The Dirac equation is one of the fundamental equations in relativistic quantum mechanics, widely used in a large number of applications from physics to quantum chemistry. From the dynamical point of view, it falls within the chapter of the so called dispersive PDEs, along with other celebrated models of quantum mechanics (Schrödinger, wave, ...).

The aim of this talk, after a brief introduction on the linear theory for dispersive PDEs, will be to discuss some recent results concerning dispersive estimates (local smoothing, Strichartz, ...) for the Dirac equation perturbed with a Coulomb potential. This model happens to be particularly relevant as indeed the Coulomb potential, which is widely used in the applications to model particle interactions, is a scaling critical perturbation of the (massless) Dirac operator, and thus it provides a substantial difficulty in the analysis of linear estimates for the flow.

The talk is based on joint works with E. Danesi, E. Séré and J. Zhang.