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

Abstract

We consider a cable described by a discrete, space-homogeneous, quasi one-dimensional Schrödinger 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 $m\to \infty$.

19 pages, 1 figure

References in corpus (3)

• Random Schrodinger operators on long boxes, noise explosion and the GOE
• From the Anderson model on a strip to the DMPK equation and random matrix theory
• The random phase property and the Lyapunov spectrum for disordered multi-channel systems

Cited by in corpus (2)

• A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes
• On absolutely continuous spectrum for one-channel unitary operators