Uncertainty principles for free metaplectic transformation and associated metaplectic operators

TL;DR

This paper establishes new uncertainty principles for free metaplectic transformations and operators, extending classical results and connecting with recent literature in harmonic analysis and quantum mechanics.

Contribution

It introduces two types of uncertainty principles for free metaplectic transformations, generalizes one to the $L^p$-case, and relates these results to existing work in the field.

Findings

Uncertainty principles derived for free metaplectic transformations.

Generalization of results to the $L^p$-case for $1 \\le p \\le 2$.

New connections established with existing literature.

Abstract

In this paper, we systematically investigate the Heisenberg-Pauli-Weyl uncertainty principle for free metaplectic transformation, as well as metaplectic operators. Specifically, we obtain two different types of the uncertainty principle for free metaplectic transformations in terms of the so-called phase derivative, one of which can be generalized to the $L^p$-case with $1\le p\le 2$. The obtained results are valid not only for free metaplectic transformations but also for general metaplectic operators. In particular, we point out that our results are closely related to those given in \cite{Dias-deGosson-Prata}, and the relationship should be new and not exactly given in the existing literature.