The dielectric theorem within the Hartree-Fock-Bogoliubov framework

Excitation spectra usually reveal important features of the many-body systems. The vibrational excitations can be studied through the well-known linear response theory. This theory is realized, in the nuclear case, by means of the random-phase approximation (RPA); the generalization in the case in which one deals with open shells, and the pairing force is active, is the quasiparticle RPA (QRPA). It is useful to have at one's disposal theorems that provide information on, e.g., the sum rules and mean excitation energies associated with given external operators acting on the system. This article focuses on such theorems in the case of self-consistent QRPA based on Hartree-Fock-Bogoliubov (HFB). In particular, the so-called dielectric theorem that provides the value of the inverse-energy-weighted sum rule based on the simple knowledge of the ground state is demonstrated. This theorem is applied to the case of constrained calculations of the average excitation energy of the monopole resonance combined with the Thouless theorem. The pairing correlations are shown to have the effect of increasing the polarizability m-1. The detailed analysis of the profile of the strength functions by mean of QRPA reveals that the decrease of the average monopole excitation energies in some isotopes is associated with neutron states that emerge at an energy that is lower than the main giant resonance peak.