Uniform estimates for the solutions of the Schr\"odinger equation on the torus and regularity of semiclassical measures

Abstract

We establish uniform bounds for the solutions $e^{it\Delta}u$ of the Schr\"{o}dinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak-* limits of sequences of densities of the form $|e^{it\Delta}u_{n}|^{2}$ corresponding to highly oscillating sequences of initial data $(u_{n})$. We obtain improved regularity properties of those limits using previous results by N. Anantharaman and F. Maci\`a on the structure of semiclassical measures for solutions to the Schr\"{o}dinger equation on the torus.