Growth of Eigenfunctions and R-limits on Graphs

A characterization of the essential spectrum ${\sigma }_{\text{e}ss}$ 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 ${\sigma }_{\text{e}ss}(H)$ 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.