Gibbs measures of nonlinear Schrodinger equations as limits of many-body quantum states in dimension d <= 3

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• Vedran Sohinger, University of Warwick
• Monday 26 March 2018, 15:00-16:00
• CMS, MR13.

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Gibbs measures of nonlinear Schrodinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We prove that Gibbs measures of nonlinear Schrodinger equations arise as high-temperature limits of appropriately modified thermal states in many-body quantum mechanics. We consider bounded defocusing 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 higher dimensions with smooth non translation-invariant interactions was previously studied by Lewin, Nam, and Rougerie by means of entropy methods. In our work, we apply a perturbative expansion of the interaction, motivated by ideas from field theory. The terms of the expansion are analyzed using a diagrammatic representation and their sum is controlled using Borel resummation techniques. When d=2,3, we apply a Wick ordering renormalization procedure. Moreover, in the one-dimensional setting our methods allow us to obtain a microscopic derivation of time-dependent correlation functions for the cubic nonlinear Schrodinger equation. This is joint work with Juerg Froehlich, Antti Knowles, and Benjamin Schlein.

This talk is part of the Partial Differential Equations seminar series.

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