Scattering multichannel

To arrive at a more unique way of characterizing autoionizing states in the absence of information about the continuum phase we can recast the QDT equations into an R matrix form which is at the same time similar to the original development of QDT from multichannel scattering theory.2 The relation between the different forms of scattering matrix is discussed by Mott and Massey,11 Seaton,2 and Fano and Rau.12... 

The two-particle Boltzmann collision term if and the three-particle contribution for k = 0 were considered in Section II. It was possible to express those collision integrals in terms of the two- and three-particle scattering matrices. It is also possible to introduce the T matrix in if for the channels k = 1, 2,3, that is, in those cases where three are asymptotically bound states. Here we use the multichannel scattering theory, as outlined in Refs. 9 and 26. 

For multichannel scattering where there are two or more open channels, the S matrix is a true matrix with elements Sy and the cross section for the transition from channel i to channel j is proportional to 5y - Sy 2. The symmetry of collision processes with respect to the time reversal leads to the symmetric property of the S matrix, ST = S, which, in turn, leads to the principle of detailed balance between mutually reverse processes. The conservation of the flux of probability density for a real potential and a real energy requires that SSf = SfS = I, i.e., S is unitary. For a complex energy or for a complex potential, in general, the flux is not conserved and S is non-unitary. 

An important perspective of multichannel resonance scattering may be gained from the lifetime matrix Q(E), introduced by Smith [46] as a generalization of the time delay At of Eq. (24) in single-channel scattering. It is now more often called the time-delay matrix or the delay-time matrix. After extending At for matrix elements Qy(E) for multichannel scattering, he proved that... 

The asymptotic wavefunctions obtained from multichannel scattering calculations provide the S matrix, the eigenphases and eigenphase sum, and the cross sections. In principle, any of these quantities may be chosen for fitting the resonance formula to determine accurate values of the resonance parameters E, and T. 

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. 

If k is in atomic units ap1. the differential cross section is in units per steradian. Differential and total cross sections for multichannel scattering by an atomic target can be derived from general formulas [27, 184], The partial cross section for scattering from channel q to channel p is... 

For multichannel scattering, the situation is more complicated. As a function of magnetic field, the scattering length passes through a pole only if So (5) passes through... 

The approach discussed here has its origins in both many-body theory and multichannel scattering theory. From many-body theory it builds on the method called "configuration interaction" or "superposition of configurations" (SOC), especially as that method is viewed as a systematic way to enlarge multiconfiguration Hartree-Fock basis sets until convergence is reached. This... 

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