Variational methods in relativistic quantum mechanics

TL;DR

This review discusses variational approaches to solving stationary solutions of relativistic quantum equations involving the Dirac operator, highlighting challenges due to the operator's spectrum and the development of new mathematical methods.

Contribution

It introduces novel variational methods tailored for the indefinite energy functional in relativistic quantum mechanics with the Dirac operator.

Findings

Development of variational techniques for indefinite functionals

Handling spectral gap eigenvalue problems

Application of methods to nonlinear relativistic equations

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.