Alternative R matrix form of QDT

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 

Since cos nfxa multiplies the entire left-hand side of Eq. (20.29) we can ignore it and write the remaining equation as a matrix equation. Explicitly 

If there are no interchannel couplings, R = 0, corresponding to the fact that the i channels have quantum defects of and occur when v, = 0. In general the R matrix is a real symmetric matrix with zeroes on the diagonal. 

Another useful property of the R matrix formulation is that we can write it in blocks corresponding to the bound (closed) and continuum (open) channels. Explicitly, we can write Eq. (20.33) as3 

38) may be written in exactly the same form. In Eq. (20.41) we have replaced all the continuum phases by jit to reflect the fact that we are searching for 

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