ar X iv : m at h - ph / 9 90 60 22 v 1 2 4 Ju n 19 99 On Mathematical Structure of Effective Observables

We decompose the Hilbert space of wave functions into two subspaces, and assign to a given observable two effective representatives that act in the model space. The first serves to determine some of the eigen-values of the full observable, while the second serves to determine its matrix elements, in any basis in one of the subspaces, in terms of quantities pertaining to the model space. We also show that if the Hamiltonian of a physical system possesses symmetries then these symmetries continue to hold for its effective representatives of the first type. Maximum information about the system can be obtained in terms of two sets of effective representatives. The first set of representatives is complete. Other observables that do not commute with all members of the complete set have only one type of representative. 1. Introduction Effective operators are often used in nuclear, atomic and molecular physics. The general scheme aims to construct from the Hamiltonian of the system, acting on the Hilbert space of wave functions, an operator that acts on a low dimensional space, so that the eigenvalues of the latter operator are also eigenvalues of the full Hamiltonian of the given system [1-9]. The low dimensional space, we have mentioned, is called a model space and the operator acting on it to produce some of the eigenvalues of the full Hamiltonian is called an effective Hamiltonian, or an effective representative of the Hamiltonian. The latter requirement does not determine an effective representative uniquely. A general class of effective representatives was obtained by Suzuki [6] who also delineated forms according to the role of an arbitrary parameter, the starting energy, in the iterative method of solution.[4], or according to their Hermiticity. Hermitian forms have been introduced or adopted by many researchers [10-17]. A standard non-Hermitian form [2, 8, 18] is relatively simple, and is commonly used for implementing the scheme of effective representatives.