Observability inequality for the von Neumann equation in crystals
Thomas Borsoni, Virginie Ehrlacher
TL;DR
This work addresses observability for the von Neumann equation in a periodic crystal, aiming for a quantitative inequality that remains valid in the semiclassical limit $\hbar\to 0$. It extends the Golse–Paul stability framework to the crystal setting by employing a Bloch-decomposed phase-space metric based on a periodic optimal-transport-like cost $E_{\hbar,\lambda}$ to couple classical densities $f$ and quantum density matrices $R$. Key contributions include constructing $\mathscr{L}$-periodic traces, Schrödinger coherent states, Töplitz and Husimi transforms, and a stability estimate with explicit $\hbar$-dependent quantum corrections, enabling observability bounds under a geometric control condition (GC). The results provide a rigorous bridge between classical controllability and quantum observability in periodic media, with potential implications for controllability and sensing in crystalline quantum systems.
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 Schrödinger coherent state, periodic Töplitz operator and periodic Husimi densities.