Absolutely continuous spectrum of a Schr\"odinger operator on a tree

TL;DR

This paper establishes conditions under which a Schrödinger operator on a loopless regular tree exhibits absolutely continuous spectrum, advancing understanding of spectral properties in such graph structures.

Contribution

It provides new sufficient conditions for the absolutely continuous spectrum of Schrödinger operators on regular trees, a topic with limited prior results.

Findings

Identifies specific conditions ensuring absolutely continuous spectrum

Enhances understanding of spectral behavior on Bethe lattices

Contributes to spectral theory of operators on tree graphs

Abstract

We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schr\"odinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).