  We report a high resolution resonant inelastic x-ray scattering investigation
of the bilayer manganites La$_{2-2x}$Sr$_{1+2x}$Mn$_2$O$_7$ with $x = 0.36$ and
$0.5$. The momentum dependence along the crystallographic $(110)$ direction for
energy losses $1\,\rm{eV} \leq \Delta E \leq 15\,\rm{eV}$ has been measured in
detail with the data analysis focusing on the energy loss region $1\,\rm{eV}
\leq \Delta E \leq 5\,\rm{eV}$, which includes a strong peak located at $\Delta
E \approx 2\,\rm{eV}$. We observe a clear dispersion of up to $0.5\,\rm{eV}$ in
the measured $\textbf{q}$ range, which is direct evidence of the non-local
character of this excitation. Further, we found that the intensity in this low
energy region strongly depends on both the reduced wave vector $\textbf{q} =
(h, h, 0)$, $h = 0.1 - 0.5$, and temperature, i.e. different ordered phases.
Results can be explained via an intersite $d-d$ charge transfer excitation,
proposed for pseudo-cubic manganites, where the hopping rate is strongly
increased (decreased) by ferromagnetic (antiferromagnetic) alignment of
neighboring in-plane Mn ion core spins.
