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CATEGORIES:Partial Differential Equations seminar
SUMMARY:Dynamics\, dispersion and control of Schrödinger e
quations - Fabricio Macià\, Universidad Politécnic
a de Madrid
DTSTART;TZID=Europe/London:20180423T150000
DTEND;TZID=Europe/London:20180423T160000
UID:TALK103457AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/103457
DESCRIPTION:We are interested in the dynamics of linear Schröd
inger equations: 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 com
pact manifold\, equipped with a Riemannian metric\
, or a bounded domain in Euclidean space. More spe
cifically\, we would like to understand the struct
ure of those subsets on which high-frequency solut
ions can concentrate (in the sense of the L2 norm)
\; that is\, regions on which the position probabi
lity densities |u_n(t\,x)|^2 of a normalized seque
nce of solutions can accumulate. This problem is a
lso related to quantifying dispersion and understa
nding controllability properties for Schrödinger e
quations.\n \nWe give a detailed answer to this qu
estion for systems whose underlying classical dyna
mics (the geodesic flow or the billiard flow) is c
ompletely integrable (as flat tori\, spheres or th
e planar disk). Our analysis is based on understan
ding the structure of Wigner measures associated
to sequences of solutions. We accomplish that by a
nalysing the solutions to the corresponding Wigner
equations by means of a (second-micro)localizatio
n with respect to a partition of phase-space adapt
ed to the classical dynamical system.\n\nThis talk
is based on joint works with Nalini Anantharaman\
, Clotilde Fermanian-Kammerer\, Matthieu Léautaud
and Gabriel Rivière.
LOCATION:CMS\, MR13
CONTACT:Ivan Moyano
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