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


Fröhlich, Jürg; Knowles, Antti; Schlein, Benjamin; Sohinger, Vedran (2019). A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation. Advances in Mathematics, 353:67-115.

Abstract

We give a microscopic derivation of time-dependent correla-tion 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 impor-tant new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaus-sian 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 in-teractions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in $\X^s,b\) spaces.

Abstract

We give a microscopic derivation of time-dependent correla-tion 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 impor-tant new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaus-sian 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 in-teractions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in $\X^s,b\) spaces.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:General Mathematics
Language:English
Date:1 September 2019
Deposited On:16 Dec 2019 09:27
Last Modified:16 Dec 2019 09:28
Publisher:Elsevier
ISSN:0001-8708
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1016/j.aim.2019.06.029

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