Observability inequality for the von Neumann equation in crystals

We provide a quantitative observability inequality for the von Neumann equation on ${\mathbb{R}}^d$ in the crystal setting, uniform in small $\mathrm{\hslash }$. Following the method of Golse and Paul (2022) proving this result in the non-crystal setting, the method relies on a stability argument between the quantum (von Neumann) and classical (Liouville) dynamics and uses an optimal transport-like pseudo-distance between quantum and classical densities. Our contribution yields in the adaptation of all the required tools to the periodic setting, relying on the Bloch decomposition, notions of periodic Schrödinger coherent state, periodic Töplitz operator and periodic Husimi densities.