Partial-wave analysis of a relativistic Coulomb problem

We consider the behavior for small $r$ of a Schrödinger equation with relativistic kinematics and Coulomb potential $-\frac{\alpha }{r}$; we also discuss critical values of $\alpha$. Our solutions generalize to all the partial waves results already obtained for the $S$ wave, and are seen to match in a simple way with WKB approximate solutions in the oscillating region.