In this talk I will present recent results with M. Lewin and E. Séré about Dirac-Coulomb operators where the potential perturbing the free Dirac operator is a Coulomb-like potential of the form $-\mu\star\frac1{|x]$, where $\mu$ is a general nonnegative finite measure.
For these operators, distinguished self-adjoint extensions can be defined under natural conditions on $\mu$ and also min-max characterizations can be proved for the eigenvalues of such operators in the spectral gap. But interesting problems arise when one tries to understand the location of the lowest eigenvalue in the gap, and in particular to see what conditions has to satisfy the measure $\mu$ so that this eigenvalue does not leave the spectral gap, that is, so that the (relativistic) electron is not destabilized.
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