A dynamical Amrein-Berthier uncertainty principle

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.