Non-orthogonality and overcompleteness in direct nuclear reactions

When rearrangement reactions are too strong to be treated by a one-step DWBA, they must be described by coupled reaction channel (CRC) equations or by a multistep DWBA derived from them. The derivation of these equations requires the use of projection operators on subspaces which are in general not orthogonal and which may be linearly dependent. We consider the case of many coupled two-cluster channels, and show how solving the non-orthogonality kernel to give an orthogonalized CRC formally simplifies the structure of the equation and clarifies the relations between different methods, including connected kernel approaches. We use the requirement that the distorted Faddeev (N = 3) or LBRS equations (general N) be satisfied in the two-cluster model space. We demonstrate that this determines the distortion potentials and that the resulting pole approximation yields the orthogonalized CRC equations. A modified one- and two-step DWBA is written in which non-orthogonality corrections are summed to all orders in each step. Methods of generating the non-orthogonality correction operator are discussed.