In this talk I will present variational methods devised to compute the eigenvalues of operators with gaps, inside those gaps. This is a nontrivial problem, since those eigenvalues have infinite Morse index and therefore computing them is a very unstable matter. But we have been able to find a variational characterization which is easy to implement and which avoids all those instabilities. Our main application is for the computation of energy levels for Dirac Hamiltonians in relativistic Quantum Mechanics. The variational characterization allows to construct easy to implement algorithms which are efficient and very accurate. The presentation will contain the theoretical description of the variational methods and also a description of the computing algorithms and of the results obtained for some atomic and molecular relativistic models.
This work has been done in collaboration with J. Dolbeault, M. Lewin and E. Séré.