The one-dimensional Coulomb Problem

TL;DR

This paper investigates one-dimensional Coulomb scattering and bound states, demonstrating that transmission vanishes regardless of regularization method and identifying two types of bound states in the attractive case.

Contribution

It introduces two regularization methods for the Coulomb singularity and characterizes both standard and anomalous bound states in the attractive potential case.

Findings

Transmission probability is zero for all regularizations.

Two groups of bound states are identified in the attractive case.

Anomalous bound states relax as coherent states.

Abstract

One-dimensional scattering by a Coulomb potential V(x)=lambda/|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, respecting standard orthogonality relations. The second set consists of "anomalous"} bound states (in a sense to be clarified), which always relax as coherent states.