We have developed an approach based on the characteristic determinant ͑the Green function poles͒ to solve the Dyson equation in quasi-one-dimensional ͑Q1D͒ 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 ␦ potentials with signs and strengths determined randomly, and a tight-binding Hamiltonian with several modes and on-site disorder. We calculate

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... y 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 ϫ N ͑N is the number of scatterers͒ instead of the rank NM ϫ NM ͑M is the number of modes͒. We calculate the inverse localization lengths for the two models. They are exactly on the order of w 2 for the weak disorder regime and are valid for an arbitrary number of channels, M.