On the absence of bound states for a planar massless Brown-Ravenhall-type operator

TL;DR

This paper proves that a two-dimensional massless Dirac operator with a Bessel-Macdonald potential has no bound states in the subcritical coupling region, using a Relativistic Hardy Inequality.

Contribution

It establishes the non-existence of bound states for a massless Dirac operator with a specific potential in the subcritical regime, filling a gap in the theoretical understanding.

Findings

No bound states for $ ext{γ} extless ext{γ}_{ m crit}$

Relativistic Hardy Inequality is key to the proof

Results confirm previous conjectures in the subcritical region

Abstract

We address the question of the existence of bound states for a suitably projected two-dimensional massless Dirac operator in the presence of a Bessel-Macdonald potential (also known as $K_0$-potential potential), raised by De Lima, Del Cima and Miranda, in Eur.Phys.J. B (2020) 93, 187. Based on Relativistic Hardy Inequality, we prove that this operator has no bound states if $\gamma \leqslant \gamma_{\rm crit}$ (subcritical region), where $\gamma$ is a coupling constant.