The One-Dimensional Spinless Relativistic Coulomb Problem

TL;DR

This paper investigates the one-dimensional spinless Salpeter equation with a Coulomb potential, establishing the existence of a critical coupling constant and providing analytic bounds that challenge previous explicit solutions.

Contribution

It proves the existence of a critical coupling constant and offers analytic upper bounds on energy eigenvalues, contradicting prior explicit solutions.

Findings

Existence of a critical coupling constant.

Analytic upper bounds on energy eigenvalues.

Contradiction of previous explicit solutions.

Abstract

Motivated by a recent analysis which presents explicitly the general solution, we consider the eigenvalue problem of the spinless Salpeter equation with a ("hard-core amended") Coulomb 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.