Variational methods in relativistic quantum mechanics

Maria J. Esteban (1) , Mathieu Lewin (2) , Eric Séré (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

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 Euler-Lagrange 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 Palais-Smale 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.

Keywords: mean-field approximation; Dirac-Fock equations; Hartree-Fock equations; Bogoliubov-Dirac-Fock method; Relativistic quantum mechanics; Dirac operator; variational methods; critical points; strongly indefinite functionals; nonlinear eigenvalue problems; ground state; nonrelativistic limit; Quantum Chemistry; Quantum Electrodynamics

Domains: Mathematical Physics [math-ph]; Mathematical Physics [math-ph]; Analysis of PDEs [math.AP]

Maria J. Esteban, Mathieu Lewin, Eric Séré. Variational methods in relativistic quantum mechanics. Bull. Amer. Math. Soc. (N.S.), 2008, 45 (4), pp.535-593. ⟨hal-00156710v2⟩