Boundary triples for the Dirac operator with Coulomb-type spherically symmetric perturbations

Abstract

We determine explicitly a boundary triple for the Dirac operator $H:=-i\alpha \cdot \nabla +m\beta +\mathbb{V}(x)$ in ${\mathbb{R}}^3$, for $m\in \mathbb{R}$ and $\mathbb{V}(x)=|x{|}^{-1}(\nu {\mathbb{I}}_4+\mu \beta -i\lambda \alpha \cdot x/|x|\beta )$, with $\nu ,\mu ,\lambda \in \mathbb{R}$. Consequently we determine all the self-adjoint realizations of $H$ in terms of the behaviour of the functions of their domain in the origin. When ${\sup }_x|x||\mathbb{V}(x)|\le 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form.