Variational methods in relativistic quantum mechanics
Esteban, Maria J.; Lewin, Mathieu; Séré, Eric (2008), Variational methods in relativistic quantum mechanics, Bulletin of the American Mathematical Society, 45, 4, p. 535–593. http://dx.doi.org/10.1090/S0273-0979-08-01212-3
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00156710/en/
Journal nameBulletin of the American Mathematical Society
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Abstract (EN)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.
Subjects / KeywordsQuantum Chemistry; mean-field approximation; Dirac-Fock equations; Hartree-Fock equations; Bogoliubov-Dirac-Fock method; Quantum Electrodynamics; nonrelativistic limit; ground state; nonlinear eigenvalue problems; strongly indefinite functionals; critical points; variational methods; Dirac operator; Relativistic quantum mechanics
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