The scattering matrix is derived for the scattering of nucleons by nuclei lacking one nucleon from being doubly magic. It is assumed that an average field has been determined through a Hartree-Fock procedure (HF). The residual interaction is treated in the random-phase approximation (RPA). In contrast to previous treatments, it is not assumed that this interaction is separable. The RPA ground state of the compound system is given by a correlated wave function $|{\Psi }_0〉$. It is assumed that states of the target and residual nucleus can be described as one-hole states in this correlated ground state $|{\Psi }_0〉$. It is found that the RPA equations allow for a proper definition of asymptotic states only if the full Hamiltonian (including the c.m. energy) is used in the HF procedure. A general, yet explicit, expression for the $S$ matrix is obtained by applying to the channel-channel part of the residual interaction a method first proposed by Weinberg. The correlations contained in $|{\Psi }_0〉$ give rise to poles of the scattering matrix for real negative energies below the energy of the lowest bound state, i.e., the ground state $|{\Psi }_0〉$. In the energy region of physical interest, these poles have two effects on the scattering matrix. First, a constant background term is introduced. Second, the partial widths ${\Gamma }_{\lambda c}$ for decay of a compound state ($\lambda$) into an open channel ($c$) are complex. The sum of the partial widths, ${\Sigma }_{\lambda c}{\Gamma }_{\lambda c}$, is compared with the sum of the total widths, ${\Sigma }_{\lambda }{\Gamma }_{\lambda }$. It is found that the two sums differ by terms of second order in the admixture of correlations in the ground state. The influence of symmetry properties of the Hamiltonian on the RPA solutions is discussed. It is shown that the scattering matrix derived is that in the c.m. frame, and it is completely independent of the total momentum of the nucleus.

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