Revista Matemática Iberoamericana

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Volume 35, Issue 3, 2019, pp. 877–924
DOI: 10.4171/rmi/1074

Published online: 2019-04-15

Domains for Dirac–Coulomb min-max levels

Maria J. Esteban[1], Mathieu Lewin[2] and Éric Séré[3]

(1) Université Paris-Dauphine, France
(2) Université Paris-Dauphine, France
(3) Université Paris-Dauphine, France

We consider a Dirac operator in three space dimensions, with an electrostatic (i.e., real-valued) potential $V(x)$, having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a distinguished self-adjoint extension $D_V$. 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 $D_V$, in a range of simple function spaces independent of $V$. Our results include the critical case lim inf$_{x \to 0} |x| V(x)= -1$, with units such that $\hbar=mc^2=1$, and they are the first ones in this situation. We also give the corresponding results in two dimensions.

Keywords: Dirac–Coulomb operator, eigenvalues, distinguished self-adjoint extension, min-max methods

Esteban Maria, Lewin Mathieu, Séré Éric: Domains for Dirac–Coulomb min-max levels. Rev. Mat. Iberoam. 35 (2019), 877-924. doi: 10.4171/rmi/1074