Dispersion in relativistic and non relativistic quantum dynamics

This thesis concerns the study of Strichartz estimates for different dispersive systems and their applications. The structure is as follows. In the introduction we present the problems treated in this manuscript, describing how they insert in the existing literature. Then, we describe in simple terms the main contributions of the works on which the thesis is based on, leaving the details to the successive chapters. In Chapter 1, we prove generalized Strichartz estimates for the massless 2D and 3D Dirac equation with the Coulomb potential. Moreover, as an application, we prove local well posed- ness for an Hartree-type nonlinear system in dimension 3. In the last section on this chapter we introduce an ongoing project, in collaboration with Federico Cacciafesta and Junyong Zhang concerning the analysis of the Dirac-Coulomb equation with a positive mass. Chapter 2, based on a published work in collaboration with Federico Cacciafesta and Long Meng, is dedicated to the study of the dispersive behaviour, via local in time Strichartz estimates, of the half wave and half Klein-Gordon equations on com- pact smooth Riemannian manifolds without boundary. As an application, we derive similar estimates for the Dirac equation on the same setting, whose definition is introduced in Section 0.2.4. The strategy of the proof follows the ones introduced by Burq-Gérard-Tzvetkov and Dinh, combined with a refined version of the WKB approximation. In Chapter 3 we present joint work with Charles Collot, Anne-Sophie de Suzzoni and Cyril Malézé, submitted to a journal for publication. We study the Hartree-Fock equation, which admits homogeneous states that model infinitely many particles at equi- librium. We prove their asymptotic stability in large dimensions, under assumptions on the linearized operator. Perturbations are moreover showed to scatter to linear waves. We obtain this result for the equivalent formulation of the Hartree-Fock equation in the framework of random fields. The main novelty is to consider the full Hartree-Fock equa- tion, including for the first time the exchange term in the study of these equilibria. The proof relies on dispersive estimates for the study of the linearized operator around the equilibrium and perturbative techniques.