Multichannel scattering theory

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. 

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... 

In bound-state calculations, the Rayleigh-Ritz or Schrodinger variational principle provides both an upper bound to an exact energy and a stationary property that determines free parameters in the wave function. In scattering theory, the energy is specified in advance. Variational principles are used to determine the wave function but do not generally provide variational bounds. A variational functional is made stationary by choice of variational parameters, but the sign of the residual error is not determined. Because there is no well-defined bounded quantity, there is no simple absolute standard of comparison between different variational trial functions. The present discussion will develop a stationary estimate of the multichannel A -matrix. Because this matrix is real and symmetric for open channels, it provides the most... 

We have thus far considered the probability of superelastic scattering on a single orbit. To obtain the scattering rate, or autoionization rate, we simply multiply this probability by the orbital frequency, 1/n3.4 Once again we find that T oc 1/n3 and that T decreases with increasing Z. The scattering description we have just given is a two channel description. This picture, when many channels are present, forms the basis of multichannel quantum defect theory.5... 

Resonant enhancements of scattering cross-sections in multichannel collision physics are often described in terms of the Feshbach theory of closed-channel resonance states [57], Feshbach s general formalism involves projecting the stationary Schrddinger equation onto complementary subspaces associated with the open and closed scattering channels. This theory has been applied in the context of the nearthreshold collision physics of ultracold gases consisting of alkali-metal atoms in a variety of different approaches (e.g.. Refs. [9,30,58]). 

---