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Fine Structure for Second Order Superintegrable Systems

  • Ernie G. Kalnins
  • Jonathan M. Kress
  • Willard MillerJr
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 144)

Abstract

A classical (or quantum) superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n — 1 functionally independent constants of the motion polynomial in the momenta, the maximum possible. If the constants are all quadratic the system is second order superintegrable. Such systems have remarkable properties: multi-integrability and multi-separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions and with QES systems. For n=2 we have worked out the structure and classified the possible spaces and potentials, and for n=3 on conformally flat spaces with nondegenerate potentials we determined the structure theory and made major progress on the classification.

The quadratic algebra closes at order 6 and there is a 1–1 classical-quantum relationship. All such systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. We survey these results and announce a series of new results concerning the structure of superintegrable systems with degenerate potentials. In several cases the classification theory for such systems reduces to the study of polynomial ideals on which the symmetry group of the correspond manifold acts.

Keywords

Integrability Condition Regular Point Flat Space Quadratic Algebra Superintegrable System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Wojciechowski S., Superintegrability of the Calogero-Moser System. Phys. Lett,. 1983, A 95: 279–281.MathSciNetGoogle Scholar
  2. [2]
    Evans N.W., Superintegrability in Classical Mechanics; Phys. Rev. 1990, A 41, 5666–5676; Group Theory of the Smorodinsky-Winternitz System; J. Math Phys. 1991, 32: 3369.Google Scholar
  3. [3]
    Evans N.W., Super-Integrability of the Winternitz System; Phys. Lett. 1990, A 147: 483–486.Google Scholar
  4. [4]
    Friı J., Mandrosov V., Smorodinsky Ya.A, Uhlír M., AND Winternitz P., On Higher Symmetries in Quantum Mechanics; Phys. Lett. 1965, 16: 354–356.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Friı J., Smorodinskii Ya.A., Uhlír M., AND Winternitz P., Symmetry Groups in Classical and Quantum Mechanics; Sov. J. Nucl. Phys. 1967, 4: 444–450.Google Scholar
  6. [6]
    Makarov A.A., Smorodinsky Ya.A., Valiev Kh., AND Winternitz P., A Systematic Search for Nonrelativistic Systems with Dynamical Symmetries. Nuovo Cimento, 1967, 52: 1061–1084.CrossRefGoogle Scholar
  7. [7]
    Calogero F., Solution of a Three-Body Problem in One Dimension. J. Math. Phys. 1969, 10: 2191–2196.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Cisneros A. AND McIntosh H.V., Symmetry of the Two-Dimensional Hydrogen Atom. J. Math. Phys. 1969, 10: 277–286.CrossRefGoogle Scholar
  9. [9]
    Sklyanin E.K., Separation of variables in the Gaudin model. J. Sov. Math. 1989, 47: 2473–2488.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Faddeev L.D. AND Takhtajan L.A., Hamiltonian Methods in the Theory of Solitons Springer, Berlin 1987.MATHGoogle Scholar
  11. [11]
    Harnad J., Loop groups, R-matrices and separation of variables. In “Integrable Systems: From Classical to Quantum” J. Harnad, G. Sabidussi and P. Winternitz eds. CRM Proceedings and Lecture Notes, 26: 21–54, 2000.Google Scholar
  12. [12]
    Eisenhart L.P., Riemannian Geometry. Princeton University Press, 2nd printing, 1949.Google Scholar
  13. [13]
    Miller W.Jr., Symmetry and Separation of Variables. Addison-Wesley Publishing Company, Providence, Rhode Island, 1977.MATHGoogle Scholar
  14. [14]
    Kalnins E.G. AND Miller W.Jr., Killing tensors and variable separation for Hamilton-Jacobi and Helmholtz equations. SIAM J. Math. Anal., 1980, 11: 1011–1026.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Miller W., 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, 189: Springer-Verlag, New York, 1983.Google Scholar
  16. [16]
    Kalnins E.G., Separation of Variables for Riemannian Spaces of Constant Curvature, Pitman, Monographs and Surveys in Pure and Applied Mathematics, 28: 184–208, Longman, Essex, England, 1986.Google Scholar
  17. [17]
    Miller W.Jr., Mechanisms for variable separation in partial differential equations and their relationship to group theory. In Symmetries and Non-linear Phenomena pp. 188–221, World Scientific, 1988.Google Scholar
  18. [18]
    Kalnins E.G., Kress J.M., AND Miller W.Jr., Second order superintegrable systems in conformally flat spaces. I: 2D classical structure theory. J. Math. Phys., 2005, 46: 053509.CrossRefMathSciNetGoogle Scholar
  19. [19]
    Kalnins E.G., Kress J.M., AND Miller W.Jr., Second order superintegrable systems in conformally flat spaces. II: The classical 2D Stäckel transform. J. Math. Phys., 2005, 46: 053510.CrossRefMathSciNetGoogle Scholar
  20. [20]
    Kalnins E.G., Kress J.M., AND Miller W.Jr., Second order superintegrable systems in conformally flat spaces. III: 3D classical structure theory. J. Math. Phys., 2005, 46: 103507.CrossRefMathSciNetGoogle Scholar
  21. [21]
    Kalnins E.G., Kress J.M., AND Miller W.Jr., Second order superintegrable systems in conformally flat spaces. IV: The classical 3D Stäckel transform and 3D classification theory. J. Math. Phys., 2006, 47: 043514.CrossRefMathSciNetGoogle Scholar
  22. [22]
