In this talk, I will present several recent results, in collaboration with Nicolas Burq, on the control theory of Schrödinger propagators on tori. Our goal is to address the following conjecture: on a torus of arbitrary dimension, Schrödinger propagators with bounded potentials are observable, and therefore controllable, from arbitrary space-time domains of positive Lebesgue measure.
Using a scheme that combines (1) approximation of rough functions by continuous functions, (2) the cluster structure of lattice points near paraboloids, and (3) mathematical induction on the dimension, we reduce the conjecture to certain integrability bounds for linear Schrödinger waves. These bounds are weaker than Bourgain’s conjectured periodic Strichartz estimates but remain nontrivial. In particular, our criteria imply the observability conjecture for the one-dimensional torus.
Applications of our results include Cantor--Lebesgue type theorems and uniform nonvanishing estimates for quantum limits.
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