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Uniform estimates for the solutions of the Schrödinger equation on the torus and regularity of semiclassical measures

We establish uniform bounds for the solutions $e^{itΔ}u$ of the Schrödinger 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 $|e^{itΔ}u_{n}|^{2}$ 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. Macià on the structure of semiclassical measures for solutions to the Schrödinger equation on the torus.