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

Abstract

We study the observability of the Schrödinger equation on the $d$-dimensional torus $\T^d=\R^d/(2π\Z)^d$, $d\ge 1$, from an open subset $ω\subset \T^d$. Our main result establishes a quantitative observability estimate for the free Schrödinger equation in the regime of small times $T$ and small observation sets of the form $\omega ={\prod }_{j=1}^d(a_j,b_j)$. 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.

This version replaces the last one that was containing a gap in the proofs of the main result. We have saved one of our main result about the quantitative observability of the Schrödinger equation, but we have lost the one about the potential unfortunately