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

We study the observability of the Schrödinger equation on the $d$-dimensional torus $\T^d=\R^d/(2π\Z)^d$, $d \geq 1$, 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 ob