We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schr¨odinger operator, as well as for operators with compactly supported time-independent potentials, a sharp analog of the Hardy uncertainty principle is obtained, using an argument based on the theory of entire functions. Logarithmic convexity of weighted norms is employed in the case of general bounded potentials.
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Philippe Jaming, Yurii I. Lyubarskii, Eugenia Malinnikova, Karl-Mikael Perfekt, Uniqueness for discrete Schrödinger evolutions. Rev. Mat. Iberoam. 34 (2018), no. 3, pp. 949–966DOI 10.4171/RMI/1011