Self-Adjoint Extensions of Dirac Operator with Coulomb Potential

TL;DR

This paper reviews the current understanding of self-adjoint extensions of the Dirac operator with Coulomb potential, highlighting techniques, historical development, and open questions in the field.

Contribution

It provides a concise overview of the state-of-the-art, emphasizing the methods and conceptual progress in classifying self-adjoint realizations of the Dirac operator with Coulomb-like potentials.

Findings

Summary of existing self-adjoint extension classifications

Identification of key techniques used in the analysis

Open questions regarding multiplicity of extensions

Abstract

In this note we give a concise review of the present state-of-art for the problem of self-adjoint realisations for the Dirac operator with a Coulomb-like singular scalar potential $V(\vec x)=\phi(\vec x) I_4$. We try to follow the historical and conceptual path that leads to the present understanding of the problem and to highlight the techniques employed and the main ideas. In the final part we outline a few major open questions that concern the topical problem of the multiplicity of self-adjoint realisations of the model, and which are worth addressing in the future.