An overview on linear and nonlinear Dirac equations

Abstract

In this paper we review the variational treatment of linear and nonlinear eigenvalue problems involving the Dirac operator. These problems arise when searching for bound states and bound energies of electrons submitted to external or self-consistent interactions in a relativistic framework. The corresponding energy functionals are totally indefinite and that creates difficulties to define variational principles related to those equations. Here we describe recent works dealing with this kind of problems. In particular we describe the solutions found to linear Dirac equations with external potential, the Maxwell-Dirac equations, some nonlinear Dirac equations with local nonlinear terms and the Dirac-Fock equations arising in the description of atoms and molecules.