Convergent Theory for Effective Interaction in Nuclei

Abstract

A general equation is derived for constructing the effective interaction on the basis of the similarity transformation theory. It is proved that the equation is equivalent lo Bloch's equation for the degenerate perturbation theory and, therefore, the present approach is also equivalent to the energy independent Raylcigh-Schroclinger theory. Some iteration methods are proposed to solve the equation, and the convergence conditions for the iteration procedures are discussed. Two iteration methods~self-energy insertion and vertex renormalization~are obtained to reach the true eigenvalues of the full Hamiltonian even when there are some intruder states. It is proved that the self-energy insertion procedure produces the eigenvalues of eigenstates which have large overlap with the model space. On the other hand, the vertex renormalization procedure gives the eigenvalues nearest to the unperturbed starting energy. l. Introduction One of the most fundamental problems in the nuclear theory is the derivation of the effective interaction between nucleons in nuclei. There have been many theoretical studies on this subject, which are summarized in several articles,ll~ 4 J since Kuo and Brown's pioneering work') in applying Brueckner's reaction matrix theory 6 J to finite nuclei. The folded diagram theory 1 J.7l.sJ has been developed as a powerful perturbation method for the derivation of the effective interaction. One of the characteristics of the nuclear many-body problem is that the free nucleon-nucleon force may have a strong repulsive core at short distance. For this reason, straightforward application of the perturbation theory to the nuclear many-body problem cannot be successful. Usual treatment of the short-range corre-lation is to introduce Brueckner's reaction matrix (G matrix) . 51 • 6) Usually one