    Kalnins E.G., Kress J.M., AND Miller W.Jr., Second order superintegrable systems in conformally flat spaces. V: 2D and 3D quantum systems. J. Math. Phys., 2006, 47: 093501.CrossRefMathSciNetGoogle Scholar
  23. [23]
    Kalnins E.G., Miller W.Jr. AND Pogosyan G.S., Superintegrability in three dimensional Euclidean space. J. Math. Phys., 1999, 40: 708–725.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Kalnins E.G., Miller W.Jr., AND Pogosyan G.S., Superintegrability and associated polynomial solutions. Euclidean space and the sphere in two dimensions. J.Math.Phys., 1996, 37: 6439–6467.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Bonatos D., Daskaloyannis C., AND Kokkotas K., Deformed Oscillator Algebras for Two-Dimensional Quantum Superintegrable Systems; Phys. Rev., 1994, A 50: 3700–3709.Google Scholar
  26. [26]
    Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associate algebras of quantum superintegrable systems. J. Math. Phys., 2001, 42: 1100–1119.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Smith S.P., A class of algebras similar to the enveloping algebra of sl(2). Trans. Amer. Math. Soc, 1990, 322, 285–314.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Kalnins E.G., Miller W., AND Tratnik M.V., Families of orthogonal and biorthogonal polynomials on the n-sphere. SIAM J. Math. Anal., 1991,22: 272–294.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    Ushveridze A.G., Quasi-Exactly solvable models in quantum mechanics. Institute of Physics, Bristol, 1993.Google Scholar
  30. [30]
    Letourneau P. AND Vinet L., Superintegrable systems: Polynomial Algebras and Quasi-Exactly Solvable Hamiltonians. Ann. Phys., 1995, 243: 144–168.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    Kalnins E.G., Miller W.Jr., and Pogosyan G.S., Exact and quasi-exact solvability of second order superintegrable systems. I. Euclidean space preliminaries (submitted).Google Scholar
  32. [32]
    Grosche C., Pogosyan G.S., AND Sissakian A.N., Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two-and Three Dimensional Euclidean Space. Fortschritte der Physik, 1995, 43: 453–521.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    Kalnins E.G., Kress J.M., Miller W.Jr., AND Pogosyan G.S., Completeness of superintegrability in two-dimensional constant curvature spaces. J. Phys. A: Math. Gen., 2001, 34: 4705–4720.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    Kalnins E.G., Kress J.MN., AND Winternitz P., Superintegrability in a two-dimensional space of non-constant curvature. J. Math. Phys., 2002, 43: 970–983.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    Kalnins E.G., Kress J.M., Miller, W.Jr., AND Winternitz P., Superintegrable systems in Darboux spaces. J. Math. Phys., 2003, 44: 5811–5848.MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    Ranada M.F., Superintegrable n=2 systems, quadratic constants of motion, and potentials of Drach. J. Math. Phys., 1997, 38: 4165–4178.MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    Kalnins E.G., Miller W.Jr., Williams G.C., AND Pogosyan G.S., On superintegrable symmetry-breaking potentials in n-dimensional Euclidean space. J. Phys. A: Math. Gen., 2002, 35: 4655–4720.CrossRefMathSciNetGoogle Scholar
  38. [38]
    Kalnins E.G., Miller W.Jr., AND Pogosyan G.S., Completeness of multisepa-rable superintegrability in E2, c. J-Phys. A: Math. Gen., 2000, 33: 4105.MATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    Kalnins E.G., Miller W.Jr., AND Pogosyan G.S., Completeness of multiseparable superintegrability on the complex 2-sphere. J. Phys. A: Math. Gen., 2000, 33: 6791–6806.MATHCrossRefMathSciNetGoogle Scholar
  40. [40]
    Koenigs G., Sur les géodésiques a intégrales quadratiques. A note appearing in “Lecons sur la théorie générale des surfaces”. G. Darboux., 4: 368–404, Chelsea Publishing, 1972.Google Scholar
  41. [41]
    Daskaloyannis C. AND Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold. (Preprint) (2005).Google Scholar
  42. [42]
    Boyer C.P., Kalnins E.G., AND Miller W., Stäckel-equivalent integrable Hamiltonian systems. SIAM J. Math. Anal, 1986, 17: 778–797.MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    Hietarinta J., Grammaticos B., Dorizzi B., AND Ramani A., Coupling-constant metamorphosis and duality between integrable Hamiltonian systems. Phys. Rev. Lett, 1984, 53: 1707–1710.CrossRefMathSciNetGoogle Scholar
  44. [44]
    Calogero F., Solution to the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials. J. Math. Phys., 1971, 12: 419–436.CrossRefMathSciNetGoogle Scholar
  45. [45]
    Rauch-Wojciechowski S. AND Waksjö C., What an effective criterion of separability says about the Calogero type systems. J. Nonlinear Math. Phys., 2005, 12(1): 535–547.CrossRefMathSciNetGoogle Scholar
  46. [46]
    Horwood J.T., McLenaghan R.G., AND Smirnov R.G., Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space. Comm. Math. Phys., 2005, 259: 679–709.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Ernie G. Kalnins
    • 1
  • Jonathan M. Kress
    • 2
  • Willard MillerJr
    • 3
  1. 1.Department of MathematicsUniversity of WaikatoHamiltonNew Zealand
  2. 2.School of MathematicsThe University of New South WalesSydneyAustralia
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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