Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors

TL;DR

This paper studies the mathematical properties of a two-dimensional Dirac operator with Coulomb potential and infinite mass boundary conditions in sectors, focusing on self-adjointness, extensions, and spectral analysis.

Contribution

It provides a comprehensive analysis of self-adjoint extensions and spectrum of Dirac operators with Coulomb potentials in sectors, including explicit descriptions for all potential intensities.

Findings

Established Dirac-Hardy inequality for the operator

Described all self-adjoint extensions for Coulomb potentials

Provided spectral characterization of the operators

Abstract

We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate our results giving a description of the spectrum of these operators.