Self-Adjoint Extensions for the Dirac Operator with Coulomb-Type Spherically Symmetric Potentials

Abstract

We describe the self-adjoint realizations of the operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$, for $m\in\mathbb R $, and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/{|x|}\,β)$, for $ν,μ,λ\in \mathbb R$. We characterize the self-adjointness in terms of the behaviour of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension.