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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Quantum Mechanics (General and Nonrelativistic)

J. Math. Phys. 53, 022103 (2012); http://dx.doi.org/10.1063/1.3684955 (12 pages)

Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation

Ian Marquette

School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

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(Received 7 December 2011; accepted 20 January 2012; published online 15 February 2012)

The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a su(2) non-Abelian monopole interaction (Yang-Coulomb monopole). This system is also related to an 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this system. They generate a quadratic algebra with structure constants involving the Casimir operator of a so(4) Lie algebra. We also show that this system remains superintegrable with a su(2) non-Abelian monopole (generalized Yang-Coulomb monopole). We study this system using parabolic coordinates and obtain from Hurwitz transformation its dual that is an 8D singular oscillator. This 8D singular oscillator is also a new superintegrable system and multiseparable. We obtained its quadratic algebra that involves two Casimir operators of so(4) Lie algebras. This correspondence is used to obtain algebraically the energy spectrum of the generalized Yang-Coulomb monopole.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. QUADRATIC ALGEBRAS
    1. Higher dimensional systems and quadratic algebras
  3. GENERALIZED FIVE-DIMENSIONAL KEPLER
    1. Five-dimensional Kepler system
    2. Generalized 5D Kepler system
    3. Finite-dimensional unitary representations
  4. GENERALIZED YANG-COULOMB MONOPOLE SYSTEM AND DUALITY
    1. Yang-Coulomb monopole
    2. Yang-Coulomb monopole with Smorodinsky-Winternitz terms
    3. Hurwitz transformation and 8D singular oscillator
  5. CONCLUSION

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KEYWORDS and PACS

Keywords

Casimir effect, gauge field theory, harmonic oscillators, Lie algebras, SO(4) groups, SU(2) theory

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3684955

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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