Mathematical analysis of quantum mechanical models.

Summary This thesis gathers a set of works related to the study of minimization problems which arise in the quantum mechanical modeling of atoms and molecules, in atomic physics, on the one hand, and of nuclei, in nuclear physics, on the other hand. The two parts of this thesis deal successively with these two types of problems. In the first part, we are interested in the existence of an optimal geometry of nuclei, for a given ion or molecule, in the framework of Thomas-Fermi, Hartree and Hartree-Fock models. The second part is devoted to the study of two families of nuclear physics models: Hartree-type models and a Hartree-Fock model with a simplified Skyrme-type potential. In both cases, the questions posed are translated in terms of minimization problems in three-dimensional space, invariant by translation, by means of an (energy) functional depending on one or more functions subject to various normalization constraints. The loss of compactness of the minimizing sequences, linked to the invariance by translation, is analyzed by the concentration-compactness method.