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Uniform estimates for the solutions of the Schrödinger equation on the torus and regularity of semiclassical measures. (English) Zbl 1276.35063
Summary: We establish uniform bounds for the solutions $$\mathrm{e}^{it\Delta}u$$ of the Schrödinger equation on arithmetic flat tori, generalizing earlier results by J. Bourgain. We also study the regularity properties of weak-$$*$$ limits of sequences of densities of the form $$|\mathrm{e}^{it\Delta}u_{n}|^{2}$$ corresponding to highly oscillating sequences of initial data $$(u_{n})$$. We obtain improved regularity properties of those limits using previous results by N. Anantharaman and F. Macià [“Semiclassical measures for the Schrödinger equation on the torus.”, arxiv:1005.0296] on the structure of semiclassical measures for solutions to the Schrödinger equation on the torus.

MSC:
 35J10 Schrödinger operator, Schrödinger equation 81Q50 Quantum chaos 35Q41 Time-dependent Schrödinger equations and Dirac equations 35B45 A priori estimates in context of PDEs 42B05 Fourier series and coefficients in several variables
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