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 $\hbar$. 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.

💡 Research Summary

The paper establishes a quantitative observability inequality for the von Neumann equation in a periodic crystal setting, with constants that remain uniform as the semiclassical parameter ℏ tends to zero. The authors adapt the recent method of Golse and Paul (2022), originally developed for non‑periodic systems, to the case of an infinite periodic lattice. The key difficulty is that in a crystal the usual trace of an operator diverges because the number of electrons is infinite; therefore the authors introduce a “periodic trace” based on Bloch decomposition, which measures the number of particles per unit cell rather than the total number.

To compare quantum and classical dynamics they define periodic analogues of coherent states, Töplitz operators, and Husimi transforms. A periodic coherent state |q,p⟩ is obtained by summing the standard Gaussian coherent state over all lattice translations, yielding a function that is L‑periodic both in the spatial variable and in the centre q. The Töplitz operator TL