Solutions to the 1d Klein-Gordon equation with cutoff Coulomb potentials

TL;DR

This paper confirms analytical solutions to the 1D Klein-Gordon equation with a non-singular Coulomb-like potential through numerical methods and extends the analysis to an alternative cutoff Coulomb potential.

Contribution

It validates previous analytical results and introduces a numerical approach to solve the Klein-Gordon equation with different cutoff Coulomb potentials.

Findings

Analytical solutions are confirmed numerically.

Numerical methods successfully solve for the alternative potential.

The approach extends understanding of Klein-Gordon equations with cutoff Coulomb potentials.

Abstract

In a recent paper by Barton (J. Phys. A40, 1011 (2007)), the 1-dimensional Klein-Gordon equation was solved analytically for the non-singular Coulomb-like potential V_1(|x|) = -\alpha/(|x|+a). In the present paper, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cutoff Coulomb potential V_2(|x|) = -\alpha/|x|, ~|x| > a, and otherwise V_2(|x|) = -\alpha/a.