Eigenfunctions growth of R-limits on graphs

Abstract

A characterization of the essential spectrum of Schrödinger 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.