The Weyl Transform of a measure

TL;DR

This paper investigates the properties of the Weyl transform of smooth measures on curved hypersurfaces, showing conditions under which it is compact or belongs to Schatten classes, and explores quantum translation dependencies.

Contribution

It establishes new criteria for the Weyl transform's compactness and Schatten class membership for measures on curved hypersurfaces, and demonstrates the existence of Schatten class operators with linearly dependent quantum translates.

Findings

Weyl transform of smooth measures on curved hypersurfaces is compact in certain dimensions.

Weyl transform belongs to p-Schatten class for p>n when n≥6.

Existence of Schatten class operators with linearly dependent quantum translates.

Abstract

(1) Suppose $\mu$ is a smooth measure on a hypersurface of positive Gaussian curvature in $\R^{2n}$. If $n\ge 2$, then $W(\mu)$, the Weyl transform of $\mu$, is a compact operator, and if $p>n\ge 6$ then $W(\mu)$ belongs to the $p$-Schatten class. (2) There exist Schatten class operators with linearly dependent quantum translates.