Transfer matrices for discrete Hermitian operators and absolutely continuous spectrum

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