A simple model, in which nuclei are represented as homogeneous spheres of symmetric nuclear matter, is used to study the effects of a self-consistent pairing interaction on the nuclear response. Effects due to the finite size of nuclei are suitably t
aken into account. The semiclassical equations of motion derived in a previous paper for the time-dependent Hartree-Fock-Bogoliubov problem are solved in an improved (linear) approximation in which the pairing field is allowed to oscillate and to become complex. The new solutions are in good agreement with the old ones and also with the result of well-known quantum approaches. The role of the Pauli principle in eliminating one possible set of solutions is also discussed. The pairing-field fluctuations have two main effects: they restore the particle-number symmetry which is broken in the constant-$Delta$ approximation and introduce the possibility of collective eigenfrequencies of the system due to the pairing interaction. A numerical study with values of parameters appropriate for nuclei, shows an enhancement of the density-density strength function in the region of the low-energy giant octupole resonance, while no similar effect is present in the region of the high-energy octupole resonance and for the giant monopole and quadrupole resonances.
The solutions of the Wigner-transformed time-dependent Hartree--Fock--Bogoliubov equations are studied in the constant-$Delta$ approximation. This approximation is known to violate particle-number conservation. As a consequence, the density fluctuati
on and the longitudinal response function given by this approximation contain spurious contributions. A simple prescription for restoring both local and global particle-number conservation is proposed. Explicit expressions for the eigenfrequencies of the correlated systems and for the density response function are derived and it is shown that the semiclassical analogous of the quantum single--particle spectrum has an excitation gap of $2Delta$, in agreement with the quantum result. The collective response is studied for a simplified form of the residual interaction.
