A dynamical Amrein-Berthier uncertainty principle
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917218398240768 |
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| author | D'Ancona, Piero Fiorletta, Diego |
| author_facet | D'Ancona, Piero Fiorletta, Diego |
| contents | Given a selfadjoint magnetic Schrödinger operator \begin{equation*} H = ( i \partial + A(x) )^2 + V(x) \end{equation*} on $L^{2}(\mathbb{R}^n)$, with $V(x)$ strictly subquadratic and $A(x)$ strictly sublinear, we prove that the flow $u(t)=e^{-itH}u(0)$ satisfies an Amrein--Berthier type inequality \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,A,V} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad 0\le t\le T \end{equation*} for all compact sets $E,F \subset \mathbb{R}^{n}$. In particular, if both $u(0)$ and $u(T)$ are compactly supported, then $u$ vanishes identically. Under different assumptions on the operator, which allow for time--dependent coefficients, the result extends to sets $E,F$ of finite measure. We also consider a few variants for Schrödinger operators with singular coefficients, metaplectic operators, and we include applications to control theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_13746 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A dynamical Amrein-Berthier uncertainty principle D'Ancona, Piero Fiorletta, Diego Analysis of PDEs Mathematical Physics 35Q41, 35J10, 42B37 Given a selfadjoint magnetic Schrödinger operator \begin{equation*} H = ( i \partial + A(x) )^2 + V(x) \end{equation*} on $L^{2}(\mathbb{R}^n)$, with $V(x)$ strictly subquadratic and $A(x)$ strictly sublinear, we prove that the flow $u(t)=e^{-itH}u(0)$ satisfies an Amrein--Berthier type inequality \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,A,V} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad 0\le t\le T \end{equation*} for all compact sets $E,F \subset \mathbb{R}^{n}$. In particular, if both $u(0)$ and $u(T)$ are compactly supported, then $u$ vanishes identically. Under different assumptions on the operator, which allow for time--dependent coefficients, the result extends to sets $E,F$ of finite measure. We also consider a few variants for Schrödinger operators with singular coefficients, metaplectic operators, and we include applications to control theory. |
| title | A dynamical Amrein-Berthier uncertainty principle |
| topic | Analysis of PDEs Mathematical Physics 35Q41, 35J10, 42B37 |
| url | https://arxiv.org/abs/2504.13746 |