Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 054, 11 pages      arXiv:1102.0397
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential

Yannis Tanoudis and Costas Daskaloyannis
Mathematics Department, Aristotle University of Thessaloniki, 54124 Greece

Received February 01, 2011, in final form May 22, 2011; Published online June 03, 2011

In the three-dimensional flat space, a classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller (J. Math. Phys. 48 (2007), 113518, 26 pages) have proved that, in the case of nondegenerate potentials, i.e. potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary quadratic Poisson algebra with five generators. The superintegrability of the generalized Kepler-Coulomb potential that was investigated by Verrier and Evans (J. Math. Phys. 49 (2008), 022902, 8 pages) is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the nondegenerate case of systems with quadratic integrals. In this paper, the ternary quadratic associative algebra corresponding to the quantum Verrier-Evans system is discussed. The subalgebras structure, the Casimir operators and the the finite-dimensional representation of this algebra are studied and the energy eigenvalues of the nondegenerate Kepler-Coulomb are calculated.

Key words: superintegrable; quadratic algebra; Coulomb potential; Verrier-Evans potential; ternary algebra.

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  1. Kalnins E.G., Kress J.M., Miller W. Jr., Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties, J. Math. Phys. 48 (2007), 113518, 26 pages, arXiv:0708.3044.
  2. Kalnins E.G., Kress J.M., Miller W. Jr., Fine structure for 3D second-order superintegrable systems: three-parameter potentials, J. Phys. A: Math. Theor. 40 (2007), 5875-5892.
  3. Verrier P.E., Evans N.W., A new superintegrable Hamiltonian, J. Math. Phys. 49 (2008), 022902, 8 pages, arXiv:0712.3677.
  4. Kalnins E.G., Williams G.C., Miller W. Jr., Pogosyan G.S., Superintegrability in three-dimensional Euclidean space, J. Math. Phys. 40 (1999), 708-725.
  5. Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100-1119, math-ph/0003017.
  6. Tanoudis Y., Daskaloyannis C., The algebra of the quantum nondegenerate three-dimensional Kepler-Coulomb potential, In Proceedings of the XIIIth Conference "Symmetries in Physics" (in Memory of Professor Yurii Fedorovich Smirnov) (July 2009, Dubna), to appear.
  7. Jacobson N., General representation theory of Jordan algebras, Trans. Amer. Math. Soc. 70 (1951), 509-530.
    Lister W.G., A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217-242.
  8. Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, math-ph/0412055.
  9. Daskaloyannis C., Ypsilantis K., Quantum superintegrable systems with quadratic integrals on a two dimensional manifold, J. Math. Phys. 48 (2007), 072108, 22 pages, math-ph/0607058.
  10. Tanoudis Y., Daskaloyannis C., Quadratic algebras for three-dimensional nondegenerate superintegrable systems with quadratic integrals of motion, Contribution at the XXVII Colloquium on Group Theoretical Methods in Physics (August 2008, Yerevan, Armenia), arXiv:0902.0130.
    Daskaloyannis C., Tanoudis Y., Quadratic algebras for three-dimensional superintegrable systems, Phys. Atomic Nuclei 73 (2010), 214-221.
  11. Marquette I., Winternitz P., Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion, J. Math. Phys. 48 (2007), 012902, 16 pages, Erratum, J. Math. Phys. 49 (2008), 019901, math-ph/0608021.
  12. Marquette I., Winternitz P., Superintegrable systems with third-order integrals of motion, J. Phys. A: Math. Theor. 41 (2008), 304031, 10 pages, arXiv:0711.4783.
  13. Marquette I., Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials, J. Math. Phys. 50 (2009), 012101, 23 pages, arXiv:0807.2858.
  14. Marquette I., Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendent potentials, J. Math. Phys. 50 (2009), 095202, 18 pages, arXiv:0811.1568.
  15. Marquette I., Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion, J. Math. Phys. 50 (2009), 122102, 10 pages, arXiv:0908.1246.
  16. Marquette I., Superintegrability and higher order polynomial algebras, J. Phys. A: Math. Gen. 43 (2010), 135203, 15 pages, arXiv:0908.4399.
  17. Quesne C., Quadratic algebra approach to an exactly solvable position-dependent mass Schrödinger equation in two dimensions, SIGMA 3 (2007), 067, 14 pages, arXiv:0705.2577.
  18. Marquette I., Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras, J. Math. Phys. 51 (2010), 102105, 10 pages, arXiv:1004.4579.

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