• français
    • English
  • English 
    • français
    • English
  • Login
JavaScript is disabled for your browser. Some features of this site may not work without it.
BIRD Home

Browse

This CollectionBy Issue DateAuthorsTitlesSubjectsJournals BIRDResearch centres & CollectionsBy Issue DateAuthorsTitlesSubjectsJournals

My Account

Login

Statistics

View Usage Statistics

Variational methods in relativistic quantum mechanics

Thumbnail
Date
2008
Link to item file
http://hal.archives-ouvertes.fr/hal-00156710/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
Quantum 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
Journal issue
Bulletin of the American Mathematical Society
Volume
45
Number
4
Publication date
2008
Article pages
535–593
Publisher
American Mathematical Society
DOI
http://dx.doi.org/10.1090/S0273-0979-08-01212-3
URI
https://basepub.dauphine.fr/handle/123456789/573
Collections
  • CEREMADE : Publications
Metadata
Show full item record
Author
Esteban, Maria J.
Lewin, Mathieu
Séré, Eric
Type
Article accepté pour publication ou publié
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.

  • Accueil Bibliothèque
  • Site de l'Université Paris-Dauphine
  • Contact
SCD Paris Dauphine - Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16

 Content on this site is licensed under a Creative Commons 2.0 France (CC BY-NC-ND 2.0) license.