Landis-type results for discrete equations

TL;DR

This paper establishes Landis-type unique continuation results for semidiscrete heat and stationary discrete Schrödinger equations, demonstrating conditions under which solutions must vanish, and highlighting differences between continuum and discrete regimes.

Contribution

It provides the first Landis-type results for these discrete equations, with new quantitative estimates bridging continuum and discrete analysis.

Findings

Solutions vanish under decay conditions for the semidiscrete heat equation.

Solutions are zero if they vanish at infinity for the discrete Schrödinger equation.

Quantitative bounds reveal an interpolation between continuum and discrete scales.

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.