DOCUMENTA MATHEMATICA, Vol. 14 (2009), 463-516

Rupert L. Frank, Heinz Siedentop, and Simone Warzel

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

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 bounds on the eigenvalues of the hydrogenic Brown-Ravenhall operator up to and including the critical coupling constant.

2000 Mathematics Subject Classification: Primary 81V45; Secondary 81V55, 35Q40, 46N50, 47N50.

Keywords and Phrases: Heavy atoms, ground state energy, relativistic Coulomb system, Scott correction, Brown-Ravenhall operator.

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