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 SIGMA, 2012, Volume 8, 034, 25 pages (Mi sigma711)

Structure theory for extended Kepler–Coulomb 3D classical superintegrable systems

Ernie G. Kalninsa, Willard Miller Jr.b

a Department of Mathematics, University of Waikato, Hamilton, New Zealand
b School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

Abstract: The classical Kepler–Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler–Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler–Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis (2011) showed that in the quantum case, if a second 4th order symmetry is added to the generators, the double commutators in the symmetry algebra close polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of classical extended Kepler–Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers $(k_1,k_2)$ and reducing to the usual systems when $k_1=k_2=1$. We show these systems to be superintegrable of arbitrarily high order and work out explicitly the structure of the symmetry algebras determined by the 5 basis generators we have constructed. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations.

Keywords: superintegrability, Kepler–Coulomb system.

DOI: https://doi.org/10.3842/SIGMA.2012.034

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ArXiv: 1202.0197
MSC: 20C35, 22E70, 37J35, 81R12.
Received: March 14, 2012; in final form June 4, 2012; Published online June 7, 2012
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Citation: Ernie G. Kalnins, Willard Miller Jr., “Structure theory for extended Kepler–Coulomb 3D classical superintegrable systems”, SIGMA, 8 (2012), 034, 25 pp.

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Kalnins E.G. Kress J.M. Miller Jr. W., “Extended Kepler-Coulomb Quantum Superintegrable Systems in Three Dimensions”, J. Phys. A-Math. Theor., 46:8 (2013), 085206
2. Kalnins E.G., Kress J.M., Miller Jr. W., “Superintegrability in a Non-Conformally-Flat Space”, J. Phys. A-Math. Theor., 46:2 (2013), 022002
3. Miller Jr. Willard Post S. Winternitz P., “Classical and Quantum Superintegrability with Applications”, J. Phys. A-Math. Theor., 46:42 (2013), 423001
4. Panahi H. Alizadeh Z., “Deformed Oscillator Algebra for Quantum Superintegrable Systems in Two-Dimensional Euclidean Space and on a Complex Two-Sphere”, Chin. Phys. B, 22:6 (2013), 060304
5. Celeghini E., Kuru S., Negro J., del Olmo M.A., “A Unified Approach to Quantum and Classical Ttw Systems Based on Factorizations”, Ann. Phys., 332 (2013), 27–37
6. Coletti C., Calderini D., Aquilanti V., “D-Dimensional Kepler-Coulomb Sturmians and Hyperspherical Harmonics as Complete Orthonormal Atomic and Molecular Orbitals”, Proceedings of Mest 2012: Exponential Type Orbitals for Molecular Electronic Structure Theory, Adv. Quantum Chem., 67, ed. Hoggan P., Elsevier Academic Press Inc, 2013, 73–127
7. Markakis C., “Constants of Motion in Stationary Axisymmetric Gravitational Fields”, Mon. Not. Roy. Astron. Soc., 441:4 (2014), 2974–2985
8. Hoque M.F. Marquette I. Zhang Ya.-Zh., “Recurrence Approach and Higher Rank Cubic Algebras For the N-Dimensional Superintegrable Systems”, J. Phys. A-Math. Theor., 49:12 (2016), 125201
9. Ballesteros A. Herranz F.J. Kuru S. Negro J., “The anisotropic oscillator on curved spaces: A new exactly solvable model”, Ann. Phys., 373 (2016), 399–423
10. S. Kuru, J. Negro, O. Ragnisco, “The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories”, Phys. Lett. A, 381:39 (2017), 3355–3363
11. M. F. Ranada, “Quasi-bi-Hamiltonian structures, complex functions and superintegrability: the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems”, J. Phys. A-Math. Theor., 50:31 (2017), 315206
12. C. M. Chanu, G. Rastelli, “Extended Hamiltonians and shift, ladder functions and operators”, Ann. Phys., 386 (2017), 254–274
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