Positivity of the two-dimensional Brown-Ravenhall operator

TL;DR

This paper establishes the exact critical coupling constant for the two-dimensional Brown-Ravenhall operator with Coulomb potential and proves its positivity, correcting previous errors and extending known results from three to two dimensions.

Contribution

It precisely determines the critical coupling and proves the positivity of the two-dimensional Brown-Ravenhall operator, correcting prior inaccuracies.

Findings

Corrected the critical coupling constant for the operator

Proved the operator is positive in two dimensions

Extended the proof technique from three to two dimensions

Abstract

We determine the critical coupling of the two-dimensional Brown-Ravenhall operator with Coulomb potential. Boundedness from below has essentially been proven by Bouzouina, whose work however contains a trivial error leading to a wrong constant exactly one half of the actual critical constant. Furthermore we show that the operator is in fact positive. Our proof of that is for the most part analogous to Tix's proof of the corresponding result for the three-dimensional operator.