Given a selfadjoint magnetic Schr\"odinger 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 Schr\"{ o}dinger operators with singular coefficients, metaplectic operators, and we include applications to control theory.
Categories: math.AP math-ph math.MP