Uncertainty principle for solutions of the Schr{ö}dinger equation on the Heisenberg group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915162916651008 |
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| author | Jaming, Philippe Gosh, Somnath |
| author_facet | Jaming, Philippe Gosh, Somnath |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_14411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uncertainty principle for solutions of the Schr{ö}dinger equation on the Heisenberg group Jaming, Philippe Gosh, Somnath Classical Analysis and ODEs Complex Variables Functional Analysis 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. |
| title | Uncertainty principle for solutions of the Schr{ö}dinger equation on the Heisenberg group |
| topic | Classical Analysis and ODEs Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2502.14411 |