On Landis conjecture for positive Schrödinger operators on graphs

In this note we study Landis conjecture for positive Schrödinger operators on graphs. More precisely, we give a decay criterion that ensures when ℋ-harmonic functions for a positive Schrödinger operator ℋ with potentials bounded from above by 1 are trivial. We then specifically look at the special cases of ℤ^d and regular trees for which we get explicit decay criterion. Moreover, we consider the fractional analogue of Landis conjecture on ℤ^d. Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article.