Semiclassical measures for the Schrödinger equation on the torus

  • Nalini Anantharaman

    Université de Strasbourg, France
  • Fabricio Macià

    Universidad Politécnica de Madrid, Spain

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 -norm of a solution on any open subset of the torus controls the full -norm.

Cite this article

Nalini Anantharaman, Fabricio Macià, Semiclassical measures for the Schrödinger equation on the torus. J. Eur. Math. Soc. 16 (2014), no. 6, pp. 1253–1288

DOI 10.4171/JEMS/460