Exact factorization relations, consistency conditions, and the reduction to the ES approximation for the T-matrix

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Abstract
In this paper, we derive explicit expressions for exact factorizations for both inelastic and reactive scattering. The expressions enable us to reanalyze the ES factorization and obtain new insight into its optimum structure. In the first instance, we are able to give a rigorous basis to the scheme of factoring T-matrices in terms of input T-matrix elements with the same initial relative kinetic energy and show this is not a correction to the ES, but rather results directly from it. We also show that the optimum reference T-matrices in the ES factorization are not on-shell, but rather half-off-shell. This suggests new corrections to currently extant versions of the ES factorization. Finally, novel exact factorization of T-matrices for one potential in terms of those for another potential are given, as well as expressions for obtaining the exact T-matrix elements from approximate ones.