In this article, the structure of semiclassical measures for solutions to the linear Schrödinger 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 -norm of a solution on any open subset of the torus controls the full -norm.
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Nalini Anantharaman, Fabricio Macià, Semiclassical measures for the Schrödinger equation on the torus. J. Eur. Math. Soc. 16 (2014), no. 6, pp. 1253–1288DOI 10.4171/JEMS/460