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

TL;DR

This paper explicitly constructs boundary triples for a Dirac operator with Coulomb-type spherically symmetric perturbations, classifying all self-adjoint extensions based on boundary behavior at the origin.

Contribution

It provides an explicit boundary triple for the Dirac operator with Coulomb-type potentials and characterizes all self-adjoint realizations via boundary conditions at the origin.

Findings

Explicit boundary triple for the Dirac operator with Coulomb perturbations

Complete classification of self-adjoint extensions based on boundary behavior

Analysis of distinguished extension under boundedness condition

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 $\sup_{x} |x||\mathbb V(x)| \leq 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form.