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SUMMARY:Gibbs measures of nonlinear Schrodinger equations as limits of man
 y-body quantum states in dimension d &lt\;= 3 - Vedran Sohinger\, Universi
 ty of Warwick
DTSTART:20180326T140000Z
DTEND:20180326T150000Z
UID:TALK100891@talks.cam.ac.uk
CONTACT:Josephine Evans
DESCRIPTION:Gibbs measures of nonlinear Schrodinger equations are a fundam
 ental object used to study low-regularity solutions with random initial da
 ta. In the dispersive PDE community\, this point of view was pioneered by 
 Bourgain in the 1990s. We prove that Gibbs measures of nonlinear Schroding
 er equations arise as high-temperature limits of appropriately modified th
 ermal states in many-body quantum mechanics. We consider bounded defocusin
 g interaction potentials and work either on the d-dimensional torus or on 
 R^d with a confining potential. The analogous problem for d=1 and in highe
 r dimensions with smooth non translation-invariant interactions was previo
 usly studied by Lewin\, Nam\, and Rougerie by means of entropy methods. In
  our work\, we apply a perturbative expansion of the interaction\, motivat
 ed by ideas from field theory. The terms of the expansion are analyzed usi
 ng a diagrammatic representation and their sum is controlled using Borel r
 esummation techniques. When d=2\,3\, we apply a Wick ordering renormalizat
 ion procedure. Moreover\, in the one-dimensional setting our methods allow
  us to obtain a microscopic derivation of time-dependent correlation funct
 ions for the cubic nonlinear Schrodinger equation. This is joint work with
  Juerg Froehlich\, Antti Knowles\, and Benjamin Schlein.
LOCATION:CMS\, MR13
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