  We have derived the Bethe-Salpeter (BS) equations for the collective-mode
spectrum of a mixture of fermion atoms of two hyperfine states loaded into a
moving optical lattice. The collective excitation spectrum exhibits rotonlike
minimum and the Landau instability takes place when the energy of the rotonlike
minimum hits zero. The BS approach is compared with the other existing methods
for calculating the collective-mode spectrum. In particular, it is shown that
the spectrum obtained by the BS equations in an excellent agreement with
corresponding spectrum obtained by the perturbation theory, while the Green's
function formalism provides slightly different results for the collective
excitations.
