebras and $q$–special functions related to coherent states maps of the disk”, Commun. Mat. Phys., 192 (1998), 183–215 | DOI | MR | Zbl

[12] J.-P. Gazeau, J. R. Klauder, “Coherent states for systems with discrete and continuous spectrum”, J. Phys. A, 32:1 (1999), 123–132 | DOI | MR | Zbl

[13] K. A. Penson, A. I. Solomon, “New generalized coherent states”, J. Math. Phys., 40:5 (1999), 2354–2363 | DOI | MR | Zbl

[14] X.-G. Wang, Coherent states, displaced number states and Laguerre polynomial states for $su(1,1)$ Lie algebra, arXiv: /quant-ph/0001002 | MR

[15] J.-P. Antoine, J.-P. Gazeau, P. Monceau, J. R. Klauder, K. A. Penson, “Temporally stable coherent states for infinite well and Pöschl–Teller potentials”, J. Math. Phys., 42 (2001), 2349–2386 ; arXiv: /math-ph/0012044 | DOI | MR

[16] J. R. Klauder, K. A. Penson, J.-M. Sixdeniers, “Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems”, Phys. Rev. A, 64:1 (2001), 013817 | DOI | MR

[17] V. V. Borzov, Generalized Hermite polynomials, arXiv: /math.QA/0101216 | MR

[18] A. Jellal, Coherent States for Generalized Laguerre Functions, arXiv: /hep-th/0109028 | MR

[19] A. Odzijewicz, M. Horowski, A. Tereszkiewicz, “Integrable multi-boson systems and orthogonal polynomials”, J. Phys. A, 34 (2001), 4353–4376 | DOI | MR | Zbl

[20] N. Cotfas, “Shape invariance, raising and lowering operators in hypergeometric type equations”, J. Phys. A, 35:44 (2002), 9355–9365 ; arXiv: /quant-ph/0206129 | DOI | MR | Zbl

[21] A. H. El Kinani, M. Daoud, “Generalized coherent and intelligent states for exact solvable quantum systems”, J. Math. Phys., 43 (2002), 714–33 ; arXiv: /math-ph/0312040 | DOI | MR

[22] T. Appl, D. H. Schiller, “Generalized Hypergeometric Coherent States”, J. Phys. A, 37:7 (2004), 2731–2750 ; arXiv: /quant-ph/0308013 | DOI | MR | Zbl

[23] S. T. Ali, J.-P. Antoine, J.-P. Gazeau, U. A. Mueller, “Coherent states and their generalizations: A mathematical overview”, Rev. Math. Phys., 7:7 (1995), 1013–1104 | DOI | MR | Zbl

[24] V. V. Dodonov, ““Nonclassical” states in quantum optics: a “squeezed” review of the first 75 years”, J. Opt. B, 4:1 (2002), R1–R33 | MR

[25] V. V. Borzov, E. V. Damaskinsky, P. P. Kulish, “Construction of the spectral measure for the deformed oscillator position operator in the case of undetermined Hamburger moment problem”, Reviews in Math. Phys., 12:5 (2000), 691–710 ; arXiv: /math.QA/9803089 | DOI | MR | Zbl

[26] V. V. Borzov, E. V. Damaskinsky, “Realization of the annihilation operator for an oscillator-like system by a differential operator and Hermite–Chihara polynomials”, Integral Transforms and Special Functions, 13:6 (2002), 547–554 ; arXiv: /math.QA/0101215 | DOI | MR | Zbl

[27] V. V. Borzov, E. V. Damaskinskii, “Kogerentnye sostoyaniya dlya ostsillyatora Lezhandra”, Zap. nauchn. semin. POMI, 285, 2002, 39–52 ; arXiv: /math.QA/0307187 | MR | Zbl

[28] V. V. Borzov, E. V. Damaskinskii, “Kogerentnye sostoyaniya i polinomy Chebysheva”, Matematicheskie idei P. L. Chebysheva i ikh prilozheniya k sovremennym problemam estestvoznaniya, sb. trudov mezhdunarodnoi konferentsii (Obninsk, 14–18 maya 2002)

[29] V. V. Borzov, E. V. Damaskinskii, Kogerentnye sostoyaniya i ortogonalnye mnogochleny, Trudy konferentsii “Den Difraktsii 2002”, SPb., 2002; arXiv: /math.QA/0209181

[30] V. V. Borzov, E. V. Damaskinskii, “Kogerentnye sostoyaniya Baruta–Zhirardello dlya ostsillyatora Gegenbauera”, Zap. nauchn. semin. POMI, 291, 2002, 43–63 | MR | Zbl

[31] V. V. Borzov, E. V. Damaskinsky, “Generalized Coherent states: A Novel Approach”, Zap. nauchn. semin. POMI, 300, 2003, 65–70 | MR

[32] V. V. Borzov, E. V. Damaskinsky, Generalized Coherent States for $q$-oscillator connected with $q$-Hermite Polynomials, Day on Diffraction 2003, SPb 2003

