Sylvester theorem and the multichannel transfer matrix method for arbitrary transverse potential profile inside a wave guide

Based on the Sylvester and Frobenius theorems, we drastically enhance the feasibility of the transfer-matrix approach to deal with problems involving a large number of propagating and interfering modes, which require the solution of coupled differential equations and the evaluation of functions of matrix variables. We report closed formulas for the spectral decomposition of this type of functions. As specific example, besides the calculation of simple and well-known 1D one channel transfer matrices, we derive the multi-channel transfer matrix for an electron gas in the presence of a transverse electric field.