Uncertainty principles and differential operators on the weighted {B}ergman space

TL;DR

This paper classifies self-adjoint differential operators on weighted Bergman spaces and explores uncertainty principles using group representations, with potential for broader applications to symmetric domains.

Contribution

It introduces a classification of differential operators on weighted Bergman spaces and links uncertainty principles to group representation theory.

Findings

Classification of self-adjoint first-order differential operators

Connection between uncertainty principles and group representations

Potential generalization to other symmetric domains

Abstract

We classify self-adjoint first-order differential operators on weighted Bergman spaces on the unit disc and answer questions related to uncertainty principles for such operators. Our main tools are the discrete series representations of $\mathrm{SU}(1,1)$. This approach has the promise to generalize to other bounded symmetric domains.