A two-point generalisation of the Agmon estimate for Schr\"odinger operators on connected graphs

TL;DR

This paper extends Agmon's estimate for Schrödinger operators on graphs to a two-point version, broadening its applicability beyond the original separation of regions by potential and energy.

Contribution

It introduces a novel two-point generalization of Agmon's estimate for Schrödinger operators on graphs, expanding the theoretical framework.

Findings

Generalizes Agmon estimate to two points on graphs

Reduces to original estimate when points are in different regions

Enhances understanding of Schrödinger operators on connected graphs

Abstract

We provide in this Letter a two-point generalisation of the Agmon estimate for Schr\"odinger operators on graphs recently established by S. Steinerberger. It reduces to his estimate when the two points belong to different sets separated by the potential and the energy, i.e., the allowed and forbidden regions.