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

SIGMA 11 (2015), 038, 17 pages      arXiv:1501.06601      https://doi.org/10.3842/SIGMA.2015.038

Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

Joshua J. Capel a, Jonathan M. Kress a and Sarah Post b
a) Department of Mathematics, University of New South Wales, Sydney, Australia
b) Department of Mathematics, University of Hawai`i at Mānoa, Honolulu, HI, 96822, USA

Received February 03, 2015, in final form April 21, 2015; Published online May 08, 2015

Abstract
The invariant classification of superintegrable systems is reviewed and utilized to construct singular limits between the systems. It is shown, by construction, that all superintegrable systems on conformally flat, 3D complex Riemannian manifolds can be obtained from singular limits of a generic system on the sphere. By using the invariant classification, the limits are geometrically motivated in terms of transformations of roots of the classifying polynomials.

Key words: integrable systems; superintegrable systems; Lie algebra invariants; contractions.

pdf (363 kb)   tex (22 kb)

References

  1. Bôcher M., Über die Riehenentwickelungen der Potentialtheory, B.G. Teubner, Leipzig, 1894.
  2. Capel J.J., Invariant classification of second-order conformally flat superintegrable systems, Ph.D. Thesis, University of New South Wales, 2014.
  3. Capel J.J., Kress J.M., Invariant classification of second-order conformally flat superintegrable systems, J. Phys. A: Math. Theor. 47 (2014), 495202, 33 pages, arXiv:1406.3136.
  4. Friš J., Mandrosov V., Smorodinsky Ya.A., Uhlíř M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354-356.
  5. Genest V.X., Vinet L., The generic superintegrable system on the 3-sphere and the $9j$ symbols of $\mathfrak{su}(1,1)$, SIGMA 10 (2014), 108, 28 pages, arXiv:1404.0876.
  6. Genest V.X., Vinet L., Zhedanov A., Interbasis expansions for the isotropic 3D harmonic oscillator and bivariate Krawtchouk polynomials, J. Phys. A: Math. Theor. 47 (2014), 025202, 13 pages, arXiv:1307.0692.
  7. Genest V.X., Vinet L., Zhedanov A., Superintegrability in two dimensions and the Racah-Wilson algebra, Lett. Math. Phys. 104 (2014), 931-952, arXiv:1307.5539.
  8. Granovskii Ya.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved space. I. An oscillator, Theoret. and Math. Phys. 91 (1992), 474-480.
  9. Granovskii Ya.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved space. II. The Kepler problem, Theoret. and Math. Phys. 91 (1992), 604-612.
  10. Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., Coupling-constant metamorphosis and duality between integrable Hamiltonian systems, Phys. Rev. Lett. 53 (1984), 1707-1710.
  11. Kalnins E.G., Kress J.M., Miller Jr. W., 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.
  12. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages.
  13. Kalnins E.G., Kress J.M., Miller Jr. W., Post S., Laplace-type equations as conformal superintegrable systems, Adv. in Appl. Math. 46 (2011), 396-416, arXiv:0908.4316.
  14. Kalnins E.G., Miller Jr. W., Quadratic algebra contractions and second-order superintegrable systems, Anal. Appl. (Singap.) 12 (2014), 583-612, arXiv:1401.0830.
  15. Kalnins E.G., Miller Jr. W., Pogosyan G.S., Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions, J. Math. Phys. 37 (1996), 6439-6467.
  16. Kalnins E.G., Miller Jr. W., Post S., Models of quadratic quantum algebras and their relation to classical superintegrable systems, Phys. Atomic Nuclei 72 (2009), 801-808.
  17. Kalnins E.G., Miller Jr. W., Post S., Coupling constant metamorphosis and $N$th-order symmetries in classical and quantum mechanics, J. Phys. A: Math. Theor. 43 (2010), 035202, 20 pages, arXiv:0908.4393.
  18. Kalnins E.G., Miller Jr. W., Post S., Models for the 3D singular isotropic oscillator quadratic algebra, Phys. Atomic Nuclei 73 (2010), 359-366.
  19. Kalnins E.G., Miller Jr. W., Post S., Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials, SIGMA 9 (2013), 057, 28 pages, arXiv:1212.4766.
  20. Kress J.M., Equivalence of superintegrable systems in two dimensions, Phys. Atomic Nuclei 70 (2007), 560-566.
  21. Makarov A.A., Smorodinsky J.A., Valiev K., Winternitz P., A systematic search for nonrelativistic systems with dynamical symmetries, Nuovo Cimento A 52 (1967), 1061-1084.
  22. Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
  23. Nehorosheev N.N., Action-angle variables, and their generalizations, Trans. Moscow Math. Soc. 26 (1972), 181-198.
  24. Post S., Coupling constant metamorphosis, the Stäckel transform and superintegrability, AIP Conf. Proc. 1323 (2011), 265-274.
  25. Post S., Models of quadratic algebras generated by superintegrable systems in 2D, SIGMA 7 (2011), 036, 20 pages, arXiv:1104.0734.
  26. Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42 (2001), 4248-4257.
  27. Winternitz P., Smorodinsky Ya.A., Uhlíř M., Friš I., Symmetry groups in classical and quantum mechanics, Soviet J. Nuclear Phys. 4 (1967), 444-450.

Previous article  Next article   Contents of Volume 11 (2015)