Spin dynamics in the generalized ferromagnetic Kondo model for manganites

We calculate the dynamic spin susceptibility in the generalized Kondo model describing the interaction between the band $e_g$ electrons and the localized $t_{2g}$ electrons coupled via the antiferromagnetic exchange. We show that the spin-wave spectrum in the ferromagnetic phase has acoustic and optical branches in the mean-field approximation. We evaluate the self-energy corrections to the spectrum and estimate the damping of acoustic magnons.