Geodesic
Semiclassical measures for the Schrödinger equation on the torus
Journal of the European Mathematical Society, Tome 16 (2014) no. 6, pp. 1253-1288
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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},
     journal = {Journal of the European Mathematical Society},
     pages = {1253--1288},
     year = {2014},
     volume = {16},
     number = {6},
     doi = {10.4171/jems/460},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/460/}
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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

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