On the time evolution of Wigner measures for Schrödinger equations

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March 2009, 8(2): 559-585. doi: 10.3934/cpaa.2009.8.559

On the time evolution of Wigner measures for Schrödinger equations

Rémi Carles ^1, , Clotilde Fermanian-Kammerer ^2, , Norbert J. Mauser ^3, and Hans Peter Stimming ^3,

1.	Université Montpellier 2, Mathématiques, CC051, 34095 Montpellier, CNRS, UMR 5149, 34095 Montpellier, France
2.	LAMA UMR CNRS 8050, Université Paris EST, 61, avenue du Général de Gaulle 94010 Créteil Cedex, France
3.	Wolfgang Pauli Institute c/o Fak. f. Mathematik, Univ. Wien, Nordbergstr. 15, A-1090 Wien, Austria, Austria

Received February 2008 Revised August 2008 Published December 2008

Abstract
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In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrödinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrödinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.

Keywords: WKB methods, eigenvalue crossing, caustic, semi-classical limit, Wigner measures, ill-posedness, Schrödinger equation.

Mathematics Subject Classification: Primary: 81S30; Secondary: 35B40, 35C20, 35Q55, 35P25, 81Q05, 81Q20.

Citation: Rémi Carles, Clotilde Fermanian-Kammerer, Norbert J. Mauser, Hans Peter Stimming. On the time evolution of Wigner measures for Schrödinger equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 559-585. doi: 10.3934/cpaa.2009.8.559

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