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Article Dans Une Revue Annales Henri Lebesgue Année : 2021

Dirac-Coulomb operators with general charge distribution. I. Distinguished extension and min-max formulas

Résumé

This paper is the first of a series where we study the spectral properties of Dirac operators with the Coulomb potential generated by any finite signed charge distribution $\mu$. We show here that the operator has a unique distinguished self-adjoint extension under the sole condition that $\mu$ has no atom of weight larger than or equal to one. Then we discuss the case of a positive measure and characterize the domain using a quadratic form associated with the upper spinor, following earlier works by Esteban and Loss. This allows us to provide min-max formulas for the eigenvalues in the gap. In the event that some eigenvalues have dived into the negative continuum, the min-max formulas remain valid for the remaining ones. At the end of the paper we also discuss the case of multi-center Dirac-Coulomb operators corresponding to $\mu$ being a finite sum of deltas.

Dates et versions

hal-02503460 , version 1 (10-03-2020)

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Maria J. Esteban, Mathieu Lewin, Eric Séré. Dirac-Coulomb operators with general charge distribution. I. Distinguished extension and min-max formulas. Annales Henri Lebesgue, 2021, 4, pp.1421-1456. ⟨10.5802/ahl.106⟩. ⟨hal-02503460⟩
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