Uncertainty principle for solutions of the Schr{ö}dinger equation on the Heisenberg group
Philippe Jaming, Somnath Gosh
TL;DR
This paper extends dynamical uncertainty principles to the Schrödinger evolution on the Heisenberg group ${\mathbb H^n}$, where the sub-Laplacian ${\mathcal L}$ governs the dispersion. It develops and transfers Euclidean ABB/Hardy-type results to the Heisenberg setting via annihilating pairs and a central-variable Fourier transform, establishing a two-time support principle for the free flow and a Paley-Wiener type theorem in the presence of a potential $V$ independent of the central variable. The authors show that if the initial data and its state at a fixed later time have supports confined to finite-measure sets in the horizontal variables, the solution must vanish, and they derive a Euclidean-type conclusion after a series of reductions to magnetic and harmonic-oscillator problems. They also prove limitations on quantitative dynamical principles, illustrating that some forms of strong, explicit bounds do not carry over to the noncommutative setting, thereby clarifying the scope of such principles on ${\mathbb H^n}$.
Abstract
The aim of this paper is two prove two versions of the Dynamical Uncertainty Principlefor the Schrödinger 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.