Derivation of K-matrix reaction theory in a discrete basis formalism

The usual derivations of the S and K matrices for two-particle reactions proceed through the Lippmann-Schwinger equation with formal definitions of the incoming and outgoing scattering states. Here we present an alternative derivation that is more suitable to computational applications for reactions between composite particles. The derivation is carried out completely in the Hamiltonian representation, using a discrete basis of configurations for the scattering channels as well as the quasi-bound configurations of the combined fragments. We use matrix algebra to derive an explicit expression for the K matrix in terms of the Hamiltonian of the internal states of the compound system and the coupling between the channels and the internal states. The formula for the K matrix includes explicitly a real dispersive shift matrix to the internal Hamiltonian that is easily computed in the formalism. That expression is applied to derive the usual form of the S matrix as a sum over poles in the complex energy plane. Some extensions and limitations of the discrete-basis Hamiltonian formalism are discussed in the concluding remarks and in the Appendix.