Dynamical versions of Morgan's Uncertainty Principle and Electromagnetic Schrödinger Evolutions
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913677964214272 |
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| author | Huang, Shanlin Wang, Zhenqiang |
| author_facet | Huang, Shanlin Wang, Zhenqiang |
| contents | This paper investigates the unique continuation properties of solutions of the electromagnetic Schrödinger equation
$$
i\partial_{t}u(x,t)+(\nabla-i A)^{2}u(x,t)=V(x,t)u(x,t)\,\,\,\, \mbox{in} \,\,\,\mathbb{R}^{n}\times [0,1],
$$
where $A$ represents a time-independent magnetic vector potential and $V$ is a bounded, complex valued time-dependent potential. Given $1<p<2$ and $1/p+1/q=1$, we prove that if
\begin{equation*}
\int_{\mathbb{R}^{n}}|u(x,0)|^{2}e^{2α^{p}|x|^p/p}\ d x
+\int_{\mathbb{R}^{n}}|u(x,1)|^{2}e^{2β^{q}|x|^q/q}\ d x
<\infty,
\end{equation*}
for some $α,β>0$ and there exists $N_{p}>0$ such that
\begin{equation*}
αβ>N_p,
\end{equation*}
then $u\equiv 0$. These results can be interpreted as dynamical versions of the uncertainty principle of Morgan's type. Furthermore, as an application, our results extend to a large class of semi-linear Schrödinger equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02255 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamical versions of Morgan's Uncertainty Principle and Electromagnetic Schrödinger Evolutions Huang, Shanlin Wang, Zhenqiang Analysis of PDEs Classical Analysis and ODEs This paper investigates the unique continuation properties of solutions of the electromagnetic Schrödinger equation $$ i\partial_{t}u(x,t)+(\nabla-i A)^{2}u(x,t)=V(x,t)u(x,t)\,\,\,\, \mbox{in} \,\,\,\mathbb{R}^{n}\times [0,1], $$ where $A$ represents a time-independent magnetic vector potential and $V$ is a bounded, complex valued time-dependent potential. Given $1<p<2$ and $1/p+1/q=1$, we prove that if \begin{equation*} \int_{\mathbb{R}^{n}}|u(x,0)|^{2}e^{2α^{p}|x|^p/p}\ d x +\int_{\mathbb{R}^{n}}|u(x,1)|^{2}e^{2β^{q}|x|^q/q}\ d x <\infty, \end{equation*} for some $α,β>0$ and there exists $N_{p}>0$ such that \begin{equation*} αβ>N_p, \end{equation*} then $u\equiv 0$. These results can be interpreted as dynamical versions of the uncertainty principle of Morgan's type. Furthermore, as an application, our results extend to a large class of semi-linear Schrödinger equations. |
| title | Dynamical versions of Morgan's Uncertainty Principle and Electromagnetic Schrödinger Evolutions |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.02255 |