Given a selfadjoint magnetic Schrödinger operator \(\begin{aligned} H = ( i \partial + A(x) )^2 + V(x) \end{aligned}\) 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 \(\begin{aligned} \Vert u(t)\Vert _{L^{2}}\lesssim _{E,F,T,A,V} \Vert u(0)\Vert _{L^{2}(E^{c})} + \Vert u(T)\Vert _{L^{2}(F^{c})}, \qquad 0\le t\le T \end{aligned}\) 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.
