Mathematical Research Letters

Volume 19 (2012)

Number 3

Uniform estimates for the solutions of the Schrödinger equation on the torus and regularity of semiclassical measures

Pages: 589 – 599

DOI: https://dx.doi.org/10.4310/MRL.2012.v19.n3.a7

Authors

Tayeb Aïssiou (Department of Mathematics and Statistics, Concordia University, Montréal, Quebec, Canada)

Dmitry Jakobson (Department of Mathematics and Statistics, McGill University, Montréal, Quebec, Canada)

Fabricio Macià (DCAIN, ETSI Navales, Universidad Politécnica de Madrid, Spain)

Abstract

We establish uniform bounds for the solutions ${\rm e}^{it\Delta}u$ of theSchrödinger equation on arithmetic flat tori, generalizing earlierresults by J. Bourgain.We also study the regularity properties of weak-$*$ limits of sequences ofdensities of the form $|{\rm e}^{it\Delta}u_{n}|^{2}$ corresponding to highlyoscillating sequences of initial data $(u_{n})$. We obtain improved regularityproperties of those limits using previous results by N. Anantharaman and F.Macià on the structure of semiclassical measures for solutions to theSchrödinger equation on the torus.

Keywords

semiclassical limits, linear Schrödinger equation on a torus, quantum limits, semiclassical measures, dispersive estimates

2010 Mathematics Subject Classification

35B45, 35Q41, 42B05, 58-xx, 81Q50

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Published 8 November 2012