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 $Z\to \infty$, $\alpha \to 0$ 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.

Small typos corrected, new references added

References in corpus (7)

• The Ground State Energy of Heavy Atoms: Relativistic Lowering of the Leading Energy Correction
• Scott correction for large atoms and molecules in a self-generated magnetic field
• Star-product Description of Quantization in Second-class Constraint Systems
• Second order semiclassics with self-generated magnetic fields
• Local trace asymptotics in the self-generated magnetic field
• Asymptotics of the ground state energy for atoms and molecules in the self-generated magnetic field
• Global trace asymptotics in the self-generated magnetic field in the case of Coulomb-like singularities

Cited by in corpus (5)

• The Scott conjecture for large Coulomb systems: a review
• The Ground State Energy of Heavy Atoms: the Leading Correction
• Hardy-Lieb-Thirring Inequalities for Fractional Pauli Operators
• The Magnetic Scott Correction for Relativistic Matter at Criticality
• The reduced Hartree-Fock model with self-generated magnetic fields