  Background: Nuclear reactions are complex, with a large number of possible
channels. Understanding how different channels contribute to a given reaction
is investigated by perturbing the continuous spectrum.
  Purpose: To develop tools to investigate reaction mechanisms by identifying
the contributions from each reaction channel.
  Method: Cluster decomposition methods, along with the spectral theory of
proper subsystem problems, is used to identify the part of the nuclear
Hamiltonian responsible for scattering into each channel.
  Results: The result is an expression of the nuclear Hamiltonian as a sum over
all scattering channels of channel Hamiltonians. Each channel Hamiltonian is
constructed from solutions of proper subsystem problems. Retaining any subset
of channel Hamiltonians results in a truncated Hamiltonian where the scattering
wave functions for the retained channels differ from the wave functions of the
full Hamiltonian by $N$-body correlations. The scattering operator for the
truncated Hamiltonian satisfies an optical theorem in the retained channels.
Because different channel Hamiltonians do not commute, how they interact
determines their contribution to the full dynamics.
