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Preprint, Working Paper, ... |
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| Title: |
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Semiclassical measures for the Schrödinger equation on the torus |
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| Author(s): |
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Nalini Anantharaman ( ) 1, Fabricio Macià () 2 |
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| Laboratory: |
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| Abstract: |
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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. |
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| Fulltext language: |
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English |
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| Production date: |
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2011-09-12 |
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| Keyword(s): |
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equation de Schrödinger – mesures semiclassiques – analyse harmonique |
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| Classification: |
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MSC 42B37, MSC 35P20 |
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| Comment: |
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We extended Theorem 1 to the case with potential. We added Theorem 4 and modified the organisation of the paper |
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| Contract, financing: |
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MTM2007-61755 (MEC) |
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| ANR Project: |
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| Project Id |
ANR-09-JCJC-0099-01 |
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