Superintegrability on Two-Dimensional Complex Euclidean Space

  • E. G. Kalnins
  • W. MillerJr.
  • G. S. Pogosyan
Part of the CRM Series in Mathematical Physics book series (CRM)


In this work we examine the basis functions for those quantum mechanical systems on two-dimensional complex Euclidean space, with nonzero potential, that admit separation in at least two coordinate systems. We present all of these cases from a unified point of view. In particular, all of the polynomial special functions that arise via variable separation have their essential features expressed in terms of their zeros.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    CP Boyer, E.G. Kalnins, and P. Winternitz, Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces, J. Math. Phys. 24 (1983), 2022–2034.MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    L.P. Eisenhart, Enumeration of potentials for which one-particle Schrödinger equations are separable, Physical Rev. (2)74 (1948), 87–89.MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi (eds.), Higher transcendental functions, Vols. I-II, McGraw Hill, New York, 1953.Google Scholar
  4. 4.
    N.W. Evans, Superintegrability in classical mechanics, Phys. Rev. A (3)41 (1990), 5666–5676; Group theory of the Smorodinsky Winternitz system, J. Math. Phys. 32 (1991), 3369-3375.ADSGoogle Scholar
  5. 5.
    N.W. Evans, Super-integrability of the Winternitz system, Phys. Lett. A 147 (1990), 483–486.ADSGoogle Scholar
  6. 6.
    J. Fris, V. Mandrosov, Ya. A. Smorodinsky, M. Uhlir, and P. Winternitz, On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354–356.MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    J. Fris, Ya. A. Smorodinskii, M. Uhlir, and P. Winternitz, Symmetry groups in classical and quantum mechanics, Soviet J. Nuclear Phys. 4 (1967), 444–450.MathSciNetGoogle Scholar
  8. 8.
    Ya.A. Granovsky, A.S. Zhedanov, and I.M. Lutzenko, Quadratic algebra as a “hidden” symmetry of the Hartmann potential, J. Phys. A 24 (1991), 3887–3894.ADSGoogle Scholar
  9. 9.
    C. Grosche, G.S. Pogosyan, and A.N. Sissakian, Path integral approach to superintegrable potentials. The two-dimensional hyperboloid, Phys. Particles Nuclei 27 (1996), 244.ADSGoogle Scholar
  10. 10.
    C. Grosche, G.S. Pogosyan, and A.N. Sissakian, Path integral discussion for Smorodinsky—Winternitz potentials. I. Two-and threedimensional Euclidean space, Fortschr. Phys. 43 (1995), 453–521.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    E.G. Kalnins and W. Miller, Separable coordinates, integrability and the Niven equations, J. Phys. A 25 (1992), 5663.MathSciNetADSGoogle Scholar
  12. 12.
    E.G. Kalnins, W. Miller Jr., and G.S. Pogosyan, Superintegrability and associated polynomial solutions. Euclidean space and the sphere in two dimensions J. Math. Phys. 37 (1996), 6439–6467.MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    P. Letourneau and L. Vinet, Superintegrable systems: Polynomial algebras and quasiexactly solvable Hamiltonians, Ann. Physics 243 (1995), 144–168.MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    W. Miller, Jr, Symmetry and separation of variables, Addison-Wesley Publishing Company, Providence, Rhode Island, 1977.MATHGoogle Scholar
  15. 15.
    J. Patera and P. Winternitz, A new basis for the representation of the rotation group. Lamé and Heun polynomials, J. Math. Phys. 14 (1973), 1130.MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    S. Wojciechovski, Superintegrability of the Calogero-Moser System, Phys. Lett. A 95 (1983), 279.ADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • E. G. Kalnins
  • W. MillerJr.
  • G. S. Pogosyan

There are no affiliations available

Personalised recommendations