Uncertainty principle for solutions of the Schr{ö}dinger equation on the Heisenberg group

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.