Chapter III. Kernels of GCM, RGM and OCM and Their Calculation Methods

We discuss the calculational procedures of the kernels of GCM, RGM and OCM and some properties of them related to their calculation. The GCM kernels for various types of systems are treated and methods are discussed on the analytical evaluation and on the decomposition in terms of the number of nucleons exchanged between clusters. The RGM kernels are evaluated by the integral transformation of GCM kernels. Various formulas of this transformation are presented including those for the systems of clusters with unequal oscillator widths. The problems related to the RGM norm kernel (RGM-NK) are discussed; firstly on the solution of the eigen-value problem of RGM-NK for various kinds of systems, secondly on the evaluation of kernels or physical quantities obtainable from the knowledge of RGM-NK and finally on the cluster model space for whose character the solution of the eigen-value problem of RGM-NK gives an indispensable information. The projection operator onto the Pauli-allowed states in OCM is obtained directly from the solution of the eigenvalue problem of RGM-NK. IN this paper we also present another method of construction of this operator of OCM which needs not to solve the eigen-value problem of RGM-NK which is tedious for complex systems even with our present calculational techniques.