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 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 V verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that V is the Coulomb potential.

Authorship

Download reference