Convergence properties of effective interaction expansions in a modified Lipkin model

Abstract

The convergence properties of three methods for constructing the effective interaction are studied by comparing them to the exact solutions of a two-level Lipkin model modified for open-shell nuclei with a one-dimensional model space. The methods considered are the Bloch-Horowitz equation, the folded-diagram series of Kuo et al., and the sequence of partial summations of the folded-diagram series. A criterion for the maximum region of convergence of both the folded-diagram series and the sequence of partial summations is established in terms of the overlap of the exact and model-space wave functions. The sequence of partial summations is found to be more convenient for calculations than the folded-diagram series.