The one-dimensional Coulomb problem

One-dimensional scattering by a Coulomb potential V(x)=c/|x| is studied for both repulsive (c > 0) and attractive (c < 0) cases. Two methods of regularizing the singularity at x = 0 are used, yielding the same conclusion, namely, that the transmission vanishes. For an attractive potential (c < 0), two groups of bound states are found. The first one consists of regular (Rydberg) bound states, following standard orthogonality relations. The second set consists of anomalous bound states (in a sense to be clarified), which always relax as coherent states