Semiclassical measures for the Schr\"odinger equation on the torus

TL;DR

This paper investigates the structure of semiclassical measures for the Schrödinger equation on the torus, revealing their absolute continuity on invariant tori, explicit propagation laws, and implications for observability and control.

Contribution

It provides a detailed analysis of semiclassical measures on the torus, including their disintegration, Radon-Nikodym derivatives, and propagation, with applications to observability inequalities.

Findings

Semiclassical measures disintegrate absolutely continuously on invariant tori.

Explicit Radon-Nikodym derivatives satisfy a propagation law.

An observability inequality links local $L^2$-norms to global control.

Abstract

In this article, the structure of semiclassical measures for solutions to the linear Schr\"{o}dinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the $L^2$-norm of a solution on any open subset of the torus controls the full $L^2$-norm.