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We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions. The goal is to obtain sufficient conditions on a solution u, the potential V and the behavior of the solution atdoi:10.4171/jems/134 fatcat:5xi6voqe2zactewfbsxr2dqpqu