Dirac–Coulomb operators with general charge distribution II. The lowest eigenvalue

Consider the Coulomb potential (Formula presented.) generated by a non-negative finite measure (Formula presented.). It is well known that the lowest eigenvalue of the corresponding Schrödinger operator (Formula presented.) is minimized, at fixed mass (Formula presented.), when (Formula presented.) is proportional to a delta. In this paper, we investigate the conjecture that the same holds for the Dirac operator (Formula presented.). In a previous work on the subject, we proved that this operator is self-adjoint when (Formula presented.) has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min–max formulas. Here we consider the critical mass (Formula presented.), below which the lowest eigenvalue does not dive into the lower continuous spectrum for any (Formula presented.) with (Formula presented.). We first show that (Formula presented.) is related to the best constant in a new scale-invariant Hardy-type inequality. Our main result is that for all (Formula presented.), there exists an optimal measure (Formula presented.) giving the lowest possible eigenvalue at fixed mass (Formula presented.), which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators. The existence proof is based on the concentration-compactness principle.