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Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d \leqslant 3$


Fröhlich, Jürg; Knowles, Antti; Schlein, Benjamin; Sohinger, Vedran (2017). Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d \leqslant 3$. Communications in Mathematical Physics, 356(3):883-980.

Abstract

We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions d=1,2,3. The many-body quantum thermal states that we consider are the grand canonical ensemble for d = 1 and an appropriate modification of the grand canonical ensemble for d=2,3. In dimensions d = 2, 3, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

Abstract

We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions d=1,2,3. The many-body quantum thermal states that we consider are the grand canonical ensemble for d = 1 and an appropriate modification of the grand canonical ensemble for d=2,3. In dimensions d = 2, 3, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

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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
Language:English
Date:31 December 2017
Deposited On:25 Jan 2018 07:23
Last Modified:28 Apr 2018 07:29
Publisher:Springer
ISSN:0010-3616
Additional Information:This is a post-peer-review, pre-copyedit version of an article published in Communications in Mathematical Physics. The final authenticated version is available online at: https://doi.org/10.1007/s00220-017-2994-7
OA Status:Closed
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s00220-017-2994-7

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