Recursion method for deriving an energy-independent effective interaction

The effective-interaction theory has been one of the useful and practical methods for solving nuclear many-body problems based on the shell model. Various approaches have been proposed which are constructed in terms of the so-called $\widehat{Q}$ box and its energy derivatives introduced by Kuo et al. In order to find out a method of calculating them we make a decomposition of a full Hilbert space into subspaces (the Krylov subspaces) and transform a Hamiltonian to a block-tridiagonal form. This transformation brings about much simplification of the calculation of the $\widehat{Q}$ box. In the previous work a recursion method was derived for calculating the $\widehat{Q}$ box analytically on the basis of such transformation of the Hamiltonian. In the present study, by extending the recursion method for the $\widehat{Q}$ box, we derive another recursion relation to calculate the derivatives of the $\widehat{Q}$ box of arbitrary order. With the $\widehat{Q}$ box and its derivatives thus determined we apply them to the calculation of the $E$-independent effective interaction given in the so-called Lee-Suzuki (LS) method for a system with a degenerate unperturbed energy. We show that the recursion method can also be applied to the generalized LS scheme for a system with nondegenerate unperturbed energies. If the Hilbert space is taken to be sufficiently large, the theory provides an exact way of calculating the $\widehat{Q}$ box and its derivatives. This approach enables us to perform recursive calculations for the effective interaction to arbitrary order for both systems with degenerate and nondegenerate unperturbed energies.