Growth of Eigenfunctions and R-limits on Graphs

TL;DR

This paper characterizes the essential spectrum of Schrödinger operators on infinite graphs using the concept of R-limits, linking spectral properties to eigenfunctions at infinity, especially for graphs with sub-exponential growth.

Contribution

It introduces the use of R-limits to analyze the essential spectrum of Schrödinger operators on graphs, extending previous concepts to more general graph structures.

Findings

Each point in the essential spectrum corresponds to a bounded eigenfunction of an R-limit.

For graphs with uniform sub-exponential growth, the spectrum characterization is complete.

The approach generalizes the understanding of spectral behavior at infinity for infinite graphs.

Abstract

A characterization of the essential spectrum $\sigma_{\text ess}$ of Schr\"odinger operators on infinite graphs is derived involving the concept of $\mathcal{R}$-limits. This concept, which was introduced previously for operators on $\mathbb{N}$ and $\mathbb{Z}^d$ as "right-limits", captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate we show that each point in $\sigma_{\text ess}(H)$ corresponds to a bounded generalized eigenfunction of a corresponding $\mathcal{R}$-limit of $H$. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.