Abstract (EN)

This note is a written version of the talk given at the French-Italian conference which took place in Torino in July 2006. In this talk I presented an overwiew of various results about the characterization of eigenvalues of self-adjoint operators in a spectral gap and its application to relativistic quantum mechanics, in order to define and compute the eigenvalues of Coulomb- Dirac type operators, with or without external magnetif field. The talk contained information about rigorous mathematical results and also about numerical computations done for some model problems in atomic and molecular physics. All the results described here can be found in [2, 4, 3, 1] and a very complete review about all this kind of problems for strongly indefinite operators, in particular the Dirac operator, can be found in [5].