Singular perturbations of the dirac hamiltonian

Résumé

This thesis is devoted to the study of the Dirac Hamiltonian perturbed by delta-type potentials andCoulomb-type potentials. We analysed on the delta-shell interaction on bounded and smooth domainsand its approximation by the coupling of the free Dirac operator with shrinking short rangepotentials.Under certain hypothesis of smallness of a regular potential, we prove that the Dirac operatorin R^3 coupled with a suitable rescaling of the potential converges in the strong resolvent sense to theHamiltonian coupled with a delta-shell potential supported on a bounded C^2 surface. Nevertheless, thecoupling constant depends non-linearly on the potential: the Klein's Paradox comes into play.Furthermore, we pay focus on the Dirac Operator with spherical delta-shell interactions. We characterizethe eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalitiesrelated to an uncertainty principle and we explore the spectral relation between the shell interaction andits approximation by short range potentials with shrinking support, improving previous results in thespherical case. Finally, we investigate the Dirac operator perturbed by a certain class of Coulomb-typespherically symmetric potentials. We describe the self-adjoint realizations of this operator in terms of thebehaviour of the functions of the domain in the origin, and we provide Hardy-type estimates on them.Finally, we give a description of the distinguished extension.