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 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.

中文翻译：

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离散基形式主义中K矩阵反应理论的推导

两粒子反应的 S 和 K 矩阵的通常推导通过 Lippmann-Schwinger 方程进行，其中包含传入和传出散射状态的正式定义。在这里，我们提出了另一种推导，该推导完全在哈密顿表示中进行，使用散射通道的离散配置基础以及组合片段的准绑定配置。我们使用矩阵代数根据复合系统内部状态的哈密顿量以及通道与内部状态之间的耦合导出 K 矩阵的显式表达式。K 矩阵的公式显式地包含了一个到内部哈密顿量的实弥散位移矩阵，它很容易在形式主义中计算出来。该表达式用于导出 S 矩阵的通常形式，作为复能平面中极点的总和。在结束语和附录中讨论了离散基哈密顿形式主义的一些扩展和限制。