The Energy of Heavy Atoms According to Brown and Ravenhall: The Scott Correction

TL;DR

This paper analyzes the relativistic Brown-Ravenhall model for heavy atoms, demonstrating that the ground state energy includes a significant Scott correction term beyond the Thomas-Fermi approximation, valid across a wide range of coupling constants.

Contribution

It establishes the Scott correction for the Brown-Ravenhall operator in heavy atoms, including at the critical coupling, and develops new inequalities for the operator's eigenvalues.

Findings

Validation of the Scott correction in the relativistic model.

Sharp bounds on eigenvalues up to critical coupling.

Extension of inequalities to the critical Brown-Ravenhall operator.

Abstract

We consider relativistic many-particle operators which - according to Brown and Ravenhall - describe the electronic states of heavy atoms. Their ground state energy is investigated in the limit of large nuclear charge and velocity of light. We show that the leading quasi-classical behavior given by the Thomas-Fermi theory is raised by a subleading correction, the Scott correction. Our result is valid for the maximal range of coupling constants, including the critical one. As a technical tool, a Sobolev-Gagliardo-Nirenberg-type inequality is established for the critical atomic Brown-Ravenhall operator. Moreover, we prove sharp upper and lower bound on the eigenvalues of the hydrogenic Brown-Ravenhall operator up to and including the critical coupling constant.