# Transfer matrices for discrete Hermitian operators and absolutely continuous spectrum

## TL;DR

This paper develops a transfer matrix method for analyzing the spectral properties of discrete Hermitian operators on graphs, providing criteria for absolutely continuous spectrum and applying it to specific graph examples.

## Contribution

It introduces a transfer matrix approach for discrete Hermitian operators with increasing rank connections, extending spectral analysis techniques to more complex graph structures.

## Key findings

- Established a spectral averaging formula for these operators.
- Linked transfer matrix growth to the spectral measure and spectrum type.
- Provided examples of operators with absolutely continuous spectrum on stair-like graphs.

## Abstract

We introduce a transfer matrix method for the spectral analysis of discrete Hermitian operators with locally finite hopping. Such operators can be associated with a locally finite graph structure and the method works in principle on any such graph. The key result is a spectral averaging formula well known for Jacobi or 1-channel operators giving the spectral measure at a root vector by a weak limit of products of transfer matrices. Here, we assume an increase in the rank for the connections between spherical shells which is a typical situation and true on finite dimensional lattices $\mathbb{Z}^d$. The product of transfer matrices are considered as a transformation of the relations of 'boundary resolvent data' along the shells. The trade off is that at each level or shell with more forward then backward connections (rank-increase) we have a set of transfer matrices at a fixed spectral parameter. Still, considering these products we can relate the minimal norm growth over the set of all products with the spectral measure at the root and obtain several criteria for absolutely continuous spectrum. Finally, we give some example of operators on stair-like graphs (increasing width) which has absolutely continuous spectrum with a sufficiently fast decaying random shell-matrix-potential.