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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: http://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