Convexity Properties of Discrete Schr\"odinger evolutions and Hardy's Uncertainty Principle

In this paper we give log-convexity properties for solutions to discrete Schr\"odinger equations with different discrete versions of Gaussian decay at two different times. For free evolutions, we use complex analysis arguments to derive these properties, while in a perturbative setting we use a preliminar log-convexity result in order to get these properties. Then, by proving a Carleman inequality we conclude, in one of the cases under study, a discrete version of Hardy's Uncertainty Principle.