Alternative methods for restricted energy ranges

At energies below the first excitation threshold the variational principles discussed for bound states in chapter 5 can be extended to scattering (Callaway, 1978). We do not discuss this because of its restricted validity. However, there is another extension of bound-state methods into the positive-energy range that applies at least up to the ionisation threshold and somewhat beyond. This is the R-matrix method. Its possible extension to higher energies is discussed. 

Examples of the application of all these methods are given in chapters 8 and 9. In this section we outline the methods. 

The driving terms of the distorted-wave Lippmann—Schwinger equations 

In applying the distorted-wave second Born approximation we have the same difficulty as in calculating the optical potential. We must calculate the spectrum of the Green s function of (6.87). The first iteration of (6.87) is written as 

The Q-projected Green s function (7.111) in the polarisation potential (7.115) contains the collision Hamiltonian H, but practical implementation of it involves the weak-coupling approximation (7.132). A second-Born calculation is equivalent to a calculation of the polarisation potential with Q=1 and Vopul being the reduced matrix element of U. This has 

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