Published in

Lecture Notes in Chemistry, p. 211-226

DOI: 10.1007/978-3-642-57237-1_10

American Mathematical Society, Bulletin of the American Mathematical Society, 4(45), p. 535-535

DOI: 10.1090/s0273-0979-08-01212-3

Links

Tools

Export citation

Search in Google Scholar

Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues

Journal article published in 2000 by Jean Dolbeault ORCID, Maria J. Esteban, Mathieu Lewin, Eric Séré
This paper is available in a repository.
This paper is available in a repository.

Full text: Download

Green circle
Preprint: archiving allowed
Upload
Green circle
Postprint: archiving allowed
Upload
Red circle
Published version: archiving forbidden
Policy details
Data provided by SHERPA/RoMEO

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 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.