One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so called $hatQ$ box introduced by Kuo et.al. It has been one of the central problems how to calculate the $hatQ$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $hatQ$ box can be expressed in two ways: One is a form of continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures to derive an exact $hatQ$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $hatQ$ box given in this study corresponds to a non-perturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $hatZ$-box approach based on the $hatQ$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.
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