[33] V. V. Borzov, E. V. Damaskinskii, “Obobschennye kogerentnye sostoyaniya dlya $q$-ostsillyatora, assotsiirovannogo s diskretnymi $q$-polinomami Ermita”, Matemat. voprosy teorii rasprostraneniya voln, Zap. nauchn. POMI, 308, 2004, 48–66 | MR | Zbl

[34] V. V. Borzov, E. V. Damaskinskii, “Ostsillyatoro-podobnye sistemy, svyazannye s ortogonalnymi mnogochlenami, i ikh kogerentnye sostoyaniya”, Gravitatsiya, kosmologiya i elementarnye chastitsy, Sankt-Peterburg, 2004, 9–19

[35] V. V. Borzov, “Orthogonal polynomials and generalized oscillator algebras”, Integral Transf. and Special Funct., 12:2 (2001), 115–138 ; arXiv: /math.CA/0002226 | DOI | MR | Zbl

[36] A. O. Barut, L. Girardello, “New “Coherent States” Associated with Non-Compact Groups”, Commun. Math. Phys., 21:1 (1972), 41–55 | DOI | MR

[37] A. M. Perelomov, Obobschennye kogerentnye sostoyaniya i ikh primeneniya, Nauka, FM, M., 1987 | MR

[38] W. Hahn, “Über Orthogonalpolynome, die $q$-Differenzengleichungen genügen”, Math. Nachr., 2 (1949), 4–34 | DOI | MR | Zbl

[39] R. Koekoek, R. F. Swarttouw, The Askey sheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report No 94-05, Delft University of Technology, 1994 ; arXiv: /math.CA/9602214 | Zbl

[40] N. M. Atakishiyev, S. K. Suslov, “The Hahn and Meixner polynomials of an imaginary argument and some of their applications”, J. Phys., 18:10 (1985), 1583–1596 | MR

[41] R. A. Askey, “Continuous Hahn polynomials”, J. Phys. A, 18:16 (1985), L1017–L1019 | DOI | MR | Zbl

[42] C. M. Bender, L. R. Mead, S. S. Pinsky, “Continuous Hahn polynomials and the Heisenberg algebra”, J. Math. Phys., 28:3 (1987), 509–513 | DOI | MR | Zbl

[43] T. H. Koornwinder, “Meixner-Pollaczek polynomials and the Heizenberg algebra”, J. Math. Phys., 30:4 (1989), 767–769 | DOI | MR | Zbl

[44] J. Meixner, “Orthogonale Polynomsysteme mit einem besonderen Gestalt der erzeugenden Funktion”, J. London Math. Soc., 9 (1934), 6–13 | DOI

[45] F. Pollaczek, “Sur une generalisation des polynomes de Legendre”, C. R. Acad. Sci. Paris, 230 (1950), 1563 | MR | Zbl

[46] Sh. M. Nagiev, E. I. Jafarov, R. M. Imanov, “The Relativistic Linear Singular Oscillator”, J. Phys., 36:28 (2003), 7813–7824 ; arXiv: /math-ph/0302042 | MR

[47] N. M. Atakishiev, “Kvazipotentsialnye volnovye funktsii relyativistskogo garmonicheskogo ostsillyatora i mnogochleny Pollacheka”, TMF, 58:2 (1984), 254–260 | MR

[48] N. M. Atakishiev, K. B. Wolf, “Generalised coherent states for a relativistic model of the linear oscillator in a homogeneous external field”, Repts. Math. Phys., 27:3 (1989), 305–311 | DOI | MR | Zbl

[49] E. D. Kagramanov, R. M. Mir-Kasimov, Sh. M. Nagiev, “The covariant linear oscillator and generalized realization of the dynamical $su(1,1)$ symmetry algebra”, J. Math. Phys., 31:7 (1990), 1733–1738 | DOI | MR | Zbl

[50] R. M. Mir-Kasimov, “$SU_q(1,1)$ and relativistic oscillator”, J. Phys. A, 24:18 (1991), 4275–4282 | DOI | MR

[51] N. M. Atakishiev, Sh. M. Nagiev, K. B. Volf, “O funktsiyakh raspredeleniya Vignera dlya relyativistskogo lineinogo ostsillyatora”, TMF, 114:3 (1998), 410–425 | MR | Zbl

[52] N. M. Atakishiev, R. M. Mir-Kasimov, Sh. M. Nagiev, TMF, 44 (1980), 47–53 | MR

[53] G. Beitman, A. Erdeii, Vysshie transtsendentnye funktsii, vyp. 2, Nauka, FM, M., 1966

[54] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii, Izd.5-e, Nauka, M., 1971

[55] N. I. Akhiezer, Klassicheskaya problema momentov i nekotorye voprosy analiza, svyazannye s neyu, FM, M., 1961

[56] B. Simon, “The classical moment problem as a self-adjoint finite difference operator”, Advances in Mathematics, 137, 1998, 82–203 | MR | Zbl