  The self-consistent mean-field (SCMF) theory describes many properties of the
ground state and excited states of the atomic nucleus, such as masses, radii,
deformations and giant resonance energies. SCMF models are based on the
independent particle picture where nucleons are assumed to move in a
self-generated average potential. In the first part of this work, we apply a
state-of-the-art SCMF approach, based on the Skyrme effective interaction, to
two different excitations (viz. the pygmy dipole resonance and the isovector
giant quadrupole resonance), investigating their relation with the nuclear
matter symmetry energy, which corresponds to the energy cost for changing
protons into neutrons and is a key parameter for the nuclear equation of state.
However, SCMF models present well known limitations which require the inclusion
of further dynamical correlations, e.g. the ones coming from the interweaving
between single-particle and collective degrees of freedom (particle-vibration
coupling - PVC). In the second part of this work, we report on the application
to inclusive (namely, the strength function of giant resonances) and exclusive
(the gamma decay of giant resonances) observables of a new self-consistent
model based on the PVC idea in the Skyrme framework. In our model we use an
effective interaction fitted at mean-field level to some selected experimental
data. In principle, when these interactions are used in beyond mean-field
theories, one would need to re-determine the parameters of the interaction at
the same level of approximation. Moreover, due to the zero-range nature of the
employed interaction, divergences arise. In the last and most innovative part
of this thesis, we develop, for the first time in finite nuclei, a possible way
to cure the divergences, paving the way to the possibility of producing an
effective interaction fitted at PVC level.
