The one-dimensional spinless relativistic Coulomb problem

Abstract

Motivated by a recent analysis that presents explicitly the general solution, we consider the eigenvalue problem of the spinless Salpeter equation with a (“hard-core amended”) Coulomb interaction potential in one dimension. We prove the existence of a critical coupling constant (which contradicts the assertions of the previous analysis) and give analytic upper bounds on the energy eigenvalues. These upper bounds seem to disprove the previous explicit solution.