On the observability of the Schr\"odinger equation in the torus from open sets

Abstract

We study the observability of the Schr\"odinger equation on the $d$-dimensional torus $\mathbb T^d$, $d \geq 1$, from an open subset $\omega \subset \mathbb T^d$. Our first main result establishes a quantitative observability estimate for the free Schr\"odinger equation in the regime of small times $T$ and for small observation sets of the form $\omega = \prod_{j=1}^{d}(a_j,b_j)$. Our second main result shows that observability holds for the Schr\"odinger equation with a merely bounded potential $V \in L^{\infty}(\mathbb T^d)$, in any dimension $d \geq 1$, for every time $T>0$ and every nonempty open subset $\omega$. This resolves a well-known conjecture in the field. A central ingredient in the proof is a cluster decomposition method combined with an induction scheme introduced by Bourgain and further developed by Burq and Zhu.