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

[摘要] 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.

[关键词] superintegrable;quadratic algebra;Coulomb potential;Verrier–Evans potential;ternary algebra