  Based on the Luttinger-Kohn Hamiltonian in the axial approximation, the
transcendental equations determining the hole subband dispersions in a
cylindrical Ge nanowire are analytically derived. These equations are more
general than that derived using the spherical approximation, and are suitable
to study the growth direction dependence of the subband dispersions. The axial
approximation almost gives rise to the accurate low-energy subband dispersions
for high-symmetry nanowire growth directions [001] and [111]. The perturbation
correction from the non-axial term is negligible for these two directions. The
lowest two subband dispersions can be regarded as two shifted parabolic curves
with an energy gap at $k_{z}=0$ for both growth directions [001] and [111]. At
the position of the energy gap, the eigenstates for growth direction [111] are
inverted in comparison with the normal eigenstates for growth direction [001].
A nanowire growth direction where the energy gap closes at $k_{z}=0$ is
predicted to exist between directions [001] and [111].
