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Bethe-Salpeter approach to the collective-mode spectrum of a superfluid Fermi gas in a moving optical lattice

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.