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The Schrödinger Flow in a Compact Manifold: High-frequency Dynamics and Dispersion

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Modern Aspects of the Theory of Partial Differential Equations

Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 216))

Abstract

We discuss various aspects of the dynamics of the Schrödinger flow on a compact Riemannian manifold that are related to the behavior of highfrequency solutions. In particular we show that dispersive (Strichartz) estimates fail on manifolds whose geodesic flow is periodic (thus generalizing a well-known result for spheres proved via zonal spherical harmonics). We also address the issue of the validity of observability estimates. We show that the geometric control condition is necessary in manifolds with periodic geodesic flow and we give a new, geometric, proof of a result of Jaffard on the observability for the Schrödinger flow on the two-torus. All our proofs are based on the study of the structure of semiclassical (Wigner) measures corresponding to solutions to the Schrödinger equation.

Mathematics Subject Classification (2000). Primary 35Q40; Secondary 58J40.

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Authors and Affiliations

  1. Universidad Politécnica de Madrid DEBIN, ETSI Navales, Avda. Arco de la Victoria s/n., E-28040, Madrid, Spain

    Fabricio Macià

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  1. Fabricio Macià
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Correspondence to Fabricio Macià .

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Editors and Affiliations

  1. Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, United Kingdom

    Michael Ruzhansky

  2. , Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

    Jens Wirth

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Communicated by V. Smyshlyaev

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Macià, F. (2011). The Schrödinger Flow in a Compact Manifold: High-frequency Dynamics and Dispersion. In: Ruzhansky, M., Wirth, J. (eds) Modern Aspects of the Theory of Partial Differential Equations. Operator Theory: Advances and Applications(), vol 216. Springer, Basel. https://doi.org/10.1007/978-3-0348-0069-3_16

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