Observability inequality for the von Neumann equation in crystals

TL;DR

This paper establishes a quantitative observability inequality for the von Neumann equation in crystalline structures, adapting methods from non-crystal settings using periodic analysis tools, with results uniform in small Planck's constant.

Contribution

It extends the observability inequality to the crystal setting by adapting the stability method and tools like Bloch decomposition and periodic densities, previously developed for non-crystal cases.

Findings

Derived a uniform observability inequality in the crystal setting.

Adapted stability and transport methods to periodic structures.

Utilized Bloch decomposition and periodic densities for analysis.

Abstract

We provide a quantitative observability inequality for the von Neumann equation on $\mathbb{R}^d$ in the crystal setting, uniform in small $\hbar$. 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 Schr\"odinger coherent state, periodic T\"oplitz operator and periodic Husimi densities.