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

TL;DR

This paper proves uniform bounds for Schrödinger solutions on tori and investigates the regularity of their semiclassical measures, extending previous results and providing new insights into the behavior of highly oscillating quantum states.

Contribution

It generalizes earlier uniform bounds to arithmetic tori and improves understanding of the regularity of semiclassical measures for oscillating solutions.

Findings

Established uniform bounds for Schrödinger solutions on tori.

Proved enhanced regularity properties of semiclassical measures.

Extended previous results by Bourgain, Anantharaman, and Macià.

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.