Landis-type results for discrete equations

Abstract

We prove Landis-type results for both the semidiscrete heat and the stationary discrete Schr\"odinger 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 Schr\"odinger 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 $(h\mathbb{Z})^d$. These estimates manifest an interpolation phenomenon between continuum and discrete scales, showing that close-to-continuum and purely discrete regimes are different in nature.