On crystal ground state in the Schr\"odinger-Poisson model: point ions

A space-periodic ground state is shown to exist for lattices of point ions in ${\mathbb{R}}^3$ coupled to the Schr"odinger and scalar fields. The coupling requires the renormalization of the selfaction because of the singularity of the Coulomb potential. The ground state is constructed by minimization of the renormalized energy per cell. This energy is bounded from below when the charge of each ion is positive. The elementary cell is necessarily neutral.