Critical magnetic field for 2d magnetic Dirac-Coulomb operators and Hardy inequalities

TL;DR

This paper investigates the critical magnetic field strength for which Hardy inequalities hold for 2D magnetic Dirac-Coulomb operators with Aharonov-Bohm potential, identifying conditions for self-adjointness and ground state energies.

Contribution

It characterizes the maximal magnetic field allowing Hardy inequalities and self-adjoint extensions for 2D magnetic Dirac-Coulomb operators with Aharonov-Bohm potential.

Findings

Identified the critical magnetic field threshold for Hardy inequalities.

Established the existence of a distinguished self-adjoint extension below this threshold.

Defined the ground state energy as the lowest eigenvalue in the spectral gap.

Abstract

This paper is devoted to the study of the two-dimensional Dirac-Coulomb operator in presence of an Aharonov-Bohm external magnetic potential. We characterize the highest intensity of the magnetic field for which a two-dimensional magnetic Hardy inequality holds. Up to this critical magnetic field, the operator admits a distinguished self-adjoint extension and there is a notion of ground state energy, defined as the lowest eigenvalue in the gap of the continuous spectrum.