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We prove a sharp version of the Hardy uncertainty principle for Schr\"odingerequations with external bounded electromagnetic potentials, based onlogarithmic convexity properties of Schr\"odinger evolutions. We provide, inaddition, an example of a real electromagnetic potential which produces theexistence of solutions with critical gaussian decay, at two distinct times.

Sharp Hardy uncertainty principle and gaussian profiles of covariant Schr\"odinger evolutions