Uniqueness, Nondegeneracy, and Asymptotic Expansions of Ground States for the Dirac-Coulomb System

We study ground states of the following Dirac-Coulomb system $${\mathcal{D}}_c{u}_c-(\text{abs}{x}^{-1}\ast \text{abs}{u_c}^2)u_c={\omega }_c{u}_c,$$ under the constraint $\text{norm}[L^2]{u_c}^2=1.$For sufficiently large $c$, we prove uniqueness of ground states modulo translations, phase transformations and spinorial rotations, and determine the kernel of the linearized operator at ground state. We also obtain asymptotic expansions of ground state in the nonrelativistic limit regime.