Abstract:
A study is made of the separation of variables in a spheroconical coordinate system associated
with the existence of an elliptic coordinate system on a three-dimensional sphere. In
the class of admissible potentials, interest attaches to a potential of the form $qr^{-4}[3(\boldsymbol\alpha\mathbf r)
(\boldsymbol\beta\mathbf r)-(\boldsymbol{\alpha\beta})\mathbf r^2]$, where $\boldsymbol\alpha$ and $\boldsymbol\beta$ are two arbitrary unit vectors. The angular part of this potential has the
form of a noncentral interaction similar to the angular part of the interaction between two
magnetic dipoles. After the angular part has been reduced to principal axes, the solution
of the Schrödinger equation with such a potential leads to the Lamé wave equation. Solutions
are found in the first order of perturbation theory, and a study is made of the splitting
of the energy levels of a centrally symmetric field when a noncentral potential of this
kind is presented. In particular, the energy level splitting is calculated in the presence
of such a potential in the case of the Coulomb potential and a potential with a quadratic
dependence on the radius.
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