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Microscopic derivation of Gibbs measures for the focusing one dimensional nonlinear Schrödinger equation
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Rout, Andrew (2023) Microscopic derivation of Gibbs measures for the focusing one dimensional nonlinear Schrödinger equation. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3973267
Abstract
In this thesis, we give a mircoscopic derivation of Gibbs measures for the focusing cubic and quintic (nonlocal) nonlinear Schr¨odinger equations (NLS) on T from many body quantum Gibbs states. In the cubic case, this corresponds to taking a twobody interaction, whereas the quintic case corresponds to a three-body interaction. Since we are not making positivity assumptions on the interaction potential, it is necessary to truncate in the mass in the classical setting and the rescaled particle number in the quantum setting. Our methods are based on the perturbative expansion developed in the work of Fr¨ohlich, Knowles, Schlein, and Sohinger [29]. We obtain results in both the time independent and time dependent cases. These are the first known results in the focusing regime and for any quintic regime. In particular, we give the first microscopic derivation of time-dependent correlation functions for Gibbs measures corresponding to the quintic NLS, as studied in the work of Bourgain [9]. In the quintic case, we can only study a suitable nonlocal quintic NLS, preventing us from obtaining a derivation of the local NLS in the quintic case.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Schrödinger equation, Gibbs' equation, Differential equations, Partial., Feynman diagrams | ||||
Official Date: | September 2023 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Sohinger, Vedran | ||||
Format of File: | |||||
Extent: | vii, 148 pages : illustrations | ||||
Language: | eng |
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