12-11U

The boson and fermion spectra of the Bose-Fermi-Hubbard model are investigated. The boson Green's function is calculated for the case of localized fermions and the fermion Green's function is calculated for the case of localized bosons; the formalism of Hubbard operators is used. Phase diagrams determining the regions of existence superfluid phase and the Mott insulator are built in the plane ($\mu_{b}$, $t^{b}$) (the boson chemical potential vs. boson hopping parameter) in the regimes of fixed chemical potential or fixed concentration of fermions. For  different states of the Mott insulator  phase depicted on the phase diagram the  boson and fermion spectra are built at different temperatures.