Abstract (EN)

Consider the Coulomb potential −μ∗|x|−1 generated by a non-negative finite measure μ. It is well known that the lowest eigenvalue of the corresponding Schrödinger operator −Δ/2−μ∗|x|−1 is minimized, at fixed mass μ(R3)=ν, when μ 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. In a previous work on the subject we proved that this operator is self-adjoint when μ 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 ν1, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all μ≥0 with μ(R3)<ν1. We first show that ν1 is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all 0≤ν<ν1, there exists an optimal measure μ≥0 giving the lowest possible eigenvalue at fixed mass μ(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. The existence proof is based on the concentration-compactness principle.