On the observability of the Schrödinger equation in the torus from open sets

We study the observability of the Schrödinger equation on the $d$-dimensional torus ${\mathbb{T}}^d$, $d\ge 1$, from an open subset $\omega \subset {\mathbb{T}}^d$. Our first main result establishes a quantitative observability estimate for the free Schrödinger 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 Schrödinger equation with a merely bounded potential $V\in L^{\mathrm{\infty }}({\mathbb{T}}^d)$, in any dimension $d\ge 1$, 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.