On the Classical Limit of the Schrödinger Equation

• Published in: Discrete and Continuous Dynamical Systems
• Main Authors: Bardos, Claude, Golse, François, Markowich, Peter, Paul, Thierry
• Other Authors: 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 (KAUST)
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2015
• Subjects: 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]; laptev
• Online Access: https://hal-polytechnique.archives-ouvertes.fr/hal-01074071; https://hal-polytechnique.archives-ouvertes.fr/hal-01074071/document; https://hal-polytechnique.archives-ouvertes.fr/hal-01074071/file/WKB.pdf; https://doi.org/10.3934/dcds.2015.35.5689

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.