Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman; Fabricio Macià

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arXiv · 1005.0296

Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman·Fabricio Macià

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Abstract

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 $L^2$-norm of a solution on any open subset of the torus controls the full $L^2$-norm.

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BibTeX RIS

Nalini Anantharaman, Fabricio Macià. 2011-09-13. Semiclassical measures for the Schrödinger equation on the torus. https://arxiv.org/abs/1005.0296

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