On Landis' conjecture for positive Schrödinger operators on graphs

2024/08/04 by Das, Ujjal, Keller, Matthias, Pinchover, Yehuda · 2 citations
#35B53 #35J10 #35R02 #39A12 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2408.02149

Abstract

In this note we study the Landis conjecture for positive Schrödin\-ger operators on graphs. More precisely, we prove a Landis-type result in the form of a decay criterion that ensures when H-harmonic functions for a positive Schrödinger operator 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 ℤ^d and regular trees for which we get a explicit decay criterion. Moreover, we consider the fractional analogue of the Landis conjecture on ℤ^d . Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article.

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• A Liouville-type theorem for Schrödinger equations with nonnegative potential
• On the Landis Conjecture for Positive Quasi-linear Operators on Graphs

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