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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
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
Published 8 November 2012