  The non-perturbative domain of QCD contains confinement, chiral symmetry
breaking, and the bound state spectrum. For the calculation of the latter, the
Coulomb gauge is particularly well-suited. Access to these non-perturbative
properties should be possible by means of the Green's functions. However,
Coulomb gauge is also very involved, and thus hard to tackle. We introduce a
novel BRST-type operator r, and show that the left-hand side of Gauss' law is
r-exact.
  We investigate a possible truncation scheme of the Dyson-Schwinger equations
in first-order formalism for the propagators based on an instantaneous
approximation. We demonstrate that this is insufficient to obtain solutions
with the expected property of a linear-rising Coulomb potential. We also show
systematically that a class of possible vertex dressings does not change this
result.
