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arXiv: 1005.0296

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v1 (2010-04-29)

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Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman 1, Fabricio Macià 2

(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 Mathematics Mathematics/Analysis of PDEs

Keyword(s): equation de Schrödinger – mesures semiclassiques – analyse harmonique

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From: Nalini Anantharaman <>

Submitted on: Tuesday, 27 April 2010 13:37:47

Updated on: Wednesday, 5 May 2010 16:51:08