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arXiv · 1702.04976

Domains for Dirac-Coulomb min-max levels

Maria J. Esteban·Mathieu Lewin·Eric Séré
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Abstract

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.

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Maria J. Esteban, Mathieu Lewin, Eric Séré. 2019-05-13. Domains for Dirac-Coulomb min-max levels. https://doi.org/10.4171/rmi%2F1074

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