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On Landis' conjecture for positive Schrödinger operators on graphs

Ujjal Das, Matthias Keller, Yehuda Pinchover

TL;DR

The paper proves a discrete Landis-type result for positive Schrödinger operators on graphs by combining a Liouville comparison principle with a directional decay criterion. It establishes that if $V\le1$ and a comparison to a critical operator via Agmon ground states holds, then any $\mathcal{H}$-harmonic function decaying relative to the Green function $G_1$ must vanish. The authors derive an abstract theorem and several corollaries, and specialize the results to the Euclidean lattice $\mathbb{Z}^d$ and to regular trees, yielding explicit decay thresholds. They also extend the framework to fractional Laplacians $\Delta^{\sigma}$ on $\mathbb{Z}^d$, using sharp Green-function asymptotics to obtain corresponding Landis-type conclusions. Overall, the work provides sharp, graph-specific Landis criteria and connects discrete and fractional settings through a common positivity-driven approach.

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 $\mathcal{H}$-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 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.

On Landis' conjecture for positive Schrödinger operators on graphs

TL;DR

The paper proves a discrete Landis-type result for positive Schrödinger operators on graphs by combining a Liouville comparison principle with a directional decay criterion. It establishes that if and a comparison to a critical operator via Agmon ground states holds, then any -harmonic function decaying relative to the Green function must vanish. The authors derive an abstract theorem and several corollaries, and specialize the results to the Euclidean lattice and to regular trees, yielding explicit decay thresholds. They also extend the framework to fractional Laplacians on , using sharp Green-function asymptotics to obtain corresponding Landis-type conclusions. Overall, the work provides sharp, graph-specific Landis criteria and connects discrete and fractional settings through a common positivity-driven approach.

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 -harmonic functions for a positive Schrödinger operator with potentials bounded from above by 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 and regular trees for which we get a explicit decay criterion. Moreover, we consider the fractional analogue of the Landis conjecture on . Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article.
Paper Structure (8 sections, 14 theorems, 96 equations)

This paper contains 8 sections, 14 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Set up and a Liouville comparison theorem
  3. General Landis type theorem
  4. The Euclidean lattice and regular trees
  5. Euclidean lattice
  6. Regular trees
  7. On the Fractional Laplacian
  8. Estimates of norms on $\mathbb{Z}^d$ for the resolvent asymptotics

Key Result

Theorem 2.1

Suppose $\mathcal{H}$ and $\mathcal{H}'$ are positive Schrödinger operators associated to connected graphs $b$, $b'$ and potentials $V, V'$ over $(X,m)$. Let $u\in \mathcal{F}_b$ and $v\in \mathcal{F}_{b'}$ be such that Then, $\mathcal{H}$ is critical and $u>0$ is an Agmon ground state of $\mathcal{H}$.

Theorems & Definitions (33)

  • Remark
  • Theorem 2.1: Liouville comparison principle
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['t:Landis_general']}
  • Remark
  • Corollary 3.3
  • proof
  • ...and 23 more