On the Landis Conjecture for Positive Quasi-linear Operators on Graphs

Abstract

We prove a Landis type unique continuation result for positive quasi-linear operators on graphs. Specifically, we give decay criteria that ensures when a harmonic function for a positive quasilinear Schr\"odinger operator with potential less than 1 is trivially zero. The assumption of positivity of the operator allows the application of criticality theory such as the Liouville comparison theorem. Furthermore, our results fundamentally build on the so called simplified energy. As an application we discuss the case of model graphs and in particular regular trees.