Relativistic Strong Scott Conjecture: A Short Proof

TL;DR

This paper provides a concise proof demonstrating that the ground state density of heavy relativistic atoms converges to the relativistic hydrogenic density as atomic number and speed of light go to infinity with their ratio fixed.

Contribution

The authors present a short, simplified proof of a recent result on the density convergence in relativistic heavy atoms, extending previous non-relativistic analyses.

Findings

Density converges to relativistic hydrogenic density in the limit

Validates the relativistic Scott conjecture in a simplified manner

Establishes behavior of heavy atoms in the relativistic regime

Abstract

We consider heavy neutral atoms of atomic number $Z$ modeled with kinetic energy $(c^2p^2+c^4)^{1/2}-c^2$ used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale $Z^{-1}$ in the limit $Z,c\to\infty$ keeping $Z/c$ fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale.