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

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 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.