  Covariant density functional theory, which has so far been applied only
within the framework of static and time dependent mean field theory is extended
to include Particle-Vibration Coupling (PVC) in a consistent way. Starting from
a conventional energy functional we calculate the low-lying collective
vibrations in Relativistic Random Phase Approximation (RRPA) and construct an
energy dependent self-energy for the Dyson equation. The resulting
Bethe-Salpeter equation in the particle-hole ($ph$) channel is solved in the
Time Blocking Approximation (TBA). No additional parameters are used and double
counting is avoided by a proper subtraction method. The same energy functional,
i.e. the same set of coupling constants, generates the Dirac-Hartree
single-particle spectrum, the static part of the residual $ph$-interaction and
the particle-phonon coupling vertices. Therefore a fully consistent description
of nuclear excited states is developed. This method is applied for an
investigation of damping phenomena in the spherical nuclei with closed shells
$^{208}$Pb and $^{132}$Sn. Since the phonon coupling terms enrich the RRPA
spectrum with a multitude of $ph\otimes$phonon components a noticeable
fragmentation of the giant resonances is found, which is in full agreement with
experimental data and with results of the semi-phenomenological
non-relativistic approach.
