Mathematical questions about the computation of eigenvalues of Dirac operators with critical potentials in atomic and molecular physics

This Note describes various analytical and computational results concerning the calculation of Dirac eigenvalues, or more generally, of operators with gaps. An algorithm based on an abstract theorem characterizing the eigenvalues in gaps was found years ago, but it is only recently that a delicate analysis to identify and study the domains of those operators has allowed to put that algorithm on a firm basis concerning the choice of approximation basis sets, and this both for light and for heavy atoms. The works described here concern joint papers with several collaborators: J. Dolbeault, M. Lewin, M. Loss, E. Séré and M. Vanbreugel.