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In this paper we present a new formalism to implement the nuclearparticle-vibration coupling (PVC) model. The key issue is the proper treatmentof the continuum, that is allowed by the coordinate space representation. Ourformalism, 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 neutronsingle-particle states in $^{40}$Ca, $^{208}$Pb and $^{24}$O. The first twocases are meant to illustrate the comparison with the usual (i.e., discrete)PVC model. However, we stress that the present approach allows to calculateproperly the effect of PVC on resonant states. We compare our results withthose from experiments in which the particle transfer in the continuum regionhas been attempted. The latter case, namely $^{24}$O, is chosen as an exampleof a weakly-bound system. Such a nucleus, being double-magic and not displayingcollective low-lying vibrational excitations, is characterized by quite pureneutron single-particle states around the Fermi surface.

Continuum particle-vibration coupling method in coordinate-space representation for finite nuclei