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

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.