In this talk, I will discuss recent joint work with Nicolas Burq on the controllability and observability of the linear Schrödinger equation on the torus. We establish controllability and observability results for a wide class of measurable observation domains and rough potentials. Our approach uses mathematical induction on the dimension and relies on a cluster structure of the space-time frequencies. This structure, originally introduced by Granville and Spencer and later applied by Bourgain in his study of the Schrödinger equation on the torus, plays an important role in our analysis.
