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A microscopic derivation of Gibbs measures for nonlinear Schrodinger equations with unbounded interaction potentials
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Sohinger, Vedran (2022) A microscopic derivation of Gibbs measures for nonlinear Schrodinger equations with unbounded interaction potentials. International Mathematics Research Notices, 2022 (19). pp. 14964-15063. doi:10.1093/imrn/rnab132 ISSN 1073-7928.
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Official URL: http://dx.doi.org/10.1093/imrn/rnab132
Abstract
We study the derivation of the Gibbs measure for the nonlinear Schrödinger (NLS) equation from many-body quantum thermal states in the mean-field limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded Lp interaction potentials on Td for d=1,2,3. This extends the author’s earlier joint work with Fröhlich et al. [ 45], where the regime of defocusing and bounded interaction potentials was considered. When d=1, we give an alternative proof of a result previously obtained by Lewin et al. [ 69]. Our proof is based on a perturbative expansion in the interaction. When d=1, the thermal state is the grand canonical ensemble. As in [ 45], when d=2,3, the thermal state is a modified grand canonical ensemble, which allows us to estimate the remainder term in the expansion. The terms in the expansion are analysed using a graphical representation and are resummed by using Borel summation. By this method, we are able to prove the result for the optimal range of p and obtain the full range of defocusing interaction potentials, which were studied in the classical setting when d=2,3 in the work of Bourgain [ 15].
| 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): | Probability measures, Schrèodinger equation, Nonlinear theories, Quantum field theory | ||||||||
| Journal or Publication Title: | International Mathematics Research Notices | ||||||||
| Publisher: | Oxford University Press | ||||||||
| ISSN: | 1073-7928 | ||||||||
| Official Date: | October 2022 | ||||||||
| Dates: |
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| Volume: | 2022 | ||||||||
| Number: | 19 | ||||||||
| Page Range: | pp. 14964-15063 | ||||||||
| DOI: | 10.1093/imrn/rnab132 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Reuse Statement (publisher, data, author rights): | This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Vedran Sohinger, A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials, International Mathematics Research Notices, 2021;, rnab132, is available online at: http://dx.doi.org/10.1093/imrn/rnab132 | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 27 April 2021 | ||||||||
| Date of first compliant Open Access: | 14 June 2022 | ||||||||
| RIOXX Funder/Project Grant: |
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