A two-channel problem is considered within a method based on first orderdifferential equations that are equivalent to the corresponding Schr\"odingerequation but are more convenient for dealing with resonant phenomena. Usingthese equations, it is possible to directly calculate the Jost matrix forpractically any complex value of the energy. The spectral points (bound andresonant states) can therefore be located in a rigorous way, namely, as zerosof the Jost matrix determinant. When calculating the Jost matrix, thedifferential equations are solved and thus, at the same time, the wave functionis obtained with the correct asymptotic behavior that is embedded in thesolution analytically. The method offers very accurate way of calculating notonly total widths of resonances but their partial widths as well. For each poleof the S-matrix, its residue can be calculated rather accurately, which makesit possible to obtain the Mittag-Leffler type expansion of the S-matrix as asum of the singular terms (representing the resonances) and the background term(contour integral). As an example, the two-channel model by Noro and Taylor isconsidered. It is demonstrated how the contributions of individual resonancepoles to the scattering cross section can be analyzed using the Mittag-Lefflerexpansion and the Argand plot technique. This example shows that even polessituated far away from the physical real axis may give significantcontributions to the cross section.
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