Relativistic Scott correction in self-generated magnetic fields

TL;DR

This paper extends the analysis of the relativistic Scott correction in large molecules to include self-generated magnetic fields, demonstrating that the correction remains unchanged and establishing new inequalities for relativistic kinetic energy with magnetic fields.

Contribution

It proves the Scott correction for relativistic molecules with self-generated magnetic fields and shows the correction function is unaffected by magnetic fields, extending previous results.

Findings

The leading Thomas-Fermi energy is unaffected by magnetic fields.

The Scott correction term remains unchanged with magnetic fields.

New Lieb-Thirring inequalities for relativistic kinetic energy with magnetic fields are established.

Abstract

We consider a large neutral molecule with total nuclear charge $Z$ in a model with self-generated classical magnetic field and where the kinetic energy of the electrons is treated relativistically. To ensure stability, we assume that $Z \alpha < 2/\pi$, where $\alpha$ denotes the fine structure constant. We are interested in the ground state energy in the simultaneous limit $Z \rightarrow \infty$, $\alpha \rightarrow 0$ such that $\kappa=Z \alpha$ is fixed. The leading term in the energy asymptotics is independent of $\kappa$, it is given by the Thomas-Fermi energy of order $Z^{7/3}$ and it is unchanged by including the self-generated magnetic field. We prove the first correction term to this energy, the so-called Scott correction of the form $S(\alpha Z) Z^2$. The current paper extends the result of \cite{SSS} on the Scott correction for relativistic molecules to include a self-generated magnetic field. Furthermore, we show that the corresponding Scott correction function $S$, first identified in \cite{SSS}, is unchanged by including a magnetic field. We also prove new Lieb-Thirring inequalities for the relativistic kinetic energy with magnetic fields.