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


SIGMA 6 (2010), 066, 23 pages      arXiv:1006.0864      http://dx.doi.org/10.3842/SIGMA.2010.066

Tools for Verifying Classical and Quantum Superintegrability

Ernest G. Kalnins a, Jonathan M. Kress b and Willard Miller Jr. c
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
b) School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia
c) School of Mathematics, University of Minnesota, Minneapolis, Minnesota,55455, USA

Received June 04, 2010, in final form August 06, 2010; Published online August 18, 2010

Abstract
Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n−1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the constructions to date are for n=2 but cases where n>2 are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mechanisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stäckel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum superintegrable for all rational values of a parameter k in the potential.

Key words: superintegrability; hidden algebras; quadratic algebras.

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References

  1. Tempesta P., Winternitz P., Harnad J., Miller W. Jr., Pogosyan G. (Editors), Superintegrability in classical and quantum systems (Montréal, 2002), CRM Proceedings and Lecture Notes, Vol. 37, American Mathematical Society, Providence, RI, 2004.
  2. Eastwood M., Miller W. Jr. (Editors), Symmetries and overdetermined systems of partial differential equations, Summer Program (Minneapolis, 2006), The IMA Volumes in Mathematics and its Applications, Vol. 144, Springer, New York, 2008.
  3. 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.
  4. Kalnins E.G., Kress J.M., Pogosyan G.S., Miller W. Jr., Completeness of superintegrability in two-dimensional constant curvature spaces, J. Phys. A: Math Gen. 34 (2001), 4705-4720, math-ph/0102006.
  5. 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.
  6. Kalnins E.G., Kress J.M., Miller W. Jr., Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties, J. Math. Phys. 48 (2007), 113518, 26 pages, arXiv:0708.3044.
  7. 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.
  8. Kalnins E.G., Miller W. Jr., Post S., Quantum and classical models for quadratic algebras associated with second order superintegrable systems, SIGMA 4 (2008), 008, 21 pages, arXiv:0801.2848.
  9. Chanu C., Degiovanni L., Rastelli G., Superintegrable three-body systems on the line,  J. Math. Phys. 49 (2008), 112901, 10 pages, arXiv:0802.1353.
  10. Rodriguez M.A., Tempesta P., Winternitz P., Reduction of superintegrable systems: the anisotropic harmonic oscillator, Phys. Rev. E 78 (2008), 046608, 6 pages, arXiv:0807.1047.
    Rodriguez M.A., Tempesta P., Winternitz P., Symmetry reduction and superintegrable Hamiltonian systems, J. Phys. Conf. Ser. 175 (2009), 012013, 8 pages, arXiv:0906.3396.
  11. Verrier P.E., Evans N.W., A new superintegrable Hamiltonian, J. Math. Phys. 49 (2008), 022902, 8 pages, arXiv:0712.3677.
  12. Tremblay F., Turbiner V.A., Winternitz P., An infinite family of solvable and integrable quantum systems on a plane, J. Phys. A: Math. Theor. 42 (2009), 242001, 10 pages, arXiv:0904.0738.
  13. Tremblay F., Turbiner A., Winternitz P., Periodic orbits for an infinite family of classical superintegrable systems, J. Phys. A: Math. Theor. 43 (2010), 015202, 14 pages, arXiv:0910.0299.
  14. Quesne C., Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd k, J. Phys. A: Math. Theor. 43 (2010), 082001, 10 pages, arXiv:0911.4404.
  15. Tremblay F., Winternitz P., Third order superintegrable systems separating in polar coordinates, J. Phys. A: Math. Theor. 43 (2010), 175206, 17 pages, arXiv:1002.1989.
  16. Post S., Winternitz P., An infinite family of superintegrable deformations of the Coulomb potential, J. Phys. A: Math. Theor. 43 (2010), 222001, 11 pages, arXiv:1003.5230.
  17. Kalnins E.G., Miller W. Jr., Post S., Coupling constant metamorphosis and Nth-order symmetries in classical and quantum mechanics, J. Phys. A: Math. Theor. 43 (2010), 035202, 20 pages, arXiv:0908.4393.
  18. Kalnins E.G., Kress J.M., Miller W. Jr., Families of classical subgroup separable superintegrable systems, J. Phys. A: Math. Theor. 43 (2010), 092001, 8 pages, arXiv:0912.3158.
  19. Kalnins E.G., Kress J.M., Miller W. Jr., Superintegrability and higher order integrals for quantum systems, J. Phys. A: Math. Theor. 43 (2010), 265205, 21 pages, arXiv:1002.2665.
  20. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Superintegrability and higher order constants for classical and quantum systems, Phys. Atomic Nuclei, to appear, arXiv:0912.2278.
  21. Eisenhart L.P., Separable systems of Stäckel, Ann. of Math. (2) 35 (1934), 284-305.
  22. 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.
  23. 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.
  24. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, UK, 1999.

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