Article Dans Une Revue Proceedings of the American Mathematical Society Année : 2025

Uncertainty principle for solutions of the Schrödinger equation on the Heisenberg group

Somnath Gosh
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Résumé

The aim of this paper is two prove two versions of the Dynamical Uncertainty Principle for 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.
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Dates et versions

hal-04957979 , version 1 (20-02-2025)

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  • HAL Id : hal-04957979 , version 1

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Philippe Jaming, Somnath Gosh. Uncertainty principle for solutions of the Schrödinger equation on the Heisenberg group. Proceedings of the American Mathematical Society, inPress. ⟨hal-04957979⟩

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