On the classical limit of the Schrödinger equation

This paper provides an elementary proof of the classical limit of the Schrödinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigal simple and elegant construction of a parametrix for Schrödinger type equat...

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Published in:Discrete and Continuous Dynamical Systems
Main Authors: Bardos, Claude, Golse, François, Markowich, Peter A., Paul, Thierry
Other Authors: Applied Mathematics and Computational Science Program, Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, Université Paris-Diderot, Laboratoire J.-L. Lions, BP187, 4 place Jussieu, Paris Cedex 05, France, Ecole Polytechnique, CMLS, Palaiseau Cedex, France
Format: Article in Journal/Newspaper
Language:unknown
Published: American Institute of Mathematical Sciences (AIMS) 2015
Subjects:
Caustic
Classical limit
Fourier integral operators
Lagrangian manifold
Maslov index
Schrödinger equation
WKB expansion
laptev
Online Access:http://hdl.handle.net/10754/594257
https://doi.org/10.3934/dcds.2015.35.5689
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spelling ftkingabdullahun:oai:repository.kaust.edu.sa:10754/594257 2023-12-31T10:09:02+01:00 On the classical limit of the Schrödinger equation Bardos, Claude Golse, François Markowich, Peter A. Paul, Thierry Applied Mathematics and Computational Science Program Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division Université Paris-Diderot, Laboratoire J.-L. Lions, BP187, 4 place Jussieu, Paris Cedex 05, France Ecole Polytechnique, CMLS, Palaiseau Cedex, France 2015-05 application/pdf http://hdl.handle.net/10754/594257 https://doi.org/10.3934/dcds.2015.35.5689 unknown American Institute of Mathematical Sciences (AIMS) https://arxiv.org/pdf/1410.4030 1410.4030 Bardos C, Golse F, Markowich P, Paul T (2015) On the classical limit of the Schrödinger equation. DCDS-A 35: 5689–5709. Available: http://dx.doi.org/10.3934/dcds.2015.35.5689. doi:10.3934/dcds.2015.35.5689 1078-0947 Discrete and Continuous Dynamical Systems http://hdl.handle.net/10754/594257 This is a pre-copy-editing, author-produced PDF of an article accepted for publication in [JournalTitle] following peer review. The definitive publisher-authenticated version is available online at: http://doi.org/10.3934/dcds.2015.35.5689 This file is an open access version redistributed from: https://arxiv.org/pdf/1410.4030 Caustic Classical limit Fourier integral operators Lagrangian manifold Maslov index Schrödinger equation WKB expansion Article 2015 ftkingabdullahun https://doi.org/10.3934/dcds.2015.35.5689 2023-12-02T20:21:29Z This paper provides an elementary proof of the classical limit of the Schrödinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigal simple and elegant construction of a parametrix for Schrödinger type equations [A. Laptev, I. Sigal, Review of Math. Phys. 12 (2000), 749-766]. We also explain in detail how the phase shifts across caustics obtained when using the Laptev-Sigal parametrix are related to the Maslov index. Article in Journal/Newspaper laptev King Abdullah University of Science and Technology: KAUST Repository Discrete and Continuous Dynamical Systems 35 12 5689 5709
institution Open Polar
collection King Abdullah University of Science and Technology: KAUST Repository
op_collection_id ftkingabdullahun
language unknown
topic Caustic
Classical limit
Fourier integral operators
Lagrangian manifold
Maslov index
Schrödinger equation
WKB expansion
spellingShingle Caustic
Classical limit
Fourier integral operators
Lagrangian manifold
Maslov index
Schrödinger equation
WKB expansion
Bardos, Claude
Golse, François
Markowich, Peter A.
Paul, Thierry
On the classical limit of the Schrödinger equation
topic_facet Caustic
Classical limit
Fourier integral operators
Lagrangian manifold
Maslov index
Schrödinger equation
WKB expansion
description This paper provides an elementary proof of the classical limit of the Schrödinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigal simple and elegant construction of a parametrix for Schrödinger type equations [A. Laptev, I. Sigal, Review of Math. Phys. 12 (2000), 749-766]. We also explain in detail how the phase shifts across caustics obtained when using the Laptev-Sigal parametrix are related to the Maslov index.
author2 Applied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Université Paris-Diderot, Laboratoire J.-L. Lions, BP187, 4 place Jussieu, Paris Cedex 05, France
Ecole Polytechnique, CMLS, Palaiseau Cedex, France
format Article in Journal/Newspaper
author Bardos, Claude
Golse, François
Markowich, Peter A.
Paul, Thierry
author_facet Bardos, Claude
Golse, François
Markowich, Peter A.
Paul, Thierry
author_sort Bardos, Claude
title On the classical limit of the Schrödinger equation
title_short On the classical limit of the Schrödinger equation
title_full On the classical limit of the Schrödinger equation
title_fullStr On the classical limit of the Schrödinger equation
title_full_unstemmed On the classical limit of the Schrödinger equation
title_sort on the classical limit of the schrödinger equation
publisher American Institute of Mathematical Sciences (AIMS)
publishDate 2015
url http://hdl.handle.net/10754/594257
https://doi.org/10.3934/dcds.2015.35.5689
genre laptev
genre_facet laptev
op_relation https://arxiv.org/pdf/1410.4030
1410.4030
Bardos C, Golse F, Markowich P, Paul T (2015) On the classical limit of the Schrödinger equation. DCDS-A 35: 5689–5709. Available: http://dx.doi.org/10.3934/dcds.2015.35.5689.
doi:10.3934/dcds.2015.35.5689
1078-0947
Discrete and Continuous Dynamical Systems
http://hdl.handle.net/10754/594257
op_rights This is a pre-copy-editing, author-produced PDF of an article accepted for publication in [JournalTitle] following peer review. The definitive publisher-authenticated version is available online at: http://doi.org/10.3934/dcds.2015.35.5689
This file is an open access version redistributed from: https://arxiv.org/pdf/1410.4030
op_doi https://doi.org/10.3934/dcds.2015.35.5689
container_title Discrete and Continuous Dynamical Systems
container_volume 35
container_issue 12
container_start_page 5689
op_container_end_page 5709
_version_ 1786842014119100416

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