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


SIGMA 4 (2008), 008, 21 pages      arXiv:0801.2848      http://dx.doi.org/10.3842/SIGMA.2008.008
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D

Ernest G. Kalnins a, Willard Miller Jr. b and Sarah Post b
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
b) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

Received October 25, 2007, in final form January 15, 2008; Published online January 18, 2008

Abstract
There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study the finite and infinite irreducible representations of the quantum quadratic algebras though the construction of models in which the symmetries act on spaces of functions of a single complex variable via either differential operators or difference operators. In another paper we have already carried out parts of this analysis for the generic nondegenerate superintegrable system on the complex 2-sphere. Here we carry it out for a degenerate superintegrable system on the 2-sphere. We point out the connection between our results and a position dependent mass Hamiltonian studied by Quesne. We also show how to derive simple models of the classical quadratic algebras for superintegrable systems and then obtain the quantum models from the classical models, even though the classical and quantum quadratic algebras are distinct.

Key words: superintegrability; quadratic algebras; Wilson polynomials.

pdf (297 kb)   ps (199 kb)   tex (26 kb)

References

  1. Wojciechowski S., Superintegrability of the Calogero-Moser system, Phys. Lett. A 95 (1983), 279-281.
  2. Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 1990, 5666-5676.
    Evans N.W., Group theory of the Smorodinsky-Winternitz system, J. Math. Phys. 32 (1991), 3369-3375.
  3. Evans N.W., Super-integrability of the Winternitz system, Phys. Lett. A 147 (1990), 483-486.
  4. Fris J., Mandrosov V., Smorodinsky Ya.A., Uhlír M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354-356.
  5. Bonatos D., Daskaloyannis C., Kokkotas K., Deformed oscillator algebras for two-dimensional quantum superintegrable systems, Phys. Rev. A 50 (1994), 3700-3709, hep-th/9309088.
  6. 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.
  7. Letourneau P., Vinet L., Superintegrable systems: polynomial algebras and quasi-exactly solvable Hamiltonians, Ann. Phys. 243 (1995), 144-168.
  8. Rañada M.F., Superintegrable n = 2 systems, quadratic constants of motion, and potentials of Drach, J. Math. Phys. 38 (1997), 4165-4178.
  9. Kalnins E.G., Miller W. Jr., Williams G.C., Pogosyan G.S., On superintegrable symmetry-breaking potentials in n-dimensional Euclidean space, J. Phys. A: Math. Gen. 35 (2002), 4655-4720.
  10. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Completeness of multiseparable superintegrability in E2,C, J. Phys. A: Math. Gen. 33 (2000), 4105-4120.
  11. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Completeness of multiseparable superintegrability on the complex 2-sphere, J. Phys. A: Math. Gen. 33 (2000), 6791-6806.
  12. Koenigs G., Sur les géodésiques a intégrales quadratiques, A note appearing in "Lecons sur la théorie générale des surfaces", G. Darboux, Vol. 4, Chelsea Publishing, 1972, 368-404.
  13. Calogero F., Solution to the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436.
  14. Rauch-Wojciechowski S., Waksjö C., What an effective criterion of separability says about the Calogero type systems, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 535-547.
  15. Kalnins E.G., Kress J.M., Miller W. Jr., Pogosyan G.S., Completeness of superintegrability in two-dimensional constant curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705-4720, math-ph/0102006.
  16. Zhedanov A.S., "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.
  17. Granovskii Ya.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved spaces. I. Oscillator, Theoret. and Math. Phys. 89 (1992), 474-480.
  18. Granovskii Ya.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved spaces. II. The Kepler problem, Theoret. and Math. Phys. 91 (1992), 604-612.
  19. Quesne C., Generalized deformed parafermions, nonlinear deformations of so(3) and exactly solvable potentials, Phys. Lett. A 193 (1994), 245-250.
  20. Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42 (2001), 4248-4257.
  21. Gravel S., Winternitz P., Superintegrability with third-order integrals in quantum and classical mechanics, J. Math. Phys. 43 (2002), 5902-5912, math-ph/0206046.
  22. Ballesteros A., Herranz F., Santander M., Sanz-Gil T., Maximal superintegrability on N-dimensional curved spaces, J. Phys. A: Math. Gen. 36 (2003), L93-L99, math-ph/0211012.
  23. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. I. 2D classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
  24. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. II. The classical 2D Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
  25. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. III. 3D classical structure theory, J. Math. Phys. 46 (2005), 103507, 28 pages.
  26. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory, J. Math. Phys. 47 (2006), 043514, 26 pages.
  27. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. V. 2D and 3D quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages.
  28. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Exact and quasi-exact solvability of second order superintegrable systems. I. Euclidean space preliminaries, J. Math. Phys. 47 (2006), 033502, 30 pages, math-ph/0412035.
  29. Kalnins E.G., Kress J.M., Winternitz P., Superintegrability in a two-dimensional space of non-constant curvature, J. Math. Phys. 43 (2002), 970-983, math-ph/0108015.
  30. Kalnins E.G., Kress J.M., Miller W. Jr., Winternitz P., Superintegrable systems in Darboux spaces, J. Math. Phys. 44 (2003), 5811-5848, math-ph/0307039.
  31. 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.
  32. Horwood J.T., McLenaghan R.G., Smirnov R.G., Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space, Comm. Math. Phys. 259 (2005), 679-709, math-ph/0605023.
  33. Tempesta P., Winternitz P., Miller W., Pogosyan G. (Editors), Superintegrability in classical and quantum systems, CRM Proceedings Lecture Notes, Vol. 37, American Mathematical Society, Providence, RI, 2004.
  34. Kalnins E.G., Kress J.M., Miller W. Jr., Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties, J. Phys. A: Math. Theor. 40 (2007), 3399-3411.
  35. Kalnins E.G., Kress J.M., Miller W. Jr., Fine structure for 3D second order superintegrable systems: 3-parameter potentials, J. Phys. A: Math. Theor. 40 (2007), 5875-5892.
  36. Kalnins E.G., Kress J.M., Miller W. Jr., Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties, J. Math. Phys. 48 (2007), 113518, 26 pages, arXiv:0708.3044.
  37. 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.
  38. Daskaloyannis C., Tanoudis Y., Quantum superintegrable systems with quadratic integrals on a two dimensional manifold, J. Math. Phys. 48 (2007), 072108, 22 pages, math-ph/0607058.
  39. 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.
  40. Gravel S., Hamilton separable in Cartesian coordinates and third-order integrals of motion, J. Math. Phys. 45 (2004), 1003-1019.
  41. Ballesteros A., Herranz F.J., Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature, J. Phys. A: Math. Theor. 40 (2007), F51-F59, math-ph/0610040.
  42. Fordy A.P., Quantum super-integrable systems as exactly solvable models, SIGMA 3 (2007), 025, 10 pages, math-ph/0702048.
  43. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Exact and quasi-exact solvability of second order superintegrable quantum systems. II. Connection with separation of variables, J. Math. Phys. 48 (2007), 023503, 20 pages.
  44. Kress J.M., Equivalence of superintegrable systems in two dimensions, Phys. Atomic Nuclei 70 (2007), 560-566.
  45. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1999.
  46. Kalnins E.G., Separation of variables for Riemannian spaces of constant curvature, Pitman, Monographs and Surveys in Pure and Applied Mathematics, Vol. 28, Longman, Essex, England, 1986.
  47. Arnold V.I., Mathematical methods of classical mechanics (translated by K. Vogtmann and A. Weinstein), Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York, 1978.

Previous article   Next article   Contents of Volume 4 (2008)