Coulomb Problem in a One-Dimensional Space with Constant Positive Curvature

We construct a complex transformation $S_{1\mathbb C}\to S_1$ generalizing the known Hurwitz transformation in the Euclidean space for one-dimensional quantum mechanics. As in the case of the flat space, this transformation allows establishing the connection between the Coulomb problem and the oscillator problem with the Calogero–Sutherland potential added. We fully describe the motion in a Coulomb field in $S_1$ and determine the energy spectrum and the wave functions with the correct normalizing constant.