Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Variational methods in relativistic quantum mechanics   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.

31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol