# Domains for Dirac-Coulomb min-max levels

## TL;DR

This paper studies the spectral properties of a Dirac operator with Coulomb singularities, establishing new domain characterizations and min-max formulas for eigenvalues, including critical cases in three and two dimensions.

## Contribution

It provides the first analysis of Dirac operators with Coulomb singularities at the critical limit, extending previous domain results and eigenvalue formulas to these challenging cases.

## Key findings

- New domain descriptions for Dirac-Coulomb operators
- Valid min-max eigenvalue formulas at the critical Coulomb strength
- Extension of results to two-dimensional cases

## 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.