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

2018/10/02 by Cassano, Biagio, Pizzichillo, Fabio, Vega, Luis
#35P05 #47B25 (Secondary) #47N20 #81Q10(Primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1810.01309

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.

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