Integral formulas for the Weyl and anti-Wick symbols

2018/06/13 by Laurent Amour, Amour, Laurent, Jean Nourrigat +1
Mathematics · #35S05 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35S05

paper · pdf · doi:10.48550/arxiv.1806.04898

arxiv created 2018/06/13 · arxiv updated 2018/06/14

Abstract

The first purpose of this article is to provide conditions for a bounded operator in L²(\Rⁿ) to be the Weyl (resp. anti-Wick) quantization of a bounded continuous symbol on \R²ⁿ. Then, explicit formulas for the Weyl (resp. anti-Wick) symbol are proved. Secondly, other formulas for the Weyl and anti-Wick symbols involving a kind of Campbell Hausdorff formula are obtained. A point here is that these conditions and explicit formulas depend on the dimension n only through a Gaussian measure on \R²ⁿ of variance 1/2 in the Weyl case (resp. variance 1 in the anti-Wick case) suggesting that the infinite dimension setting for these issues could be considered. Besides, these conditions are related to iterated commutators recovering in particular the Beals characterization Theorem.

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