Abstract
This paper is concerned with the three-dimensional potentialVq =ησ2 (2a0/r−q ηa02/r2 sin2θ) e0 which comprises as particular cases the ring-shaped potential (q = 1) and the Coulomb potential (q = 0). The Schrodinger equation for the potentialV q is transformed via a nonbijective canonical transformation, viz., the Kustaanheimo-Stiefel transformation, into a coupled pair of Schrodinger equations for two-dimensional harmonic oscillators with inverse-square potentials. As a consequence, the discrete spectrum for the potentialV q is obtained in a straightforward way. A special attention is paid to the caseq = 0. In particular, the coupled pair of Schrodinger equations for two-dimensional harmonic oscillators is tackled in the situations where the spectrum for the potentialV0 is discrete, continuous, or reduced to the zero point. Finally, some group-theoretical questions about the potentialV q are mentioned as well as a connection, via the Kustaanheimo-Stiefel and the Levi-Civita transformations, between the quantum-mechanical problems for the potentialV q and the Sommerfeld and Kratzer potentials.
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