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 at

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... wo different times, t 0 = 0 and t 1 = 1, which guarantee that u ≡ 0 in R n × [0, 1]. One of our motivations comes from a well known result due to G. H. Hardy [16, pp. 131] (see also [1] for a recent survey on this topic), which concerns the decay of a function f and its Fourier transform, f (ξ ) = (2π ) −n/2 R n e −iξ ·x f (x) dx,