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Bulletin of the American Mathematical Society

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Variational methods in relativistic quantum mechanics


Authors: Maria J. Esteban, Mathieu Lewin and Eric Séré
Journal: Bull. Amer. Math. Soc. 45 (2008), 535-593
MSC (2000): Primary 49S05, 35J60, 35P30, 35Q75, 81Q05, 81V70, 81V45, 81V55.
DOI: https://doi.org/10.1090/S0273-0979-08-01212-3
Published electronically: June 25, 2008
MathSciNet review: 2434346
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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 $ \mathbb{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.

In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow us to describe the localized state of a spin-$ 1/2$ particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or Klein-Gordon-Dirac equations.

The second part is devoted to the presentation of min-max principles allowing us to characterize and compute the eigenvalues of linear Dirac operators with an external potential in the gap of their essential spectrum. Many consequences of these min-max characterizations are presented, among them are new kinds of Hardy-like inequalities and a stable algorithm to compute the eigenvalues.

In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of $ N$ interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit.

In the last part, we present a more involved relativistic model from Quantum Electrodynamics in which the behavior of the vacuum is taken into account, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.


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Additional Information

Maria J. Esteban
Affiliation: CNRS and Ceremade (UMR 7534), Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
Email: esteban@ceremade.dauphine.fr

Mathieu Lewin
Affiliation: CNRS and Laboratoire de Mathématiques (UMR 8088), Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, 95 302 Cergy-Pontoise Cedex, France
Email: Mathieu.Lewin@math.cnrs.fr

Eric Séré
Affiliation: Ceremade (UMR 7534), Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
Email: sere@ceremade.dauphine.fr

DOI: https://doi.org/10.1090/S0273-0979-08-01212-3
Keywords: Relativistic quantum mechanics, Dirac operator, variational methods, critical points, strongly indefinite functionals, nonlinear eigenvalue problems, ground state, nonrelativistic limit, Quantum Chemistry, mean-field approximation, Dirac-Fock equations, Hartree-Fock equations, Bogoliubov-Dirac-Fock method, Quantum Electrodynamics
Received by editor(s): June 22, 2007
Published electronically: June 25, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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