Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras. (English) Zbl 1284.81156
Summary: We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.{
©2013 American Institute of Physics}
©2013 American Institute of Physics}
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 17B62 | Lie bialgebras; Lie coalgebras |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |
References:
| [1] | Zhedanov, A. S., Hidden symmetry of Askey-Wilson polynomials, Theor. Math. Phys., 89, 2, 1146-1157 (1991) · Zbl 0782.33012 · doi:10.1007/BF01015906 |
| [2] | Granovskii, Ya. I.; Zhedanov, A. S.; Lutzenko, I. M., Quadratic algebra as a “hidden” symmetry of the Hartmann potential, J. Phys. A, 24, 3887-3894 (1991) · doi:10.1088/0305-4470/24/16/024 |
| [3] | Granovskii, Ya. I.; Zhedanov, A. S.; Lutzenko, I. M., Quadratic algebras and dynamics in curved spaces. II. The Kepler problem, Theor. Math. Phys., 91, 604-612 (1992) · doi:10.1007/BF01017335 |
| [4] | Granovskii, Ya. I.; Lutzenko, I. M.; Zhedanov, A. S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Phys., 217, 1-20 (1992) · Zbl 0875.17002 · doi:10.1016/0003-4916(92)90336-K |
| [5] | Granovskii, Ya. I.; Zhedanov, A. S.; Lutsenko, I. M., Quadratic algebras and dynamics in curved space. I. Oscillator, Theor. Math. Phys., 91, 474-480 (1992) · doi:10.1007/BF01018846 |
| [6] | Zhedanov, A. S., Hidden symmetry algebra and overlap coefficients for two ring-shaped potentials, J. Phys. A, 26, 4633-4641 (1993) · doi:10.1088/0305-4470/26/18/027 |
| [7] | Bonatsos, D.; Daskaloyannis, C.; Kokkotas, K., Quantum-algebraic description of quantum superintegrable systems in two dimensions, Phys. Rev. A, 48, R3407-R3410 (1993) · doi:10.1103/PhysRevA.48.R3407 |
| [8] | Bonatsos, D.; Daskaloyannis, C.; Kokkotas, K., Deformed oscillator algebras for two-dimensional quantum superintegrable systems, Phys. Rev. A, 50, 3700-3709 (1994) · doi:10.1103/PhysRevA.50.3700 |
| [9] | Létourneau, P.; Vinet, L., Superintegrable systems: Polynomial algebras and quasi-exactly solvable Hamiltonians, Ann. Phys. (N.Y.), 243, 144-168 (1995) · Zbl 0843.58062 · doi:10.1006/aphy.1995.1094 |
| [10] | Bonatsos, D.; Daskaloyannis, C., Quantum groups and their applications in nuclear physics, Prog. Part. Nucl. Phys., 43, 537-618 (1999) · doi:10.1016/S0146-6410(99)00100-3 |
| [11] | Daskaloyannis, C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys., 42, 1100-1119 (2001) · Zbl 1053.37033 · doi:10.1063/1.1348026 |
| [12] | Kalnins, E. G.; Kress, J. M.; Miller, W. Jr., Second order superintegrable systems in conformally flat spaces. 1. 2D classical structure theory, J. Math. Phys., 46, 053509 (2005) · Zbl 1110.37054 · doi:10.1063/1.1897183 |
| [13] | Kress, J. M., Equivalence of superintegrable systems in two dimensions, Phys. At. Nucl., 70, 560-566 (2007) · doi:10.1134/S1063778807030167 |
| [14] | Daskaloyannis, C.; Ypsilantis, K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold, J. Math. Phys., 47, 042904 (2006) · Zbl 1111.37041 · doi:10.1063/1.2192967 |
| [15] | Daskaloyannis, C.; Tanoudis, Y., Quantum superintegrable systems with quadratic integrals on a two dimensional manifold, J. Math. Phys., 48, 072108 (2007) · Zbl 1144.81336 · doi:10.1063/1.2746132 |
