The energy of heavy atoms according to Brown and Ravenhall: the Scott correction

  • Rupert L. Frank

    Dept. of Mathematics, Mathematisches Institut Princeton University, LMU München Princeton, NJ 08544-1000, Theresienstraße 39 USA 80333 München, Germany
  • Heinz Siedentop

    Dept. of Mathematics, Mathematisches Institut Princeton University, LMU München Princeton, NJ 08544-1000, Theresienstraße 39 USA 80333 München, Germany
  • Simone Warzel

    TU München Zentrum Mathematik - M7 Boltzmannstraße 3 85747 Garching, Germany
The energy of heavy atoms according to Brown and Ravenhall: the Scott correction cover
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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 bounds on the eigenvalues of the hydrogenic Brown-Ravenhall operator up to and including the critical coupling constant.

Cite this article

Rupert L. Frank, Heinz Siedentop, Simone Warzel, The energy of heavy atoms according to Brown and Ravenhall: the Scott correction. Doc. Math. 14 (2009), pp. 463–516

DOI 10.4171/DM/279