Landis-type results for discrete equations

We prove Landis-type results for both the semidiscrete heat and the stationary discrete Schrödinger equations. For the semidiscrete heat equation we show that, under the assumption of two-time spatial decay conditions on the solution u, then necessarily $u\equiv 0$. For the stationary discrete Schrödinger equation we deduce that, under a vanishing condition at infinity on the solution u, then $u\equiv 0$. In order to obtain such results, we demonstrate suitable quantitative upper and lower estimates for the $L^2$-norm of the solution within a spatial lattice ${\left(hZ\right)}^d$. These estimates manifest an interpolation phenomenon between continuum and discrete scales, showing that close-to-continuum and purely discrete regimes are different in nature.