Journal of Mathematical Physics
Feedback | Help | Exit
   
 
 
 
Previous Article
Resonances of polyatomic molecules in the Born-Oppenheimer approximation
We study the spectral properties of polyatomic molecules near a scattering level. We prove that in the Born-Oppenheimer approximation this study can be reduced to the one of a family of finite matrice...
Next Article
A recursive parametrization of unitary matrices
A simple recursive scheme for parametrization of n-by-n unitary matrices is presented. The n-by-n matrix is expressed as a product containing the (n–1)-by-(n–1) matrix and a unitary matrix t...

Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory

J. Math. Phys. 46, 103507 (2005); doi:10.1063/1.2037567

Published 11 October 2005

You are not logged in to this journal. Log in

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

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

W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
This paper is part of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in real or complex conformally flat spaces. Here we consider classical superintegrable systems with nondegenerate potentials in three dimensions. We show that there exists a standard structure for such systems, based on the algebra of 3×3 symmetric matrices, and that the quadratic algebra always closes at order 6. We show that the spaces of truly second-, third-, fourth-, and sixth-order constants of the motion are of dimension 6, 4, 21, and 56, respectively, and we construct explicit bases for the fourth- and sixth order constants in terms of products of the second order constants. ©2005 American Institute of Physics
History: Received 8 July 2005; accepted 22 July 2005; published 11 October 2005
Permalink: http://link.aip.org/link/?JMAPAQ/46/103507/1
BUY THIS ARTICLE   (US$28)
Download PDF (221 kB) View Cart

KEYWORDS and PACS

Keywords
PACS
  • 04.20.Gz
    Spacetime topology, causal structure, spinor structure in general relativity
  • 02.10.Yn
    Matrix theory
  • 02.30.Cj
    Measure and integration
  • YEAR: 2005

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. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., J. Math. Phys. 46, 053509 (2005).
  2. E. G. Kalnins, J. M. Kress, and W. Miller, Jr., J. Math. Phys. 46, 053510 (2005).
  3. S. Wojciechowski, Phys. Lett. A 95, 279 (1983).
  4. N. W. Evans, Phys. Rev. A 41, 5666 (1990)
    5. J. Math. Phys. 32, 3369 (1991).
  5. N. W. Evans, Phys. Lett. A 147, 483 (1990).
  6. J. Fri[s-caron], V. Mandrosov, Ya. A. Smorodinsky, M. Uhlír, and P. Winternitz, Phys. Lett. 16, 354 (1965).
  7. J. Fri[s-caron], Ya. A. Smorodinskii, M. Uhlír, and P. Winternitz, Sov. J. Nucl. Phys. 4, 444 (1967).
  8. A. A. Makarov, Ya. A. Smorodinsky, Kh. Valiev, and P. Winternitz, Nuovo Cimento A 52, 1061 (1967).
  9. F. Calogero, J. Math. Phys. 10, 2191 (1969).
  10. A. Cisneros and H. V. McIntosh, J. Math. Phys. 10, 277 (1969).
  11. E. K. Sklyanin, J. Sov. Math. 47, 2473 (1989).
  12. L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 1987).
  13. A.G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics (Institute of Physics, Bristol, 1993).
  14. J. Harnad, "Loop groups, R-matrices and separation of variables," in Integrable systems: from classical to quantum, edited by J. Harnad, G. Sabidussi, and P. Winternitz, CRM Proceedings and Lecture Notes, Vol. 26 (American Mathematical Society, Providence, Rhode Island, 2000), pp. 21–54.
  15. M. Karlovini and K. Rosquist, J. Math. Phys. 41, 370 (2000).
  16. L.P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, NJ, 1949).
  17. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Providence, RI, 1977).
  18. E. G. Kalnins and W. Miller, Jr., SIAM J. Math. Anal. 11, 1011 (1980).
  19. W. Miller, "The technique of variable separation for partial differential equations," Proceedings of School and Workshop on Nonlinear Phenomena, Oaxtepec, Mexico, November 29–December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer-Verlag, New York, 1983), pp. 184–208.
  20. E.G. Kalnins, Separation of Variables for Riemannian Spaces of Constant Curvature, Pitman, Monographs and Surveys in Pure and Applied Mathematics 28 (Longman, Essex, England, 1986).
  21. W. Miller, Jr., Mechanisms for variable separation in partial differential equations and their relationship to group theory," Symmetries and Nonlinear Phenomena (World Scientific, Singapore, 1988), pp. 188–221.
  22. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, J. Math. Phys. 40, 708 (1999).
  23. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, J. Math. Phys. 37, 6439 (1996).
  24. D. Bonatos, C. Daskaloyannis, and K. Kokkotas, Phys. Rev. A 50, 3700 (1994).
  25. C. Daskaloyannis, J. Math. Phys. 42, 1100 (2001).
  26. S. P. Smith, Trans. Am. Math. Soc. 322, 285 (1990).
  27. E. G. Kalnins, W. Miller, and M. V. Tratnik, SIAM J. Math. Anal. 22, 272 (1991).
  28. P. Letourneau and L. Vinet, Ann. Phys. (N.Y.) 243, 144 (1995).
  29. C. Grosche, G. S. Pogosyan, and A. N. Sissakian, Fortschr. Phys. 43, 453 (1995).
  30. E. G. Kalnins, J. M. Kress, W. Miller, Jr., and G. S. Pogosyan, J. Phys. A 34, 4705 (2001).
  31. E. G. Kalnins, J. M. Kress, and P. Winternitz, J. Math. Phys. 43, 970 (2002).
  32. E. G. Kalnins, J. M. Kress, W. Miller, Jr., and P. Winternitz, J. Math. Phys. 44, 5811 (2003).
  33. M. F. Rañada, J. Math. Phys. 38, 4165 (1997).
  34. E. G. Kalnins, W. Miller, Jr., G. C. Williams, and G. S. Pogosyan, J. Phys. A 35, 4655 (2002).
  35. C. P. Boyer, E. G. Kalnins, and W. Miller, SIAM J. Math. Anal. 17, 778 (1986).
  36. J. Hietarinta, B. Grammaticos, B. Dorizzi, and A. Ramani, Phys. Rev. Lett. 53, 1707 (1984).

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