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

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Fröhlich, Jürg, Knowles, Antti, Schlein, Benjamin and Sohinger, Vedran (2019) A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation. Advances in Mathematics, 353 . pp. 67-115. doi:10.1016/j.aim.2019.06.029 ISSN 0001-8708.

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Official URL: http://dx.doi.org/10.1016/j.aim.2019.06.029

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Abstract

We give a microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation (NLS) from many-body quantum theory. The starting point of our proof is [11] on the time-independent problem and [16] on the corresponding problem on a finite lattice. An important new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaussian structure of the free quantum field and prevents the use of the Wick theorem. We overcome it by means of complex analytic methods. Our methods apply to the nonlocal NLS with bounded convolution potential. In the periodic setting, we also consider the local NLS, arising from short-range interactions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in spaces.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Q Science > QC Physics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Schrödinger equation, Gross-Pitaevskii equations, Nonlinear wave equations
Journal or Publication Title: Advances in Mathematics
Publisher: Academic Press
ISSN: 0001-8708
Official Date: 7 September 2019
Dates:
DateEvent
7 September 2019Published
1 July 2019Available
19 June 2019Accepted
Volume: 353
Page Range: pp. 67-115
DOI: 10.1016/j.aim.2019.06.029
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 5 July 2019
Date of first compliant Open Access: 1 July 2020

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