Variational methods in relativistic quantum mechanics
Abstract
This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the EulerLagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R^3, the PalaisSmale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems.
 Publication:

arXiv eprints
 Pub Date:
 June 2007
 arXiv:
 arXiv:0706.3309
 Bibcode:
 2007arXiv0706.3309E
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs
 EPrint:
 Bull. Amer. Math. Soc. (N.S.) 45, 4 (2008) 535593