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