On Landis' conjecture for positive Schr\"odinger operators on graphs

TL;DR

This paper investigates a Landis-type decay criterion for positive Schrödinger operators on graphs, establishing conditions under which harmonic functions must be trivial, with specific results for lattices and trees.

Contribution

It introduces a decay criterion for harmonic functions on graphs with positive Schrödinger operators, including explicit criteria for lattices and trees, and extends to fractional cases.

Findings

Decay criterion ensures triviality of harmonic functions under certain conditions

Explicit decay criteria derived for $\

a0 ext{ and regular trees

Abstract

In this note we study the Landis conjecture for positive Schr\"odin\-ger 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 Schr\"odinger 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 a 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.