A Hardy-type inequality and some spectral characterizations for the Dirac-Coulomb operator

TL;DR

This paper establishes a sharp Hardy-type inequality for the Dirac operator and uses it to analyze the spectral properties of the Dirac-Coulomb operator with matrix-valued potentials, characterizing eigenvalues and spectral bounds.

Contribution

It introduces a new Hardy inequality for the Dirac operator and applies it to characterize eigenvalues and spectral bounds of the Dirac-Coulomb operator with Coulomb-type potentials.

Findings

Spectral properties of the Dirac operator with Coulomb-type potentials are characterized.

The ground-state energy is attained only under specific rigidity conditions on the potential.

The Coulomb potential is uniquely identified in the electrostatic case.

Abstract

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials $\mathbf V$ of Coulomb type: we characterise its eigenvalues in terms of the Birman-Schwinger principle and we bound its discrete spectrum from below, showing that the \emph{ground-state energy} is reached if and only if $\mathbf V$ verifies some {rigidity} conditions. In the particular case of an electrostatic potential, these imply that $\mathbf V$ is the Coulomb potential.