Unbounded operators and the uncertainty principle

TL;DR

This paper explores a variant of the uncertainty principle using annihilation and creation operators on generalized Segal Bargmann spaces, with applications to Berezin transforms and Szeg"H{o} kernels in complex analysis.

Contribution

It introduces a new approach to the uncertainty principle in the context of generalized Segal Bargmann spaces and connects it to Berezin transforms and Szeg"H{o} kernels.

Findings

Derived formulas for Berezin transforms of specific operators.

Established links between entire function spaces and Szeg"H{o} kernels.

Extended the uncertainty principle framework to generalized Bargmann spaces.

Abstract

We study a variant of the uncertainty principle in terms of the annihilation and creation operator on generalized Segal Bargmann spaces, which are used for the FBI-Bargmann transform. In addition, we compute the Berezin transform of these operators and indicate how to use spaces of entire functions in one variable to study the Szeg\H{o} kernel for hypersurfaces in $\mathbb C^2.$