Singular Perturbation of the Dirac Hamiltonian
This thesis is devoted to the study of the Dirac Hamiltonian perturbed by delta-type potentials and Coulomb-type potentials. We analysed the delta-shell interaction on bounded and smooth domains and its approximation by the coupling of the free Dirac operator with shrinking short-range potentials. Under certain hypothesis of smallness of a regular potential, in three dimensions, we prove that the Dirac operator coupled with a suitable rescaling of the potential converges to the Hamiltonian coupled with the delta-shell potential, in the strong resolvent sense. Nevertheless, the coupling constant depends non-linearly on the potential: Klein's Paradox comes into play. As a particular case, we pay to focus on the Dirac Operator with spherical delta-shell interactions. We characterise the eigenstates of the couplings by finding sharp constants and minimisers of some precise inequalities related to an uncertainty principle. We also explore the spectral relation between the shell interaction and its approximation by short-range potentials with shrinking support, improving previous results. Finally, we investigate the Dirac operator perturbed by a particular class of Coulomb-type spherically symmetric potentials. We describe the self-adjoint realisations of this operator in terms of the behaviour of the functions of the domain in the origin, and we provide Hardy-type estimates on them. At the end, we describe the distinguished extension.