21778 articles – 15587 references  [version française]
 HAL: hal-00476829, version 1
 arXiv: 1005.0296
 Available versions: v1 (2010-04-29) v2 (2011-09-13)
 Semiclassical measures for the Schrödinger equation on the torus
 (2010-04-27)
 Our main result is the following : let $(u_{n})$ be a sequence in $L^{2}(\mathbb{T}^{d})$, such that $\norm{u_n}_{L^{2}(\mathbb{T}^{d})}=1$ for all $n$. Consider the sequence of probability measures $\nu_{n}$ on $\mathbb{T}^{d}$, defined by $\nu_{n}(dx)=\left( \int_{0}^{1}|e^{it\Delta /2}u_{n}(x)|^{2}dt\right) dx.$ Let $\nu$ be any weak-$\ast$ limit of the sequence $(\nu_{n})$~: then $\nu$ is absolutely continuous. This generalizes a former result of Bourgain and Jakobson, who considered the case when the functions $u_{n}$ are eigenfunctions of the Laplacian. Our approach is different from theirs, it relies on the notion of (two-microlocal) semiclassical measures, and the properties of the geodesic flow on the torus.
 1: Laboratoire de Mathématiques d'Orsay (LM-Orsay) CNRS : UMR8628 – Université Paris XI - Paris Sud 2: Universidad Politécnica de Madrid (UPM) Universidad Politécnica de Madrid
 Subject : Mathematics/General MathematicsMathematics/Analysis of PDEs
 Keyword(s): equation de Schrödinger – mesures semiclassiques – analyse harmonique
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 hal-00476829, version 1 http://hal.archives-ouvertes.fr/hal-00476829 oai:hal.archives-ouvertes.fr:hal-00476829 From: Nalini Anantharaman <> Submitted on: Tuesday, 27 April 2010 13:37:47 Updated on: Wednesday, 5 May 2010 16:51:08