  We study the Hamiltonian approach to 1+1 dimensional Yang-Mills theory in
Coulomb gauge, considering both the pure Coulomb gauge and the gauge where in
addition the remaining constant gauge field is restricted to the Cartan
algebra. We evaluate the corresponding Faddeev-Popov determinants, resolve
Gauss' law and derive the Hamiltonians, which differ in both gauges due to
additional zero modes of the Faddeev-Popov kernel in the pure Coulomb gauge. By
Gauss' law the zero modes of the Faddeev-Popov kernel constrain the physical
wave functionals to zero colour charge states. We solve the Schroedinger
equation in the pure Coulomb gauge and determine the vacuum wave functional.
The gluon and ghost propagators and the static colour Coulomb potential are
calculated in the first Gribov region as well as in the fundamental modular
region, and Gribov copy effects are studied. We explicitly demonstrate that the
Dyson-Schwinger equations do not specify the Gribov region while the
propagators and vertices do depend on the Gribov region chosen. In this sense,
the Dyson-Schwinger equations alone do not provide the full non-abelian quantum
gauge theory, but subsidiary conditions must be required. Implications of
Gribov copy effects for lattice calculations of the infrared behaviour of
gauge-fixed propagators are discussed. We compute the ghost-gluon vertex and
provide a sensible truncation of Dyson-Schwinger equations. Approximations of
the variational approach to the 3+1 dimensional theory are checked by
comparison to the 1+1 dimensional case.
