  We present a detailed comparison of the self-consistent calculations based on
the Hartree-Fock and the spin density functional theory for a spit-gate quantum
wire in the IQH regime. We demonstrate that both approaches provide
qualitatively (and in most cases quantitatively) similar results for the
spin-resolved electron density, spin polarization, spatial spin separation at
the edges and the effective $g$ factor. The both approach give the same values
of the magnetic fields corresponding to the successive subband depopulation and
qualitatively similar evolution of the magnetosubbands. Quantitatively,
however, the HF and the DFT subbands are different (even though the
corresponding total electron densities are practically the same). In contrast
to the HF approach, the DFT calculations predict much larger spatial spin
separation near the wire edge for the low magnetic fields (when the
compressible strips for spinless electrons are not formed yet). In the opposite
limit of the large fields, the Hatree-Fock and the DFT approaches give very
similar values for the spatial spin separation.
