Scattering matrix poles

Feshbach resonances is purely model dependent since trapping well may exist on one type of adiabatic potential, say in hyperspherical coordinates, while only a barrier may exist on another type, say in natural collision coordinates. However, this is not correct since there are fundamental differences between QBS and Feshbach states. First, the pole structure of the S-matrix is intrinsically different in the two cases. A Feshbach resonance corresponds to a single isolated pole of the scattering matrix (S-matrix) below the real axis of the complex energy plane, see the discussion below. On the other hand, the barrier resonance corresponds to an infinite sequence of poles extending into the lower half plane. For a parabolic barrier, it is easy to show that the pole positions are given by... 

Of these direct approaches the complex coordinate method is the most rigorous one. In principle it yields the exact energies of the poles of the scattering matrix, which. Ignoring the background contribution to the scattering, gives the resonance position and width. 

Resonances are defined formally as poles of the scattering matrix in the complex energy or momentum plane (98-100). The pole location in the complex energy plane may be written as... 

Scattering cross sections from S-matrix poles through a Mittag-Leffler expansion. 

Figure 5 (A) Upper halfplane A structured scattering cross section as a function of energy. (B) Complex S-matrix poles in the second Riemann sheet which, together with their residues, are used to describe the cross section.

Poles in the scattering matrix occur at complex resonance energies (50,85,86)... 

The interpretation of the accurate quantal results in terms of variational and supernumerary transition states is consistent with model studies of scattering by unsym-metrical one-dimensional Eckart potentials (84). These studies show that both maxima in the unsymmetrical potentials are associated with poles of the scattering matrix, and some of these poles are associated with an increase in the transmission probability, while others are not. 

S.A. Rakityansky, N. Elander, Analyzing the contribution of individual resonance poles of the S-matrix to two-channel scattering, Int. J. Quantum Chem. 106 (2006) 1105. 

A number of closely lying resonances in multichannel scattering is a difficult problem to treat theoretically. Even the representation of the S matrix is very complex for these overlapping resonances as compared with the Breit-Wigner one-level formula. Various alternative proposals are found in the literature, as is reviewed by Belozerova and Henner [61]. This is mainly due to the formidable task of constructing an explicitly unitary and symmetric S matrix having more than one pole when analytically continued into the complex k plane. Thus, possible practical forms of the S matrix for overlapping resonances may be explicitly symmetric and implicitly unitary, or explicitly unitary and implicitly symmetric. 

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