Relations between Transfer and Scattering Matrices in the presence of Hyperbolic Channels

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 $m\to\infty$.