On the Crystal Ground State in the Schrödinger–Poisson Model with Point Ions

A space-periodic ground state is shown to exist for lattices of point ions in $\mathbb{R}^3$ coupled to the Schrödinger and scalar fields. The coupling requires renormalization due to the singularity of the Coulomb self-action. The ground state is constructed by minimizing 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.