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Advances in Mathematics

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

Jürg Martin Fröhlich, Antti Knowles, Benjamin Schlein, Vedran Sohinger

Published Jul 1, 2019• 15 citations
Publisher
DOI
10.1016/j.aim.2019.06.029

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A microscopic derivation of time-dependent correlation functions of the 1D1D1D cubic nonlinear Schrödinger equation
In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states.Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting.Our methods are based on a perturbative expansion of the interaction, similarly as in [1].Due to the presence of the truncation, the obtained series have infinite radius of convergence.We treat the case of bounded, L 1 and delta function interaction potentials, without any sign assumptions.Within this framework, we also study time-dependent correlation functions.This is the first such known result in the focusing regime.
Abstract We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx)W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)Wβ​(x)=N−1+2βW(Nβx) Wβ(x)=N-1+2βW(Nβx) , for any β>0\beta >0β>0 β>0 , or to be given by VN(x)=e2NV(eNx)V_N(x)=e^{2N} V(e^N x)VN​(x)=e2NV(eNx) VN(x)=e2NV(eNx) , for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R)W,V \in L^\infty ({\mathbb {R}}^2,{\mathbb {R}})W,V∈L∞(R2,R) W,V∈L∞(R2,R) . In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential VNV_NVN​ VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.