Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras

Abstract

We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_3$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_4$. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are in these cases function not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere $S^3$ using a quadratic algebra. These results point out also that results and explicit formula for structure functions obtained for quadratic, cubic and higher order polynomial algebras in context of two-dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles.

15 pages

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Cited by in corpus (17)

• Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
• Higher-order superintegrability of separable potentials with a new approach to the Post-Winternitz system
• Superintegrable systems with a position dependent mass : Kepler-related and Oscillator-related systems
• Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension
• Generalized five-dimensional Kepler system, Yang-Coulomb monopole and Hurwitz transformation
• Laplace-Runge-Lenz vector for arbitrary spin
• A new family of $N$ dimensional superintegrable double singular oscillators and quadratic algebra $Q(3)\oplus so(n)\oplus so(N-n)$
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• Curvature as an integrable deformation
• Family of $N$-dimensional superintegrable systems and quadratic algebra structures
• Algebraic structure underlying spherical, parabolic and prolate spheroidal bases of the nine-dimensional MICZ-Kepler problem
• Local certification of unitary operations