| [16] | Quesne, C., Quadratic algebra approach to an exactly solvable position-dependent mass Schrödinger equation in two dimensions, SIGMA, 3, 067 (2007) · Zbl 1193.81047 · doi:10.3842/SIGMA.2007.067 |
| [17] | Daskaloyannis, C.; Tanoudis, Y., Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms, Phys. At. Nucl., 71, 853-861 (2008) · doi:10.1134/S106377880805013X |
| [18] | Post, S., Models of quadratic algebras generated by superintegrable systems in 2D, SIGMA, 7, 036 (2011) · Zbl 1217.22020 · doi:10.3842/SIGMA.2011.036 |
| [19] | Kalnins, E. G.; Miller, W. Jr.; Post, S., Models for quadratic algebras associated with second order superintegrable systems in 2D, SIGMA, 4, 008 (2008) · Zbl 1157.81012 · doi:10.3842/SIGMA.2008.008 |
| [20] | Kalnins, E. G.; Miller, W. Jr.; Post, S., Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials · Zbl 1293.33013 |
| [21] | Sklyanin, E. K., Some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl., 16, 4, 263-270 (1982) · Zbl 0536.58007 · doi:10.1007/BF01076718 |
| [22] | Sklyanin, E. K., Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras, Funct. Anal. Appl., 17, 4, 273-284 (1983) · Zbl 0513.58028 · doi:10.1007/BF01077848 |
| [23] | Daskaloyannis, C., Generalized deformed oscillator and nonlinear algebras, J. Phys. A, 24, L789-L794 (1991) · Zbl 0752.17030 · doi:10.1088/0305-4470/24/15/001 |
| [24] | Quesne, C., Generalized deformed parafermions, nonlinear deformations of so(3) and exactly solvable potentials, Phys. Lett. A, 193, 245-250 (1994) · Zbl 0959.81552 · doi:10.1016/0375-9601(94)90591-6 |
| [25] | Marquette, I.; Winternitz, P., Polynomial Poisson algebras for superintegrable systems with a third order integral of motion, J. Math. Phys., 48, 012902 (2007) · Zbl 1121.37043 · doi:10.1063/1.2399359 |
| [26] | Marquette, I., Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I: Rational function potentials, J. Math. Phys., 50, 012101 (2009) · Zbl 1189.81094 · doi:10.1063/1.3013804 |
| [27] | Marquette, I., Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics. II. Painlevé transcendent potentials, J. Math. Phys., 50, 095202 (2009) · Zbl 1225.81071 · doi:10.1063/1.3096708 |
| [28] | Marquette, I., Superintegrability and higher order polynomial algebras, J. Phys. A: Math. Theor., 43, 135203 (2010) · Zbl 1188.81081 · doi:10.1088/1751-8113/43/13/135203 |
| [29] | Marquette, I., An infinite family of superintegrable systems from higher order ladder operators and supersymmetry, Proceeding of the GROUP28: The XXVIII International Colloquium on Group-Theoretical Methods in Physics, J. Phys.: Conf. Ser., 284, 012047 (2011) · doi:10.1088/1742-6596/284/1/012047 |
| [30] | Kalnins, E. G.; Kress, J. M.; Miller, W. Jr., A recurrence relation approach to higher order quantum superintegrability, SIGMA, 7, 31, 24 (2011) · Zbl 1217.81105 · doi:10.3842/SIGMA.2011.031 |
| [31] | Kalnins, E. G.; Kress, J. M.; Miller, W. Jr., Extended Kepler-Coulomb quantum superintegrable systems in 3 dimensions, J. Phys. A: Math. Theor., 46, 085206 (2013) · Zbl 1264.81183 · doi:10.1088/1751-8113/46/8/085206 |
