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A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation
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Rout, Andrew and Sohinger, Vedran (2023) A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation. Communications in Partial Differential Equations, 48 (7-8). pp. 1008-1055. doi:10.1080/03605302.2023.2243491 ISSN 0360-5302.
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Official URL: https://doi.org/10.1080/03605302.2023.2243491
Abstract
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 previous work of Fröhlich, Knowles, Schlein, and the second author. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, integrable, 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.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
| Library of Congress Subject Headings (LCSH): | Gibbs' equation , Schrödinger equation, Differential equations, Partial | |||||||||
| Journal or Publication Title: | Communications in Partial Differential Equations | |||||||||
| Publisher: | Taylor & Francis Inc. | |||||||||
| ISSN: | 0360-5302 | |||||||||
| Official Date: | 2023 | |||||||||
| Dates: |
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| Volume: | 48 | |||||||||
| Number: | 7-8 | |||||||||
| Page Range: | pp. 1008-1055 | |||||||||
| DOI: | 10.1080/03605302.2023.2243491 | |||||||||
| Institution: | University of Warwick | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Access rights to Published version: | Open Access (Creative Commons) | |||||||||
| Date of first compliant deposit: | 2 August 2023 | |||||||||
| Date of first compliant Open Access: | 9 August 2023 | |||||||||
| RIOXX Funder/Project Grant: |
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