  We consider a cable described by a discrete, space-homogeneous, quasi
one-dimensional Schr\"odinger operator $H_0$. We study the scattering by a
finite disordered piece (the scatterer) inserted inside this cable. For
energies $E$ where $H_0$ has only elliptic channels we use the
Lippmann-Schwinger equations to show that the scattering matrix and the
transfer matrix, written in an appropriate basis, are related by a certain
polar decomposition. For energies $E$ where $H_0$ has hyperbolic channels we
show that the scattering matrix is related to a reduced transfer matrix and
both are of smaller dimension than the transfer matrix. Moreover, in this case
the scattering matrix is determined from a limit of larger dimensional
scattering matrices, as follows: We take a piece of the cable of length $m$,
followed by the scatterer and another piece of the cable of length $m$,
consider the scattering matrix of these three joined pieces inserted inside an
ideal lead at energy $E$ (ideal means only elliptic channels), and take the
limit $m\to\infty$.
