Abstract

We consider a cable described by a discrete, space-homogeneous, quasi-one-dimensional Schrodinger 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 -> 8. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3669483]