Relativistic Scott correction in self-generated magnetic fields

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.Comment: Small typos corrected, new references adde

The Ground State Energy of Heavy Atoms: The Leading Correction

“…In fact the Scott term is not affected in the Chandrasekhar model either whereas in the non-relativistic setting-which is expected to be unphysical because of the argument mentioned above-the Scott term would be changed [16][17][18]. This and corresponding numerical results motivate us to drop the self-generated magnetic fields in this paper.…”

Eigenvalue Bounds for the Fractional Laplacian: A Review

“…Stability of matter [36,83,84,44,85,52,51]. Proof of the Scott correction without [57,99] and with (self-generated) magnetic field [43].…”

The Magnetic Scott Correction for Relativistic Matter at Criticality

“…The inclusion of (self-generated) magnetic fields was then considered by Erdős, Fournais, and Solovej [4]. Before stating their main result, we shall define the main Hamiltonian involved.…”