A Hardy's Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra

TL;DR

This paper extends Hardy's uncertainty principle to operators satisfying generalized Heisenberg-Lie algebra relations, providing new inequalities and exploring applications to time and Dirac operators.

Contribution

It introduces a generalized Hardy's inequality for operators with weak commutation relations in the Heisenberg-Lie algebra.

Findings

Established a generalized Hardy's inequality for such operators

Applied the inequality to time operators

Applied the inequality to abstract Dirac operators

Abstract

In this article we consider linear operators satisfying a generalized commutation relation of a type of the Heisenberg-Lie algebra. It is proven that a generalized inequality of the Hardy's uncertainty principle lemma follows. Its applications to time operators and abstract Dirac operators are also investigated.