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

Abstract

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.

arXiv commentContribution to the Symmetry, Separation, Super-integrability and Special Functions ($S^4$) Conference in honor of Prof. Willard Miller, Minnesota Sept. 2010

References in corpus (12)

• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve transcendent potentials
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold
• Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion
• Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds
• Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties
• Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras
• Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions
• Construction of classical superintegrable systems with higher order integrals of motion from ladder operators
• Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion
• Quadratic algebras for three dimensional non degenerate superintegrable systems with quadratic integrals of motion

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