Optimal Hardy inequalities for Schr\"odinger operators on graphs

TL;DR

This paper constructs an optimal Hardy-weight for subcritical Schr"odinger operators on infinite graphs, characterizing the operator's criticality behavior relative to the weight and providing a criticality theory framework.

Contribution

It introduces a method to construct optimal Hardy-weights for Schr"odinger operators on graphs, extending criticality theory to discrete settings.

Findings

The Hardy-weight w is optimal in the subcritical, critical, and supercritical regimes.

The operator H - λw exhibits subcritical, null-critical, and supercritical behavior depending on λ.

The results generalize criticality theory to weighted graphs.

Abstract

For a given subcritical discrete Schr\"odinger operator $H$ on a weighted infinite graph $X$, we construct a Hardy-weight $w$ which is optimal in the following sense. The operator $H - \lambda w$ is subcritical in $X$ for all $\lambda < 1$, null-critical in $X$ for $\lambda = 1$, and supercritical near any neighborhood of infinity in $X$ for any $\lambda > 1$. Our results rely on a criticality theory for Schr\"odinger operators on general weighted graphs.