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

Consider an electron moving in the Coulomb potential −μ∗|x|−1−μ∗|x|−1 generated by any non-negative finite measure μμ. It is well known that the lowest eigenvalue of the corresponding Schrodinger operator −Δ/2−μ∗|x|−1−Δ/2−μ∗|x|−1 is minimized, at fixed mass μ(R3)=νμ(R3)=ν, when μμ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator −iα⋅∇+β−μ∗|x|−1−iα⋅∇+β−μ∗|x|−1. In a previous work on the subject we proved that this operator is well defined and that its eigenvalues are given by min-max formulas. Here we show that there exists a critical number ν1ν1 below which the lowest eigenvalue does not dive into the lower continuum spectrum, for all μ≥0μ≥0 with μ(R3)<ν1μ(R3)<ν1. Our main result is that for all 0≤ν<ν10≤ν<ν1, there exists an optimal measure μ≥0 μ≥0 giving the lowest possible eigenvalue at fixed mass μ(R3)=νμ(R3)=ν, which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators.