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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Bibliographic Details
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
Description
Summary: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.

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