University of Birmingham > Talks@bham > Analysis Seminar > A microscopic derivation of time-dependent correlation functions of the 1D nonlinear Schrödinger equation

A microscopic derivation of time-dependent correlation functions of the 1D nonlinear Schrödinger equation

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If you have a question about this talk, please contact Yuzhao Wang.

The nonlinear Schrödinger equation (NLS) is a nonlinear PDE which admits an invariant Gibbs measure. The construction of these measures was given in the constructive field theory literature in the 1970s and their invariance was first rigorously proved by Bourgain in the 1990s. Since then, Gibbs measures have become an important tool in constructing solutions for low regularity random initial data.

The NLS can also be viewed as a classical limit of many-body quantum dynamics. In this context, it is natural to ask how one can obtain the Gibbs measure as a limit of many-body quantum Gibbs states. In the first part of the talk, I will review some results on this problem, obtained in earlier joint work with J. Fröhlich, A. Knowles, and B. Schlein.

The main part of the talk is devoted to the time-dependent problem. I will explain how to derive time-dependent correlation functions of the NLS in a limit from corresponding quantum objects in one dimension. This result holds for nonlocal interactions with bounded convolution potential. I will also explain how one can obtain a partial result for local interactions on the circle. This is joint work with J. Fröhlich, A. Knowles, and B. Schlein.

This talk is part of the Analysis Seminar series.

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