Journal of Mathematical Physics
Feedback | Help | Exit
   
 
 
 
Previous Article
Liouville theorems for Dirac-harmonic maps
We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space n, the hyperbolic space n, and a Riemannian manifold n (n3) with the Schwarzschild metric to any Riemannian manifold N.
Next Article
Gauge transformations for the constrained CKP and BKP hierarchies
A special chain of the gauge transformations of the two-component constrained KP is introduced, and then the transformed components and the -function of this hierarchy are presented. Based on this, by...

Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

J. Math. Phys. 48, 113518 (2007); doi:10.1063/1.2817821

Published 28 November 2007

You are not logged in to this journal. Log in

E. G. Kalnins
Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand

J. M. Kress
School of Mathematics, The University of New South Wales, Sydney, New South Wales 2052, Australia

W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA
A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n−1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials. ©2007 American Institute of Physics
History: Received 4 June 2007; accepted 2 November 2007; published 28 November 2007
Permalink: http://link.aip.org/link/?JMAPAQ/48/113518/1
BUY THIS ARTICLE   (US$28)
Download PDF (218 kB) View Cart

KEYWORDS and PACS

Keywords
PACS
  • 03.65.Ge
    Solutions of wave equations: bound states in quantum mechanics
  • 02.60.Lj
    Ordinary and partial differential equations; boundary value problems
  • YEAR: 2007

RELATED DATABASES


To view database links for this article,
you need to log in.
To view database links for this article,
you need to log in.

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (36)
View in Separate Window/Tab
For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. S. Wojciechowski, Phys. Lett. 95A, 279 (1983).
  2. N. W. Evans, Phys. Rev. A 41, 5666 (1990);
    3. J. Math. Phys. 32, 3369 (1991).
  3. N. W. Evans, Phys. Lett. A 147, 483 (1990).
  4. J. Friš, V. Mandrosov, Ya. A. Smorodinsky, M. Uhlír, and P. Winternitz, Phys. Lett. 16, 354 (1965).
  5. J. Friš, Ya. A. Smorodinskii, M. Uhlír, and P. Winternitz, Sov. J. Nucl. Phys. 4, 444 (1967).
  6. A. A. Makarov, Ya. A. Smorodinsky, Kh. Valiev, and P. Winternitz, Nuovo Cimento A 52, 1061 (1967).
  7. F. Calogero, J. Math. Phys. 10, 2191 (1969).
  8. A. Cisneros and H. V. McIntosh, J. Math. Phys. 10, 277 (1969).
  9. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., J. Math. Phys. 47, 093501 (2006).
  10. L. G. Mardoyan, G. S. Pogosyan, A. N. Sissakian, and V. M. Ter-Antonyan, Nuovo Cimento Soc. Ital. Fis., B 88, 43 (1985);
    11. Theor. Math. Phys. 61, 1021 (1984);
    J. Phys. A 18, 455 (1985).
  11. Ya. A. Granovsky, A. S. Zhedanov, and I. M. Lutzenko, J. Phys. A 24, 3887 (1991).
  12. D. Bonatos, C. Daskaloyannis, and K. Kokkotas, Phys. Rev. A 50, 3700 (1994).
  13. C. Grosche, G. S. Pogosyan, and A. N. Sissakian, Fortschr. Phys. 43, 453 (1995).
  14. E. G. Kalnins, J. M. Kress, W. Miller, Jr., and G. S. Pogosyan, J. Phys. A 34, 4705 (2001).
  15. E. G. Kalnins, J. M. Kress, and P. Winternitz, J. Math. Phys. 43, 970 (2002).
  16. E. G. Kalnins, J. M. Kress, W. Miller, Jr., and P. Winternitz, J. Math. Phys. 44, 5811 (2003).
  17. M. F. Rañada, J. Math. Phys. 38, 4165 (1997).
  18. E. G. Kalnins, W. Miller, Jr., G. C. Williams, and G. S. Pogosyan, J. Phys. A 35, 4655 (2002).
  19. A. Ballesteros, A. Enciso, F. J. Herranz, and O. Ragnisco, e-print arXiv:math-ph/0612080.
  20. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., J. Math. Phys. 46, 103507 (2005).
  21. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., Proceedings volumes of the IMA program on Symmetries and Overdetermined Systems of Partial Differential Equations, 2007 (unpublished).
  22. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., J. Math. Phys. 47, 043514 (2006).
  23. E. G. Kalnins, Separation of Variables for Riemannian Spaces of Constant Curvature, Pitman, Monographs and Surveys in Pure and Applied Mathematics Vol. 28 (Longman, Essex, 1986), pp. 184–208.
  24. W. Miller, Jr., Symmetries and Non-linear Phenomena (World Scientific, Singapore, 1988), pp. 188–221.
  25. E. G. Kalnins, J. M. Kress, W. Miller, Jr., and G. S. Pogosyan, “Non degenerate superintegrable systems in n dimensional complex Euclidean spaces,” Phys. At. Nucl. (to be published).
  26. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., J. Math. Phys. 46, 053509 (2005).
  27. M. Bôcher, Über die Riehenentwickelungen der Potentialtheory (Teubner, Leipzig, 1894).
  28. E. G. Kalnins, W. Miller, Jr., and G. K. Reid, Proc. R. Soc. London, Ser. A 394, 183 (1984).
  29. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., J. Phys. A 40, 5875 2007.
  30. E. G. Kalnins, J. M. Kress, and W. Jr. Miller, , J. Math. Phys. 46, 053510 (2005).
  31. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, J. Phys. A 33, 4105 (2000).
  32. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, J. Phys. A 33, 6791 (2000).
  33. C. Daskaloyannis and K. Ypsilantis, J. Math. Phys. 47, 042904 (2006).
  34. F. Calogero, J. Math. Phys. 12, 419 (1971).
  35. S. Rauch-Wojciechowski and C. Waksjö, J. Nonlinear Math. Phys. 12, 535 (2005).
  36. J. T. Horwood, R. G. McLenaghan, and R. G. Smirnov, Commun. Math. Phys. 259, 679 (2005).

CITING ARTICLES
For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.


Copyright © American Institute of Physics