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

0522522 - ÚJF 2021 RIV IT eng J - Journal Article
Cassano, Biagio - Pizzichillo, F. - Vega, L.
A Hardy-type inequality and some spectral characterizations for the Dirac-Coulomb operator.
Revista Mathématica Complutense. Roč. 33, č. 1 (2020), s. 1-18. ISSN 1139-1138. E-ISSN 1988-2807
R&D Projects: GA ČR GA17-01706S
Institutional support: RVO:61389005
Keywords : Dirac operator * Coulomb potential * Hardy inequality * self-adjoint operator * spectral properties * ground state
OECD category: Pure mathematics
Impact factor: 1.227, year: 2020
Method of publishing: Open access
https://doi.org/10.1007/s13163-019-00311-4

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.
Permanent Link: http://hdl.handle.net/11104/0307003