Dirac–Coulomb operators with general charge distribution II. The lowest eigenvalue. Issue 4 (6th January 2021) - Esteban, Maria J.; Lewin, Mathieu; Séré, Éric
Abstract: 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 ( R 3 ) = ν, when μ is proportional to a delta. In this paper, we investigate the conjecture that the same holds for the Dirac operator − i α · ∇ + 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 ν 1, below which the lowest eigenvalue does not dive into the lower continuous spectrum for any μ ⩾ 0 with μ ( R 3 ) < ν 1 . We first show that ν 1 is related to the best constant in a new scale‐invariant Hardy‐type inequality. Our main result is that for all 0 ⩽ ν < ν 1, there exists an optimal measure μ ⩾ 0 giving the lowest possible eigenvalue at fixed mass μ ( R 3 ) = 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.