Work

On Landis’ Conjecture for Positive Schrödinger Operators on Graphs

Year: 2025

Type: article

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.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnaf151

Authors: Ujjal Das, Matthias Keller, Yehuda Pinchover

Institutions: Technion – Israel Institute of Technology, University of Potsdam

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,534

Funders: Deutsche Forschungsgemeinschaft, Israel Academy of Sciences and Humanities, Israel Science Foundation