Integral formulas for the Weyl and anti-Wick symbols

Abstract

The first purpose of this article is to provide conditions for a bounded operator in L 2 (R n ) to be the Weyl (resp. anti-Wick) quantization of a bounded continuous symbol on R 2n . 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 R 2n 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.

On the anti-Wick symbol as a Gelfand-Shilov generalized function

“…Let us also mention the following fact as a complementary result concerning anti-Wick symbols. In [1], one provides conditions written in terms of the action of an operator A on coherent states to ensure that the anti-Wick symbol of A is a bounded continuous function on R 2n . Also note that this latter result is actually to be compared with Unterberger result [14] giving a similar necessary and sufficient condition in order that the Weyl symbol of A is a C ∞ function on R 2n , being bounded together with all of its derivatives.…”