# Effective operator theory in boson mappings

Article

The convenience of a boson basis used in conjunction with a boson mapped hamiltonian may sometimes be complicated by the appearance of spurious states in the spectrum. Under ideal circumstances where either no truncation of the boson Fock space is introduced, or where truncation involves a set of completely decoupled collective degrees of freedom only, the identification and role of such spurious states are well understood and analysed. However, for situations where truncation involves degrees of freedom which are not completely decoupled, no analogous systematic analysis has yet been given. A formalism which suggests itself for such an analysis, is effective operator theory. The energy-independent effective interaction approach of Suzuki and Lee seems to be well suited for this purpose. We first extend this approach beyond interactions only, and construct the effective operator corresponding to an arbitrary operator. We then show how the formalism is applied in the present context of identifying spurious states. In particular, we apply the extended procedure to the generalized SO(5) model, to a single-j proton-neutron model, and also to the multi-j similarity-transformed Dyson mapping. We also compare the present approach to an alternative energy-independent effective-operator approach which is based on the use of a Majorana-like operator in conjunction with the SO(2n) Dyson boson mapping. The present application is also linked to applications in the derivation of effective shell-model interactions. © 1993.