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

TL;DR

This paper proves observability estimates for the Schr"odinger equation on tori, including with bounded potentials, resolving a longstanding conjecture using cluster decomposition and induction methods.

Contribution

It establishes new quantitative observability results for Schr"odinger equations on tori, including with bounded potentials, for all times and open sets.

Findings

Quantitative observability estimate for small times and observation sets.

Observability holds for Schr"odinger with bounded potential in any dimension and for all positive times.

Resolved a well-known conjecture in the field regarding observability with potentials.

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.