Variational methods in relativistic quantum mechanics
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…
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