# Spectral Theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models

## TL;DR

This paper develops spectral theory for one-channel operators, a class of self-adjoint operators with a single wave transmission channel, and applies it to prove the presence of absolutely continuous spectrum in Anderson models on specific graphs.

## Contribution

It introduces a spectral framework for one-channel operators and extends results on absolutely continuous spectrum to Anderson models on finite graphs with this structure.

## Key findings

- Established absolutely continuous spectrum for certain Anderson models.
- Generalized spectral results from Jacobi operators to one-channel operators.
- Extended previous findings on antitrees to broader graph classes.

## Abstract

A one-channel operator is a self-adjoint operator on $\ell^2(\mathbb{G})$ for some countable set $\mathbb{G}$ with a rank 1 transition structure along the sets of a quasi-spherical partition of $\mathbb{G}$. Jacobi operators are a very special case. In essence, there is only one channel through which waves can travel across the shells to infinity. This channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells. Not all of the transfer matrices are defined for some countable set of energies. Still, many theorems from the world of Jacobi operators are translated to this setup. The results are then used to show absolutely continuous spectrum for the Anderson model on certain finite dimensional graphs with a one-channel structure. This result generalizes some previously obtained results on antitrees.