# Riesz Decompositions for Schr\"odinger Operators on Graphs

## TL;DR

This paper investigates Riesz decompositions for Schrödinger operators on weighted graphs, providing new decompositions of superharmonic functions and establishing a Brelot type theorem, advancing the understanding of potential theory in graph settings.

## Contribution

It introduces two novel Riesz decompositions for superharmonic functions on graphs and proves a Brelot type theorem, extending classical potential theory results to discrete structures.

## Key findings

- Decomposition of superharmonic functions into harmonic and potential parts.
- Decomposition into superharmonic functions with prescribed bounds.
- Establishment of a Brelot type theorem for Schrödinger operators on graphs.

## Abstract

We study superharmonic functions for Schr\"odinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.