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

Consider the Coulomb potential $-\mu\ast|x|^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that lowest eigenvalue of corresponding Schr\"odinger operator $-\Delta/2-\mu\ast|x|^{-1}$ minimized, at fixed mass $\mu(\mathbb{R}^3)=\nu$, when $\mu$ proportional to delta. In this paper we investigate conjecture same holds for Dirac $-i\alpha\cdot\nabla+\beta-\mu\ast|x|^{-1}$. previous work on subject proved self-adjoint has no atom larger than or equal 1, and its eigenvalues are given min-max formulas. Here consider critical $\nu_1$, below which does not dive into lower continuum spectrum all $\mu\geq0$ with $\mu(\mathbb{R}^3)<\nu_1$. We first show $\nu_1$ related best constant in new scaling-invariant Hardy-type inequality. Our main result $0\leq\nu<\nu_1$, there exists an optimal giving possible concentrates compact set Lebesgue zero. The last property shown using unique continuation principle operators. existence proof based concentration-compactness principle.