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The Atomic Density on the Thomas--Fermi Length Scale for the Chandrasekhar Hamiltonian

We consider a large neutral atom of atomic number $Z$, modeled by a pseudo-relativistic Hamiltonian of Chandrasekhar. We study its suitably rescaled one-particle ground state density on the Thomas--Fermi length scale $Z^{-1/3}$. Using an observation by Fefferman and Seco (1989), we find that the density on this scale converges to the minimizer of the Thomas--Fermi functional of hydrogen as $Z\to\infty$ when $Z/c$ is fixed to a value not exceeding $2/π$. This shows that the electron density on the Thomas--Fermi length scale does not exhibit any relativistic effects.