In the article The Ellipse and the Atom, the author mentions, but does not precisely prove, that the equation
$$\Psi(\mathbf{s}) - \frac{m}{2s^2} \frac{k}{2 \pi^2 \hbar} \int_{s S^3} \frac{\Psi(\mathbf{s'})}{|\mathbf{s}-\mathbf{s'}|^2} d\Omega' = 0$$
on the 3-sphere $s S^3$ of radius $s > 0$ is equivalent to the Laplace equation on the sphere. Where can I find a more thorough discussion of that equivalence?