| [32] | Post, S.; Tsujimoto, S.; Vinet, L., Families of superintegrable Hamiltonians constructed from exceptional polynomials, J. Phys. A: Math. Theor., 45, 405202 (2012) · Zbl 1255.81145 · doi:10.1088/1751-8113/45/40/405202 |
| [33] | Marquette, I.; Quesne, C., New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials, J. Math. Phys., 54, 042102 (2013) · Zbl 1282.70036 · doi:10.1063/1.4798807 |
| [34] | Marquette, I.; Quesne, C., New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems · Zbl 1267.81174 |
| [35] | Grabowski, J.; Marmo, G.; Perelomov, A. M., Poisson structures: Towards a classification, Mod. Phys. Lett. A, 08, 1719-1733 (1993) · Zbl 1020.37529 · doi:10.1142/S0217732393001458 |
| [36] | Kalnins, E. G.; Miller, W. Jr.; Pogosyan, G. S., Superintegrability in three dimensional Euclidean space, J. Math. Phys., 40, 708-725 (1999) · Zbl 1059.81525 · doi:10.1063/1.532699 |
| [37] | Kalnins, E. G.; Kress, J. M.; Miller, W. Jr., Second order superintegrable systems in conformally flat spaces. 3. 3D classical structure theory, J. Math. Phys., 46, 103507 (2005) · Zbl 1111.37055 · doi:10.1063/1.2037567 |
| [38] | Daskaloyannis, C.; Tanoudis, Y., Quadratic algebras for three-dimensional nondegenerate superintegrable systems with quadratic integrals of motion, Talk XXVII Colloquium on Group Theoretical Methods in Physics, Yerevan, Armenia, August, 2008 · Zbl 1144.81336 |
| [39] | Daskaloyannis, C.; Tanoudis, Y., Ternary Poisson algebra for the nondegenerate three dimensional Kepler-Coulomb potential, Proceedings of the Fourth International Workshop on Group Analysis of Differential Equations and Integrable Systems, 173-181 (2009) · Zbl 1226.37033 |
| [40] | Daskaloyannis, C.; Tanoudis, Y., Quadratic algebras for three-dimensional superintegrable systems, Phys. At. Nucl., 73, 214-221 (2010) · doi:10.1134/S106377881002002X |
| [41] | Marquette, I., Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras, J. Math. Phys., 51, 102105 (2010) · Zbl 1314.81075 · doi:10.1063/1.3496900 |
| [42] | Marquette, I., Generalized five-dimensional Kepler system, Yang-Coulomb monopole and Hurwitz transformation, J. Math. Phys., 53, 022103 (2012) · Zbl 1274.81088 · doi:10.1063/1.3684955 |
| [43] | Lee, Y.-H.; Yang, W.-L.; Zhang, Y.-Z., Polynomial algebras and exact solutions of general quantum nonlinear optical models I: Two-mode boson systems, J. Phys. A: Math. Theor., 43, 185204 (2010) · Zbl 1187.81151 · doi:10.1088/1751-8113/43/18/185204 |
| [44] | Lee, Y.-H.; Yang, W.-L.; Zhang, Y.-Z., Polynomial algebras and exact solutions of general quantum nonlinear optical models: II. Multi-mode boson systems, J. Phys. A: Math. Theor., 43, 375211 (2010) · Zbl 1198.81226 · doi:10.1088/1751-8113/43/37/375211 |
| [45] | Lee, Y.-H.; Links, J.; Zhang, Y.-Z., Exact solutions for a family of spin-boson systems, Nonlinearity, 24, 1975-1986 (2011) · Zbl 1236.82009 · doi:10.1088/0951-7715/24/7/004 |
| [46] | Terwilliger, P., The universal Askey-Wilson algebra, SIGMA, 7, 69, 24 (2011) · Zbl 1244.33015 · doi:10.3842/SIGMA.2011.069 |
| [47] | Lavrenov, A. N., Deformation of the Askey-Wilson algebra with three generators, J. Phys. A, 28, L503-L506 (1995) · Zbl 0880.17011 · doi:10.1088/0305-4470/28/20/001 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
