On the classical limit of the Schrödinger equation

Claude Bardos; François Golse; Peter Markowich; Thierry Paul

doi:10.3934/dcds.2015.35.5689

On the classical limit of the Schrödinger equation

Claude Bardos
, François Golse
, Peter Markowich
, Thierry Paul

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	5689-5709
Number of pages	21
Journal	Discrete and Continuous Dynamical Systems
Volume	35
Issue number	12
DOIs	https://doi.org/10.3934/dcds.2015.35.5689
Publication status	Published - 1 Dec 2015

Keywords

Caustic
Classical limit
Fourier integral operators
Lagrangian manifold
Maslov index
Schrödinger equation
WKB expansion

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10.3934/dcds.2015.35.5689

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title = "On the classical limit of the Schr{\"o}dinger equation",

abstract = "This paper provides an elementary proof of the classical limit of the Schr{\"o}dinger 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 Schr{\"o}dinger 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.",

keywords = "Caustic, Classical limit, Fourier integral operators, Lagrangian manifold, Maslov index, Schr{\"o}dinger equation, WKB expansion",

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AU - Bardos, Claude

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AU - Markowich, Peter

AU - Paul, Thierry

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N2 - 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.

AB - 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.

KW - Caustic

KW - Classical limit

KW - Fourier integral operators

KW - Lagrangian manifold

KW - Maslov index

KW - Schrödinger equation

KW - WKB expansion

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SP - 5689

EP - 5709

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 12

ER -

On the classical limit of the Schrödinger equation

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Keywords

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