  We analyze the resonant x-ray scattering (RIXS) spectra from Sr$_2$IrO$_4$ in
an itinerant electron approach. Employing a multi-orbital tight-binding model
on the square lattice, we calculate the one-electron energy band within the
Hartree-Fock approximation, which leads to an antiferromagnetic ground state.
We then evaluate the two-particle Green's functions for the particle-hole pair
excitations within the random phase approximation, which are connected to the
RIXS spectra within the fast collision approximation. The calculated RIXS
spectra exhibit two-peak structure with slightly different energies in the
low-energy region, which are originated from the bound states in the
two-particle Green's function. They may be interpreted as the split modes of
magnon. We also obtain several $\delta$-function peaks, which arise from the
bound states around the bottom of energy continuum. They may be called as the
exciton modes. These characteristics are in qualitative agreement with the RIXS
experiment, demonstrating that the weak coupling theory could explain both the
magnon and the exciton peaks in the RIXS spectra on an equal footing.
