%0 Journal Article %T On the Classical Limit of the Schrödinger Equation %+ Laboratoire Jacques-Louis Lions (LJLL) %+ Centre de Mathématiques Laurent Schwartz (CMLS) %+ MCSE Division %A Bardos, Claude %A Golse, François %A Markowich, Peter %A Paul, Thierry %Z 21 pages %< avec comité de lecture %@ 1078-0947 %J Discrete and Continuous Dynamical Systems - Series A %I American Institute of Mathematical Sciences %V 35 %N 12 %P 5689-5709 %8 2015-12 %D 2015 %R 10.3934/dcds.2015.35.5689 %K Schrödinger equation %K Classical limit %K WKB expansion %K Caustic %K Fourier integral operators %K Lagrangian manifold %K Maslov index %Z MSC 35Q41, 81Q20 (35S30, 53D12) %Z Mathematics [math]/Analysis of PDEs [math.AP] %Z Mathematics [math]/Mathematical Physics [math-ph]Journal articles %X 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. %G English %2 https://polytechnique.hal.science/hal-01074071/document %2 https://polytechnique.hal.science/hal-01074071/file/WKB.pdf %L hal-01074071 %U https://polytechnique.hal.science/hal-01074071 %~ UNIV-PARIS7 %~ X %~ UPMC %~ CMLS %~ CNRS %~ INSMI %~ X-CMLS %~ X-DEP %~ X-DEP-MATH %~ LJLL %~ TDS-MACS %~ UNIV-PARIS-SACLAY %~ X-SACLAY %~ UPMC_POLE_1 %~ SORBONNE-UNIVERSITE %~ SU-SCIENCES %~ UNIV-PARIS %~ SU-TI %~ ALLIANCE-SU