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Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion

  • Ian Marquette
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 144)

Abstract

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general associative cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in cartesian coordinates with a third order integral are known. The general formalism is applied to one of the quantum potentials.

Keywords

Associative Algebra Casimir Operator Polynomial Algebra Quantum Potential 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]
    V. Bargmann, Zur Theorie des Wasserstoffsatoms, Z. Phys. 99: 576–582 (1936).CrossRefGoogle Scholar
  2. [2]
    D. Bonatsos, C. Daskaloyannis, AND K. Kokkotas, Deformed oscillator algebras for two-dimensional quantum superintegrable systems. Phys. Rev. A 50: 3700–3709 (1994).MathSciNetGoogle Scholar
  3. [3]
    D. Bonatsos, C. Daskaloyannis, AND K. Kokkotas, Quantum-algebraic description of quantum superintegrable systems in two dimensions, Phys. Rev. A 48: R3407–R3410 (1993).MathSciNetGoogle Scholar
  4. [4]
    C. Daskaloyannis, Quadratic poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42: 1100–1119 (2001).MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    C. Daskaloyannis, Generalized deformed oscillator and nonlinear algebras, J. Phys. A: Math.Gen 24: L789–L794 (1991).CrossRefMathSciNetGoogle Scholar
  6. [6]
    J. Drach, Sur l’intégration logique des équations de la dynamique à deux variables: Forces conservatrices. Intégrales cubiques. Mouvements dans le plan, C.R. Acad. Sci III 200: 22–26 (1935).Google Scholar
  7. [7]
    J. Drach, Sur l’intégration logique et sur la transformation des équations de la dynamique à deux variables: Forces conservatrices. Intégrales. C.R. Acad. Sci III, 599–602 (1935).Google Scholar
  8. [8]
    N.W. Evans, Group theory of the Smorodinsky-Winternitz system. J. Math. Phys. 32: 3369–3375 (1991).MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    V. Fock, Zur Theorie des Wasserstoffsatoms, Z. Phys. 98: 145–154 (1935).MATHCrossRefGoogle Scholar
  10. [10]
    J. Fris, V. Mandrosov, YA.A. Smorodinsky, M. Uhlir, AND P. Winternitz, On higher symmetries in quantum mechanics, Phys. Lett. 16: 354–356 (1965).CrossRefMathSciNetGoogle Scholar
  11. [11]
    Ya.I. Granovskii, A.S. Zhedanov, AND I.M. Lutzenko, Quadratic Algebra as a Hidden Symmetry of the Hartmann Potential, J. Phys. A 24: 3887–3894 (1991).MathSciNetGoogle Scholar
  12. [12]
    S. Gravel AND P. Winternitz, Superintegrability with third-order integrals in quantum and classical mechanics J. Math. Phys. 43: 5902–5912 (2002).MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    S. Gravel, Hamiltonians separable in Cartesian coordinates and thirdorder integrals of motion, J. Math. Phys. 45: 1003–1019 (2004).MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    S. Gravel, Superintegrability, isochronicity, and quantum harmonic behavior, ArXiv:math-ph/0310004 (2004).Google Scholar
  15. [15]
    J.M Jauch AND E.L Hill, On the problem of degeneracy in quantum mechanics, Phys Rev 57: 641–645 (1940).MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    P. Létourneau AND L. Vinet, Superintegrable systems: Polynomial algebras and quasi-exactly solvable hamiltonians, Ann. Physics 243: 144–168 (1995).MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    I. Marquette AND P. Winternitz, Polynomial Poissons algebras for superintegrable systems with a third order integral of motion, ArXiv:math-ph/0608021 (2006).Google Scholar
  18. [18]
    P. Winternitz, Y.A. Smorodinsky, AND M. Uhlir ET I. Fris, Symmetry groups in classical and quantum mechanics, Yad. Fiz. 4: 625–635 (1966). (English translation: Sov. J. Nucl. Phys. 4: 444–450 (1967).)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Ian Marquette
    • 1
  1. 1.Département de physique et Centre de recherche mathématiqueUniversité de MontréalSuccursale Centre-Ville, MontréalCanada

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