Eigenfunctions effective Hamiltonians, mapping

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

Norm-preserving mappings are denoted by K, fC) and, as discussed in Section II.B, generate a Hermitian effective Hamiltonian K HK = i. The orthonormalized model eigenfunctions of H are written as a)o and the corresponding true eigenfunctions are designated by I Pa). Thus, Eq. (2.2) specializes to... 

Q is the normal-ordered wave operator, mapping the eigenfunctions of the effective Hamiltonian onto the exact ones, = 1, It satisfies intermediate normalization,... 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

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