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