Wigners resonance scattering formula

Thus R, like S to which it is directly related, is a function of energy E and relative velocity u. Since the absolute value of S is 1, R is real. Through (8.3) and (8.4), kaR is the tangent of the phase shift 6, whence 

So aR is a length, which turns out to be related to the scattering radius of the colliding particles. 

To solve (8.13), Wigner [370] notes that the left hand side is real he sets 

The quantity -2 = kat is interpreted as the time taken for the particles separating from the origin to appear at the surface r = a. 

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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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