A dynamical Amrein-Berthier uncertainty principle
Piero D'Ancona, Diego Fiorletta
TL;DR
The paper develops a dynamical Amrein–Berthier type uncertainty principle for Schrödinger evolutions generated by $H=( i\partial + A(x) )^{2}+V(x)$ with subquadratic $V$ and sublinear $A$, including time-dependent coefficients and magnetic effects. The core method is to prove compactness of localized propagators and to use a contraction argument on $S=\mathbf{1}_{E}e^{iTH}\mathbf{1}_{F}e^{-iTH}$, together with finite propagation speed and unique continuation, to derive the AB inequality $\ orm{u(t)}_{L^{2}} \lesssim_{E,F,T,A,V} \\norm{u(0)}_{L^{2}(E^{c})} + \\norm{u(T)}_{L^{2}(F^{c})}$. The paper then extends these results to singular potentials, time-dependent symmetric cases, and metaplectic operators, and connects them to observability and exact controllability in control theory. It also develops a metaplectic analogue of the AB principle, clarifying when compactness-based arguments persist under metaplectic transforms and when they fail. Overall, the findings provide a robust framework for understanding mass localization and controllability in dispersive quantum dynamics under broad, physically relevant assumptions.
Abstract
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.