Weinberg's Separation of the Amplitude in theK-Matrix Formalism

The Weinberg method of obtaining a separable amplitude by the use of the eigenfunctions of the kernel of the Lippmann-Schwinger equation is given in the $K$-matrix formalism. This can be carried out in a way parallel to, but much simpler than, the formulation given previously in the $T$-matrix formalism. Relations between quantities in the $K$-matrix and $T$-matrix formalisms are obtained, and it is shown that the analytic extension into the complex energy plane is assured through these relations. Furthermore, it is shown that the formulations in the $K$-matrix formalism are very useful in applications to the composite-particle scattering problem.