  We prove a logarithmic convexity result for exponentially weighted
$L^2$-norms of solutions to electromagnetic Schr\"odinger equation, without
needing to assume smallness of the magnetic potential. As a consequence, we can
prove a unique continuation result in the style of the Hardy uncertainty
principle, which generalizes the analogous theorems which have been recently
proved by Escauriaza, Kenig, Ponce and Vega.
