On the Classical Limit of the Schrödinger Equation
21 pages 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 t...
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ftepunivpsaclay:oai:HAL:hal-01074071v1 2024-05-19T07:43:39+00:00 On the Classical Limit of the Schrödinger Equation Bardos, Claude Golse, François Markowich, Peter Paul, Thierry Laboratoire Jacques-Louis Lions (LJLL) Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) Centre de Mathématiques Laurent Schwartz (CMLS) École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS) MCSE Division King Abdullah University of Science and Technology Saudi Arabia (KAUST) 2015-12 https://polytechnique.hal.science/hal-01074071 https://polytechnique.hal.science/hal-01074071/document https://polytechnique.hal.science/hal-01074071/file/WKB.pdf https://doi.org/10.3934/dcds.2015.35.5689 en eng HAL CCSD American Institute of Mathematical Sciences info:eu-repo/semantics/altIdentifier/doi/10.3934/dcds.2015.35.5689 hal-01074071 https://polytechnique.hal.science/hal-01074071 https://polytechnique.hal.science/hal-01074071/document https://polytechnique.hal.science/hal-01074071/file/WKB.pdf doi:10.3934/dcds.2015.35.5689 info:eu-repo/semantics/OpenAccess ISSN: 1078-0947 Discrete and Continuous Dynamical Systems - Series A https://polytechnique.hal.science/hal-01074071 Discrete and Continuous Dynamical Systems - Series A, 2015, 35 (12), pp.5689-5709. ⟨10.3934/dcds.2015.35.5689⟩ Schrödinger equation Classical limit WKB expansion Caustic Fourier integral operators Lagrangian manifold Maslov index MSC 35Q41 81Q20 (35S30 53D12) [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] info:eu-repo/semantics/article Journal articles 2015 ftepunivpsaclay https://doi.org/10.3934/dcds.2015.35.5689 2024-04-25T00:59:50Z 21 pages 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 École Polytechnique, Université Paris-Saclay: HAL Discrete and Continuous Dynamical Systems 35 12 5689 5709 |
institution |
Open Polar |
collection |
École Polytechnique, Université Paris-Saclay: HAL |
op_collection_id |
ftepunivpsaclay |
language |
English |
topic |
Schrödinger equation Classical limit WKB expansion Caustic Fourier integral operators Lagrangian manifold Maslov index MSC 35Q41 81Q20 (35S30 53D12) [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] |
spellingShingle |
Schrödinger equation Classical limit WKB expansion Caustic Fourier integral operators Lagrangian manifold Maslov index MSC 35Q41 81Q20 (35S30 53D12) [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] Bardos, Claude Golse, François Markowich, Peter Paul, Thierry On the Classical Limit of the Schrödinger Equation |
topic_facet |
Schrödinger equation Classical limit WKB expansion Caustic Fourier integral operators Lagrangian manifold Maslov index MSC 35Q41 81Q20 (35S30 53D12) [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] |
description |
21 pages 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 |
Laboratoire Jacques-Louis Lions (LJLL) Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) Centre de Mathématiques Laurent Schwartz (CMLS) École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS) MCSE Division King Abdullah University of Science and Technology Saudi Arabia (KAUST) |
format |
Article in Journal/Newspaper |
author |
Bardos, Claude Golse, François Markowich, Peter Paul, Thierry |
author_facet |
Bardos, Claude Golse, François Markowich, Peter 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 |
HAL CCSD |
publishDate |
2015 |
url |
https://polytechnique.hal.science/hal-01074071 https://polytechnique.hal.science/hal-01074071/document https://polytechnique.hal.science/hal-01074071/file/WKB.pdf https://doi.org/10.3934/dcds.2015.35.5689 |
genre |
laptev |
genre_facet |
laptev |
op_source |
ISSN: 1078-0947 Discrete and Continuous Dynamical Systems - Series A https://polytechnique.hal.science/hal-01074071 Discrete and Continuous Dynamical Systems - Series A, 2015, 35 (12), pp.5689-5709. ⟨10.3934/dcds.2015.35.5689⟩ |
op_relation |
info:eu-repo/semantics/altIdentifier/doi/10.3934/dcds.2015.35.5689 hal-01074071 https://polytechnique.hal.science/hal-01074071 https://polytechnique.hal.science/hal-01074071/document https://polytechnique.hal.science/hal-01074071/file/WKB.pdf doi:10.3934/dcds.2015.35.5689 |
op_rights |
info:eu-repo/semantics/OpenAccess |
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 |
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1799483387618000896 |