Integral formulas for the Weyl and anti-Wick symbols

TL;DR

This paper establishes conditions and explicit formulas for Weyl and anti-Wick symbols of bounded operators on $L^2( ^n)$, highlighting dimension-independent aspects and connections to commutator characterizations.

Contribution

It provides new conditions and explicit formulas for Weyl and anti-Wick symbols that depend only on Gaussian measures, enabling potential extension to infinite dimensions.

Findings

Conditions for bounded operators to be Weyl or anti-Wick quantizations.

Explicit formulas for these symbols involving Gaussian measures.

Connections to iterated commutators and Beals' characterization.

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.