  Effective interactions have been used to compute the pairing gap for nuclear
and neutron matter in several schemes. In this work we analyze the impact of
phase-shift equivalent interactions within the BCS theory on the
$^1S_0$-channel pairing gap for a translational invariant many-fermion system
such as nuclear and neutron matter. We solve the BCS pairing gap equation on a
finite momentum grid for a toy model separable Gaussian potential in the
$^1S_0$-channel explicitly evolved through the Similarity Renormalization Group
(SRG) transformation and show that in the on-shell and continuum limits the
pairing gap vanishes. For finite size systems the momentum is quantized and the
on-shell limit is realized for SRG cutoffs comparable to the momentum
resolution. In this case the pairing gap can be computed directly from the
scattering phase-shifts by an energy-shift formula. While the momentum grid is
usually used as an auxiliary way of solving the BCS pairing gap equation, we
show that it actually encodes some relevant physical information, suggesting
that in fact finite grids may represent the finite size of the system.
