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The dependence on the single-particle states of the pairing matrix elementsof the Gogny force and of the bare low-momentum nucleon-nucleon potential$v_{low-k}$ is studied in the semiclassical approximation for the case of atypical finite, superfluid nucleus ($^{120}$Sn). It is found that the matrixelements of $v_{low-k}$ follow closely those of $v_{Gogny}$ on a wide range ofenergy values around the Fermi energy $e_F$, those associated with $v_{low-k}$being less attractive. This result explains the fact that around $e_F$ thepairing gap $\Delta_{Gogny}$ associated with the Gogny interaction (and with adensity of single-particle levels corresponding to an effective $k$-mass$m_k\approx 0.7 m$) is a factor of about 2 larger than $\Delta_{low-k}$,beingin agreement with $\Delta_{exp}$= 1.4 MeV. The exchange of low-lying collectivesurface vibrations among pairs of nucleons moving in time-reversal states givesrise to an induced pairing interaction $v_{ind}$ peaked at $e_F$. Theinteraction $(v_{low-k}+ v_{ind})Z_{\omega}$ arising from the renormalizationof the bare nucleon-nucleon potential and of the single-particle motion($\omega-$mass and quasiparticle strength $Z_{\omega}$) due to theparticle-vibration coupling leads to a value of the pairing gap at the Fermienergy $\Delta_{ren}$ which accounts for the experimental value

Pairing matrix elements and pairing gaps with bare, effective, and induced interactions