Quelques modèles non linéaires en mécanique quantique

Abstract : This thesis is devoted to the study of three nonlinear models from quantum mechanics. In the first part, we prove the existence of a minimizer and of approximate excited states for the multiconfiguration methods, which aim at describing electrons in molecules. These so-defined critical points can be computed numerically by a totally new algorithm. Numerical results are provided for the first excited state of two-electron systems. In the second part, we study a mountain pass lemma modelling adiabatic reactions in the Schrödinger time-independent framework. We prove the existence of a mountain pass point, assuming that the molecules at infinity are charged or polarized. Our last part is devoted to the study of the polarization of the vacuum with the Bogoliubov-Dirac-Fock model, a relativistic mean-field theory deduced from quantum electrodynamics. Our energy is bounded from below and has a minimizer which can be interpreted as the polarized vacuum.
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Submitted on : Tuesday, June 22, 2004 - 3:45:31 PM
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Mathieu Lewin. Quelques modèles non linéaires en mécanique quantique. Mathématiques [math]. Université Paris Dauphine - Paris IX, 2004. Français. ⟨tel-00006306⟩

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