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

TL;DR

This paper characterizes all self-adjoint extensions of a Dirac operator with Coulomb-like potentials, providing a detailed analysis of boundary behaviors and identifying the unique distinguished extension.

Contribution

It offers a comprehensive description of self-adjoint realizations for Dirac operators with Coulomb-type potentials, including the distinguished extension, using Hardy estimates and trace lemmas.

Findings

Complete classification of self-adjoint extensions.

Identification of the distinguished extension.

Analysis of boundary behavior at the origin.

Abstract

We describe the self-adjoint realizations of the operator $H:=-i\alpha\cdot \nabla + m\beta + \mathbb V(x)$, for $m\in\mathbb R $, and $\mathbb V(x)= |x|^{-1} ( \nu \mathbb{I}_4 +\mu \beta -i \lambda \alpha\cdot{x}/{|x|}\,\beta)$, for $\nu,\mu,\lambda \in \mathbb R$. 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.