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 (Q) over cap 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 (Q) over cap box. In the previous work a recursion method was derived for calculating the (Q) over cap box analytically on the basis of such transformation of the Hamiltonian. In the present study, by extending the recursion method for the (Q) over cap box, we derive another recursion relation to calculate the derivatives of the (Q) over cap box of arbitrary order. With the (Q) over cap 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 (Q) over cap 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.