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

TL;DR

This paper derives algebraic formulas for the energy levels of the nondegenerate three-dimensional Kepler-Coulomb system using quadratic associative algebra methods, extending superintegrability concepts to quantum systems.

Contribution

It introduces an algebraic approach to compute energy eigenvalues for the nondegenerate Kepler-Coulomb potential via quadratic associative algebra analysis.

Findings

Derived explicit energy eigenvalues algebraically.

Analyzed the subalgebra structure and Casimir operators.

Provided finite-dimensional representations of the algebra.

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.