Domains for Dirac-Coulomb min-max levels

Maria J. Esteban (1) , Mathieu Lewin (1) , Eric Séré (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 $\liminf_{x \to 0} |x| V(x)= -1$, 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.

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]

Maria J. Esteban, Mathieu Lewin, Eric Séré. Domains for Dirac-Coulomb min-max levels. Revista Matemática Iberoamericana, In press. ⟨hal-01467766v3⟩