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

TL;DR

This paper proves the existence of a space-periodic ground state for a lattice of point ions in the Schrödinger-Poisson model, addressing the challenges of Coulomb singularities through renormalization.

Contribution

It introduces a method to construct ground states for point ion lattices in the Schrödinger-Poisson framework, including a renormalization technique for Coulomb singularities.

Findings

Ground state exists for positive ion charges.

Energy per cell is bounded below and minimized.

Elementary cell must be neutral.

Abstract

A space-periodic ground state is shown to exist for lattices of point ions in $\R^3$ 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.