Rice University Department of Mathematics Colloquium

From quantum many particle systems to nonlinear dispersive PDE, and back

4:00 pm Thursday, April 13, 2017
Natasa Pavlovic (UT Austin)

Abstract: The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\"{o}dinger (NLS) from many particle quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE that describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE. We will discuss the process of going from a quantum many particle system of bosons to the NLS via the GP. The most involved part in such a derivation of NLS consists in establishing uniqueness of solutions to the GP. In the talk we will focus on approaches to the uniqueness step that are motivated by the perspective coming from nonlinear dispersive PDE, including the approach that we developed with Chen, Hainzl and Seiringer based on the quantum de Finetti's theorem. Also we will look into what else the nonlinear PDE such as the NLS can tell us about the GP hierarchy, and will present recent results on infinitely many conserved quantities for the GP hierarchy that are obtained with Mendelson, Nahmod and Staffilani.

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