  We linearize the neutrino mean-field evolution equations describing the
neutrino propagation in a background of matter and of neutrinos, using
techniques from many-body microscopic approaches. The procedure leads to an
eigenvalue equation that allows to identify instabilities in the evolution,
associated with a change of the curvature of the neutrino energy-density
surface. Our result includes all contributions from the neutrino Hamiltonian
and is generalizable to linearize the equations of motion at an arbitrary point
of the evolution. We then consider the extended equations that comprise the
normal mean field as well as the abnormal mean field that is associated with
neutrino-antineutrino pairing correlations. We first re-derive the extended
neutrino Hamiltonian and show that such a Hamiltonian can be diagonalized by
introducing a generalized Bogoliubov-Valatin transformation with quasi-particle
operators that mix neutrinos and antineutrinos. We give the eigenvalue
equations that determine the energies of the quasi-particles eigenstates.
Finally we derive the eigenvalue equation of the extended equations of motion,
valid in the small amplitude approximation. Our results apply to an arbitrary
number of neutrino families.
