Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 036, 20 pages      arXiv:1104.0734      http://dx.doi.org/10.3842/SIGMA.2011.036
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

Sarah Post
Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128 succ. Centre-Ville, Montréal (QC) H3C 3J7, Canada

Received February 01, 2011, in final form March 24, 2011; Published online April 05, 2011

Abstract
In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of Stäckel equivalent systems for both degenerate and nondegenerate systems. In almost all cases, the models can be used to determine the quantization of energy and eigenvalues for integrals associated with separation of variables in the original system.

Key words: quadratic algebras; superintegrability; special functions; representation theory.

pdf (394 kb)   tex (21 kb)

References

  1. Miller W. Jr., Lie theory and special functions, Mathematics in Science and Engineering, Vol. 43, Academic Press, New York - London, 1968.
  2. Fris I., Mandrosov V., Smorodinsky Ya.A., Uhlír M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354-356.
  3. Makarov A.A., Smorodinsky Ya.A., Valiev Kh., Winternitz P., A systematic search for non-relativistic system with dynamical symmetries, Nuovo Cim. A 52 (1967), 1061-1084.
  4. Kalnins E.G., Pogosyan G.S., Miller W. Jr., Completeness of multiseparable superintegrability in two dimensions, Phys. Atomic Nuclei 6 (2002), 1033-1035.
  5. Kalnins E.G., Kress J.M., Miller W. Jr., Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
  6. Kalnins E.G., Kress J.M., Miller W. Jr., Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages.
  7. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
  8. Post S., Coupling constant metamorphosis, the Stäckel transform and superintegrability, in Symmetries in Nature: Symposium in Memoriam Marcos Moshinsky (August 9-14, 2010, Cuernavaca, Mexico), Editors L. Benet, P.O. Hess, J.M. Torres and K.B. Wolf, AIP Conf. Proc. 1323 (2011), 265-274.
  9. 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 (2006), 042904, 38 pages, math-ph/0412055.
  10. Daskaloyannis C., Ypsilantis K., Quantum superintegrable systems with quadratic integrals on a two dimensional manifold, J. Math. Phys. 48 (2007), 072108, 22 pages, math-ph/0607058.
  11. Kress J.M., Equivalence of superintegrable systems in two dimensions, Phys. Atomic Nuclei 70 (2007), 560-566.
  12. Daskaloyannis C., Tanoudis Y., Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms, Phys. Atomic Nuclei 71 (2008), 853-861.
  13. Kalnins E.G., Kress J.M., Miller W. Jr., Post S., Structure theory for second order 2D superintegrable systems with 1-parameter potentials, SIGMA 5 (2009), 008, 24 pages, arXiv:0901.3081.
  14. Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100-1119, math-ph/0003017.
  15. Daskaloyannis C., Tanoudis Y., Ternary Poisson algebra for the nondegenerate three dimensional Kepler-Coulomb potential, in Proceedings of the Fourth International Workshop on Group Analysis of Differential Equations and Integrable Systems (October 26-30, 2008, Protaras, Cyprus), University of Cyprus, Nicosia, 2009, 173-181, arXiv:0902.0259.
  16. Kalnins E.G., Miller W. Jr., Post S., Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere, arXiv:1010.3032.
  17. Marquette I. Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras, J. Math. Phys. 51 (2010), 102105, 10 pages, arXiv:1004.4579.
  18. Quesne C., Quadratic algebra approach to an exactly solvable position-dependent mass Schrödinger equation in two dimensions, SIGMA 3 (2007), 067, 14 pages, arXiv:0705.2577.
  19. Magnus W., Oberhettinger F., Soni R.P., Formula and theorems for the special functions of mathematical physics, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966.
  20. Kalnins E.G., Miller W. Jr., Post S., Models for quadratic algebras associated with second order superintegrable systems in 2D, SIGMA 4 (2008), 008, 21 pages, arXiv:0801.2848.
  21. Kalnins E.G., Miller W. Jr., Post S., Models of quadratic quantum algebras and their relation to classical superintegrable systems, Phys. Atomic Nuclei 72 (2009), 801-808.
  22. Post S., Models of second-order superintegrable systems, PhD Thesis, University of Minnesota, 2009.
  23. Kalnins E.G., Miller W. Jr., Post S., Wilson polynomials and the generic superintegrable system on the 2-sphere, J. Phys. A: Math. Theor. 40 (2007), 11525-11538.
  24. Kalnins E.G., Miller W. Jr., Reid G.J., Separation of variables for complex Riemannian spaces of constant curvature. I. Orthogonal separable coordinates for SnC and EnC, Proc. Roy. Soc. London Ser. A 39 (1984), 183-206.
  25. Kalnins E.G., Kress J.M., Miller W. Jr., Pogosyan G., Nondegenerate superintegrable systems in n-dimensional Euclidean spaces, Phys. Atomic Nuclei 70 (2007), 545-553.
  26. Daskaloyannis C., Tanoudis Y., Quadratic algebras for three dimensional non degenerate superintegrable systems with quadratic integrals of motion, arXiv:0902.0130.
  27. Kalnins E.G., Miller W. Jr., Post S., Models for the 3D singular isotropic oscillator quadratic algebra, Phys. Atomic Nuclei 73 (2009), 359-366.

Previous article   Next article   Contents of Volume 7 (2011)