Non-commutative polynomials and the transport properties in multichannel-multilayer systems

Abstract

Using the transfer-matrix technique to describe transport properties in multichannel-multilayer systems, a three-term non-commutative matrix recurrence relation is deduced and solved. The matrix polynomials obtained in this way allow one to write compact expressions for the n-cell transmission amplitudes, from channel to channel i. In the one-dimensional, one-channel limit, the non-commutative polynomials reduce to the well known Chebyshev orthogonal polynomials. To illustrate the role of these polynomials in the resonant tunnelling and channel-mixing behaviour, we discuss one- and two-channel examples.