Eigenfunctions growth of \(\mathcal{R}\)-limits on graphs (Q2078949)

Summary: 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.