Variational methods in relativistic quantum mechanics.(English)Zbl 1288.49016

Summary: 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.

MSC:

 49S05 Variational principles of physics 35J60 Nonlinear elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35Q75 PDEs in connection with relativity and gravitational theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81V70 Many-body theory; quantum Hall effect 81V45 Atomic physics 81V55 Molecular physics
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