Uncertainty principle for solutions of the Schr{\"o}dinger equation on the Heisenberg group

TL;DR

This paper proves two versions of the Dynamical Uncertainty Principle for the Schrödinger equation on the Heisenberg group, extending Euclidean results to a non-commutative setting and exploring limitations of these principles.

Contribution

It introduces dynamical uncertainty principles for Schrödinger equations on the Heisenberg group, including versions with and without potential, extending classical Euclidean results to this setting.

Findings

Proves a version of the Amrein-Berthier-Benedicks Uncertainty Principle on the Heisenberg group.

Establishes a dynamical Paley-Wiener type theorem with potential.

Identifies limitations of dynamical uncertainty principles in this context.

Abstract

The aim of this paper is two prove two versions of the Dynamical Uncertainty Principlefor the Schr\"odinger equation $i\partial_s u=\mathcal{L}u+Vu$, $u(s=0)=u_0$ where$\mathcal{L}$ is the sub-Laplacian on the Heisenberg group.We show two results of this type. For the first one, the potential $V=0$, we establish a dynamical version of Amrein-Berthier-Benedicks's Uncertainty Principle that shows that if $u_0$ and $u_1=u(s=1)$ have both small support then $u=0$. For the second result, we add some potential to the equation and we obtain a dynamical version of the Paley-Wiener Theorem in the spirit of the result of Kenig, Ponce, Vega \cite{KPV}. Both results are obtained by suitably transfering results from the Euclidean setting.We also establish some limitations to Dynamical Uncertainty Principles.