Asymptotic behaviour of small solutions for the discrete nonlinear Schrodinger and Klein-Gordon equations

We show decay estimates for the propagator of the discrete Schrodinger and Klein-Gordon equations in the form parallel to U (t) f parallel to(l infinity) <= C(1 + vertical bar t vertical bar)-d/3 parallel to f parallel to(l1). This implies a corresponding (restricted) set of Strichartz estimates. Applications of the latter include the existence of excitation thresholds for certain regimes of the parameters and the decay of small initial data for relevant l(p) norms. The analytical decay estimates are corroborated with numerical results.