On the 1D Coulomb Klein-Gordon equation

For a single particle of mass m experiencing the potential -alpha/vertical bar x vertical bar, the 1D Klein-Gordon equation is mathematically underdefined even when alpha << 1: unique solutions require some physically motivated prescription for handling the singularity at the origin. The procedure appropriate in most cases is to soften the singularity by means of a cutoff. Here we study the bound states of spin-zero particles in the potential -alpha/(vertical bar x vertical bar + R), extending the nonrelativistic results of Loudon (1959 Am. J. Phys. 27 649) to allow for relativistic effects, which become appreciable and eventually dominant for small enough m R: they are totally different from conclusions based hitherto on mathematically simple-seeming matching conditions on the wavefunction at x = 0. For realizable R, all relativistic effects remain very small; but with mR decreasing to order alpha(2) the ground-state energy E decreases through zero, and soon after that m R reaches a finite critical value below which E becomes complex, signalling a breakdown of the single-particle theory. At this critical point of the curve E(m R) the Klein-Gordon norm changes sign: the curve has a lower branch describing a bound antiparticle state, with positive energy - E, which exists for mR between the critical and some higher value where E reaches - m. Though apparently unanticipated in this context, similar scenarios are in fact familiar for strong short-range potentials (1D or 3D), and also for strong 3D Coulomb potentials with a of order unity.