The effect of compound states on collective-state singularities in the optical potential

From an exact form of the self-energy of a nucleon in a nucleus, which is equivalent to the optical potential, two approximation schemes, which are both based on a weak-coupling picture, are presented. They lead to expressions which can be interpreted physically as the coupling of a dressed single-particle (single-hole) state to the exact particle-hole (particle-particle) propagator. The residues of the many poles of the self-energy are divided into two categories: (i) residues which are distributed statistically, and (ii) residues which vary systematically. The first set is interpreted as compound nuclear structure and its influence on the single particle motion is investigated. The second set leads to intermediate structure. This structure is further investigated within an extension of the random phase approximation where all particle and hole lines are dressed by compound nuclear effects. In this way the spreading width of intermediate resonances, which is found to be the imaginary part of the pole position in the self-energy, is calculated. The escape width for these resonances is obtained by solving the Lippmann-Schwinger equation in which the calculated self-energy appears. This width depends on the structure of the continuum as well as on the residues of the poles of the self-energy. It is shown that these residues are complex, one of the reasons for this being that corrections to the vertex for the excitation of a collective state, appear. The similarity between the complex form factors introduced in the macroscopic model for inelastic scattering and the complex vertex functions above, is discussed. It is also pointed out that the results obtained here may be used in a phenomenological analysis.