  We calculate the Hubbard bands for the half-filled Hubbard model on a Bethe
lattice with infinite coordination number up to and including third order in
the inverse Hubbard interaction. We employ the Kato--Takahashi perturbation
theory to solve the self-consistency equation of the Dynamical Mean-Field
Theory analytically for the single-impurity Anderson model in multi-chain
geometry. The weight of the secondary Hubbard sub-bands is of fourth order so
that the two-chain geometry is sufficient for our study. Even close to the
Mott--Hubbard transition, our results for the Mott--Hubbard gap agree very well
with those from numerical Dynamical Density-Matrix Renormalization Group
(DDMRG) calculations. The density of states of the lower Hubbard band also
agrees very well with DDMRG data, apart from a resonance contribution at the
upper band edge which cannot be reproduced in low-order perturbation theory.
