Inhomogeneous Random Phase Approximation for Nuclear and Atomic Reactions

Abstract

A random phase approximation (RPA) of the inhomogenous time-independent mean field (TIMF) equations for reactions is developed. The TIMF method is based on a general variational principle for calculating matrix elements of operator inverses as, for example, matrix elements of the resolvent of a Hamiltonian or the scattering T-operator. In the case of nuclear or atomic reactions, these matrix elements are calculated with respect to the asymptotic channel wave functions which also define the inhomogeneous terms in the variational equations. For inhomogeneous equations a direct RPA-like treatment, analogous to the standard RPA used for diagonalization problems, is not possible. Hence the starting point to obtain an RPA extension for the inhomogeneous mean field equations is a transformation of the inhomogeneous into homogeneous variational equations. The problem of operator inversion is then transformed into the inversion of a function which is obtained by the diagonalization of an auxilliary operator. This auxilliary operator consists of the operator to be inverted and an additional term incorporating the inhomogeneities. Taking into account particle-hole correlations for the nonhermitian diagonalization problem, the standard random phase approximation for hermitian operators can be generalized to a nonhermitian RPA built on top of TIMF. For that purpose, the antisymmetric TIMF theory for operator inversion is formulated in second quantization. Complications are discussed, which result from the use of several biorthogonal bases and of only intra-fragment antisymmetrized channel wave functions, as needed in antisymmetric scattering theories. In this framework it is shown how particle-hole correlations can be used to generalize the inhomogeneous TIMF method, thus going beyond the mean field approximation.