Upper limit on the critical strength of central potentials in relativistic quantum mechanics

TL;DR

This paper develops a method to establish upper bounds on the critical coupling strength for bound state formation in relativistic quantum mechanics using the spinless Salpeter equation.

Contribution

It introduces a new approach to derive upper limits on the critical coupling constant for central potentials in relativistic quantum mechanics.

Findings

Derived a class of upper bounds on the critical coupling constant.

Applicable to a wide range of central potentials.

Enhances understanding of bound state thresholds in relativistic settings.

Abstract

In the context of relativistic quantum mechanics, where the Schr\"odinger equation is replaced by the spinless Salpeter equation, we show how to construct a large class of upper limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant, $g$, of the central potential, $V(r)=-g v(r)$. This critical value is the value of $g$ for which a first $\ell$-wave bound state appears.