On the Essential Spectrum of Schr\"odinger Operators on Trees

TL;DR

This paper explores the relationship between the essential spectrum of Schr"odinger operators on graphs, showing that on certain graphs the essential spectrum can be strictly larger than the union of spectra of right limits, unlike on regular trees.

Contribution

It generalizes the concept of right limits to arbitrary infinite graphs and demonstrates cases where the essential spectrum exceeds the union of right limit spectra.

Findings

Constructed examples of graphs with larger essential spectrum

Proved the equality holds for bounded operators on regular trees

Characterized the essential spectrum in spherically symmetric cases

Abstract

It is known that the essential spectrum of a Schr\"odinger operator $H$ on $\ell^{2}\left(\mathbb{N}\right)$ is equal to the union of the spectra of right limits of $H$. The natural generalization of this relation to $\mathbb{Z}^{n}$ is known to hold as well. In this paper we generalize the notion of right limits to general infinite connected graphs and construct examples of graphs for which the essential spectrum of the Laplacian is strictly bigger than the union of the spectra of its right limits. As these right limits are trees, this result is complemented by the fact that the equality still holds for general bounded operators on regular trees. We prove this and characterize the essential spectrum in the spherically symmetric case.