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

Consider the Coulomb potential $-\mu \ast \mathrm{\mid }x{\mathrm{\mid }}^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schr"odinger operator $-\mathrm{\Delta }\mathrm{/}2-\mu \ast \mathrm{\mid }x{\mathrm{\mid }}^{-1}$ is minimized, at fixed mass $\mu ({\mathbb{R}}^3)=\nu$, when $\mu$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-i\alpha \cdot \mathrm{\nabla }+\beta -\mu \ast \mathrm{\mid }x{\mathrm{\mid }}^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $\mu$ 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 ${\nu }_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $\mu \ge 0$ with $\mu ({\mathbb{R}}^3)<{\nu }_1$. We first show that ${\nu }_1$ is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all $0\le \nu <{\nu }_1$, there exists an optimal measure $\mu \ge 0$ giving the lowest possible eigenvalue at fixed mass $\mu ({\mathbb{R}}^3)=\nu$, 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.