  The gap equation in the $^1S_0$-channel is solved for a nuclear slab with the
separable form of the Paris potential. The gap equation is considered in the
model space in terms of the effective pairing interaction which is found in the
complementary subspace. The absolute value of the gap $\Delta$ turned out to be
very sensitive to the cutoff $K_{max}$ in the momentum space in the equation
for the effective interaction. It is necessary to take $K_{max}=160-180
fm^{-1}$ to guarantee 1% accuracy for $\Delta$. The gap equation itself is
solved directly, without any additional approximations. The solution reveals
the surface enhancement of the gap $\Delta$ which was earlier found with an
approximate consideration. A strong surface-volume interplay was found also
implying a kind of the proximity effect. The diagonal matrix elements of
$\Delta$ turned out to be rather close to the empirical values for heavy atomic
nuclei.

Nuclear Theory