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 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 verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that is the Coulomb potential.
A Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator
Cassano B.;
2020
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 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 verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that is the Coulomb potential.File in questo prodotto:
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