On Landis’ Conjecture for Positive Schrödinger Operators on Graphs

Abstract In this note we study the Landis conjecture for positive Schrödinger operators on graphs. More precisely, we prove a Landis-type result in the form of a decay criterion that ensures when $ \mathcal{H} $-harmonic functions for a positive Schrödinger operator $ \mathcal{H} $ with potentials bounded from above by $ 1 $ are trivial. The positivity assumption on the operator allows us to impose slow decay across the entire graph, while requiring fast decay in only one direction, rather than throughout the whole graph. We then specifically look at the special cases of $ \mathbb{Z}^{d} $ and regular trees for which we get an explicit decay criterion. Moreover, we consider the fractional analogue of the Landis conjecture on $ \mathbb{Z}^{d} $. Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article.