  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.
