Solutions to the 1d Klein–Gordon equation with cut-off Coulomb potentials

In a recent paper by Barton [G. Barton, J. Phys. A: Math. Gen. 40 (2007) 1011], the 1-dimensional Klein–Gordon equation was solved analytically for the non-singular Coulomb-like potential V1(|x|)=−α/(|x|+a)V1(|x|)=−α/(|x|+a). In the present Letter, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cut-off Coulomb potential V2(|x|)=−α/|x|V2(|x|)=−α/|x|, |x|>a|x|>a, and otherwise V2(|x|)=−α/aV2(|x|)=−α/a.