Domains for Dirac–Coulomb min-max levels

  • Maria J. Esteban

    Université Paris-Dauphine, France
  • Mathieu Lewin

    Université Paris-Dauphine, France
  • Éric Séré

    Université Paris-Dauphine, France
Domains for Dirac–Coulomb min-max levels cover
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We consider a Dirac operator in three space dimensions, with an electrostatic (i.e., real-valued) potential , having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a distinguished self-adjoint extension . In a first part we obtain new results on the domain of this extension, complementing previous works of Esteban and Loss. Then we prove the validity of min-max formulas for the eigenvalues in the spectral gap of , in a range of simple function spaces independent of . Our results include the critical case lim inf, with units such that , and they are the first ones in this situation. We also give the corresponding results in two dimensions.

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Maria J. Esteban, Mathieu Lewin, Éric Séré, Domains for Dirac–Coulomb min-max levels. Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 877–924

DOI 10.4171/RMI/1074