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

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

2008-05-28

arxiv.org/abs/0805.4441v2

Documenta Math. 14 (2009) 463--516

Physics

Mathematical Physics

42 pages

Scientific paper

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.

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