  In this article we present exact calculations that substantiate a clear
picture relating the confining force of QCD to the zero-modes of the
Faddeev-Popov (FP) operator $\mathcal{M}(A) = - \partial \cdot D(A)$. This is
done in two steps. First we calculate the spectral decomposition of the FP
operator and show that the ghost propagator $\mathcal{G}(k; A) = \langle
\vec{k}| \mathcal{M}^{-1}(A) | \vec{k} \rangle$ in an external gauge potential
$A$ is enhanced at low $k$ in Fourier space for configurations $A$ on the
Gribov horizon. This results from the new formula in the low-$k$ regime
$\mathcal{G}^{ab}(k,A) = \delta^{ab} \lambda_{|\vec{k}|}^{-1}(gA)$, where
$\lambda_{|\vec{k}|}(gA)$ is the eigenvalue of the FP operator that emerges
from $\lambda_{|\vec{k}|}(0) = \vec{k}^2$ at $A$ = 0. Next we derive a strict
inequality signaling the divergence of the color-Coulomb potential at low
momentum $k$ namely, $\widetilde{\mathcal{V}}(k) \geq k^2 G^2(k)$ for $k \to
0$, where $\widetilde{\mathcal{V}}(k)$ is the Fourier transform of the
color-Coulomb potential $\mathcal{V}(r)$ and $G(k)$ is the ghost propagator in
momentum space. The first result holds in the Landau and Coulomb gauges,
whereas the second holds in the Coulomb gauge only. We propose a new numerical
lattice gauge fixing that should be closer to the present analytic approach
than other numerical gauges.
