Renormalization Constants in the Random-Phase Approach

The renormalized random-phase-approximation equations are solved for a density-dependent particle-hole force emerging from the nucleon-nucleon interaction. Migdal's renormalization constants are determined for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ assuming Woods-Saxon states for the quasiparticle and quasihole propagation, and compared with the renormalization constants for a harmonic-oscillator propagation. In both cases the assumption of good isospin is made. Furthermore, we determined the renormalization constants in the ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$, and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ problem in order to test the assumption of a mass-number-independent renormalization constant. For this purpose we used the standard harmonic-oscillator states for the quasi-single-particle (-hole) propagation (no good isospin).