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SUMMARY:Dynamics\, dispersion and control of Schrödinger equations - Fabr
 icio Macià\, Universidad Politécnica de Madrid
DTSTART:20180423T140000Z
DTEND:20180423T150000Z
UID:TALK103457@talks.cam.ac.uk
CONTACT:Ivan Moyano
DESCRIPTION:We are interested in the dynamics of linear Schrödinger equat
 ions: i\\partial_t u(t\,x)+\\Delta_x u(t\,x)-V(t\,x)u(t\,x)=0\,\\quad (t\,
 x)\\in \\mathbb{R}\\times M\, in bounded geometries such as a compact mani
 fold\, equipped with a Riemannian metric\, or a bounded domain in Euclidea
 n space. More specifically\, we would like to understand the structure of 
 those subsets on which high-frequency solutions can concentrate (in the se
 nse of the L2 norm)\; that is\, regions on which the position probability 
 densities |u_n(t\,x)|^2 of a normalized sequence of solutions can accumula
 te. This problem is also related to quantifying dispersion and understandi
 ng controllability properties for Schrödinger equations.\n \nWe give a de
 tailed answer to this question for systems whose underlying classical dyna
 mics (the geodesic flow or the billiard flow) is completely integrable (as
  flat tori\, spheres or the planar disk). Our analysis is based on underst
 anding the structure of  Wigner measures associated to sequences of soluti
 ons. We accomplish that by analysing the solutions to the corresponding Wi
 gner equations by means of a (second-micro)localization with respect to a 
 partition of phase-space adapted to the classical dynamical system.\n\nThi
 s talk is based on joint works with Nalini Anantharaman\, Clotilde Fermani
 an-Kammerer\, Matthieu Léautaud and Gabriel Rivière.
LOCATION:CMS\, MR13
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