On the classical limit of the Schrödinger equation

Claude  Bardos; François  Golse; Peter  Markowich; Thierry  Paul

doi:10.3934/dcds.2015.35.5689

Article Contents

2015, Volume 35, Issue 12: 5689-5709. Doi: 10.3934/dcds.2015.35.5689

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On the classical limit of the Schrödinger equation

1.
Université Paris-Diderot, Laboratoire J.-L. Lions, BP187, 4 place Jussieu, 75252 Paris Cedex 05
2.
Ecole polytechnique, CMLS, 91128 Palaiseau Cedex
3.
King Abdullah University of Science and Technology, MCSE Division, Thuwal 23955-6900

Received: April 2014

Published: December 2015

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Abstract

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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Mathematics Subject Classification: Primary: 35Q41, 81Q20; Secondary: 35S30, 53D12.

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