  In this paper we present a new formalism to implement the nuclear
particle-vibration coupling (PVC) model. The key issue is the proper treatment
of the continuum, that is allowed by the coordinate space representation. Our
formalism, based on the use of zero-range interactions like the Skyrme forces,
is microscopic and fully self-consistent. We apply it to the case of neutron
single-particle states in $^{40}$Ca, $^{208}$Pb and $^{24}$O. The first two
cases are meant to illustrate the comparison with the usual (i.e., discrete)
PVC model. However, we stress that the present approach allows to calculate
properly the effect of PVC on resonant states. We compare our results with
those from experiments in which the particle transfer in the continuum region
has been attempted. The latter case, namely $^{24}$O, is chosen as an example
of a weakly-bound system. Such a nucleus, being double-magic and not displaying
collective low-lying vibrational excitations, is characterized by quite pure
neutron single-particle states around the Fermi surface.
