The Magnetic Scott Correction for Relativistic Matter at Criticality

Abstract

We provide a proof of the first correction to the leading asymptotics of the minimal energy of pseudo-relativistic molecules in the presence of magnetic fields, the so-called "relativistic Scott correction", when $\max{Z_k\alpha} \leq 2/\pi$, where $Z_k$ is the charge of the $k$-th nucleus and $\alpha$ is the fine structure constant. Our theorem extends a previous result by Erd\H{o}s, Fournais, and Solovej to the critical constant $2/\pi$ in the relativistic Hardy inequality $|p| - \frac{2}{\pi |x|} \geq 0$.