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CATEGORIES:Geometric Analysis and Partial Differential Equati
ons seminar
SUMMARY:Gibbs measures of nonlinear Schrodinger equations
as limits of many-body quantum states in dimension
d <\;= 3 - Vedran Sohinger\, University of Warw
ick
DTSTART;TZID=Europe/London:20180326T150000
DTEND;TZID=Europe/London:20180326T160000
UID:TALK100891AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/100891
DESCRIPTION:Gibbs measures of nonlinear Schrodinger equations
are a fundamental object used to study low-regular
ity solutions with random initial data. In the dis
persive PDE community\, this point of view was pio
neered by Bourgain in the 1990s. We prove that Gib
bs measures of nonlinear Schrodinger equations ari
se as high-temperature limits of appropriately mod
ified thermal states in many-body quantum mechanic
s. We consider bounded defocusing interaction pote
ntials and work either on the d-dimensional torus
or on R^d with a confining potential. The analogou
s problem for d=1 and in higher dimensions with sm
ooth non translation-invariant interactions was pr
eviously studied by Lewin\, Nam\, and Rougerie by
means of entropy methods. In our work\, we apply a
perturbative expansion of the interaction\, motiv
ated by ideas from field theory. The terms of the
expansion are analyzed using a diagrammatic repres
entation and their sum is controlled using Borel r
esummation techniques. When d=2\,3\, we apply a Wi
ck ordering renormalization procedure. Moreover\,
in the one-dimensional setting our methods allow u
s to obtain a microscopic derivation of time-depen
dent correlation functions for the cubic nonlinear
Schrodinger equation. This is joint work with Jue
rg Froehlich\, Antti Knowles\, and Benjamin Schlei
n.
LOCATION:CMS\, MR13
CONTACT:Josephine Evans
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