  In this work we report the results concerning calculations for
quantum-mechanical rotational transitions in molecular hydrogen, H$_2$, induced
by an ultra-cold ground state anti-hydrogen atom $\bar{H}_{1s}$. The
calculations are accomplished using a non-reactive close-coupling
quantum-mechanical approach. The H$_2$ molecule is treated as a rigid rotor.
The total elastic scattering cross section $\sigma_{el}(\epsilon)$ at energy
$\epsilon$, state-resolved rotational transition cross sections
$\sigma_{jj'}(\epsilon)$ between states $j$ and $j'$ and corresponding thermal
rate coefficients $k_{jj'}(T)$ are computed in the temperature range 0.004 K $
\lesssim T \lesssim$ 4 K. Satisfactory agreement with other calculations
(variational) has been obtained for $\sigma_{el}(\epsilon)$.
