Determinant approach to the scattering matrix elements in quasi-one-dimensional and two-dimensional disordered systems

We have developed an approach based on the characteristic determinant (the Green function poles) to solve the Dyson equation in quasi-one-dimensional (Q1D) and two-dimensional disordered systems without any restriction on the numbers of impurities and modes. We consider two different models for a disordered Q1D wire: a set of two-dimensional δ potentials with signs and strengths determined randomly, and a tight-binding Hamiltonian with several modes and on-site disorder. We calculate analytically the scattering matrix elements for particles coming both from the left and from the right without actually determining the eigenfunctions of the electrons. It is shown that the poles of the Green functions for these models can be deduced from a determinant of rank N×N (N is the number of scatterers) instead of the rank NM×NM (M is the number of modes). We calculate the inverse localization lengths for the two models. They are exactly on the order of w² for the weak disorder regime and are valid for an arbitrary number of channels, M.