Dynamics of a particle in a superposition of a Coulomb potential and certain anisotropic harmonic-oscillator potentials

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Abstract
The Hamilton-Jacobi and Schrödinger equations for a particle moving in N (>1) dimensions in a concentric superposition of a Coulomb potential and an anisotropic 2:1 axially symmetric harmonic-oscillator potential together with an axial linear potential separate in parabolic coordinates. The separation constant is an eigenvalue of an operator which is a generalization of the axial component of the Runge-Lenz vector of the Coulomb problem in the Schrödinger case. Also for the Schrödinger case, methods of solving the joint eigenvalue problem and the separated differential equations are studied and specific results for the two-dimensional case presented.