Geodesic
Semiclassical measures for the Schrödinger equation on the torus
Journal of the European Mathematical Society, Tome 16 (2014) no. 6, pp. 1253-1288.

Voir la notice de l'article provenant de la source EMS Press

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 L2-norm of a solution on any open subset of the torus controls the full L2-norm.
DOI : 10.4171/jems/460
Classification : 47-XX, 00-XX, 35-XX
Keywords: Semiclassical (Wigner) measures, linear Schrödinger equation on the torus, semiclassical limit, dispersive estimates, observability estimates 
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     author = {Nalini Anantharaman and Fabricio Maci\`a},
     title = {Semiclassical measures for the {Schr\"odinger} equation on the torus},
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Nalini Anantharaman; Fabricio Macià. Semiclassical measures for the Schrödinger equation on the torus. Journal of the European Mathematical Society, Tome 16 (2014) no. 6, pp. 1253-1288. doi : 10.4171/jems/460. http://geodesic.mathdoc.fr/articles/10.4171/jems/460/

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