A rigorous microscopic analysis of a weakly interacting Bose gas at zero temperature shows that the elementary excitation spectrum $\ensuremath{\omega}(k)$ is not an analytic function of the wave vector
k in the long-wavelength limit but has the form $k[{c}_{0}+{c}_{2}{k}^{2}+{c}_{L4}{k}^{4}\mathrm{ln}(\frac{1}{k})+\ensuremath{\cdots}]$. It is pointed out that the structure factor
S(k) is not an analytic function of
k and that the specific heat
Cv(T) at low temperatures
T has the form ${C}_{v}={T}^{3}[{C}_{0}+{C}_{2}{T}^{2}+{C}_{L4}{T}^{4}\mathrm{ln}(\frac{1}{T})+\ensuremath{\cdots}]$. The extension of these results to liquid
He4 is discussed.
We generalize the Beliaev diagrammatic theory of an interacting spinless Bose-Einstein condensate to the case of a binary mixture. We derive a set of coupled Dyson equations and find analytically the Green's functions of the system. The elementary excitation spectrum consists of two branches, one of which takes the characteristic parabolic form $\ensuremath{\omega}\ensuremath{\propto}{p}^{2}$ in the limit of a spin-independent interaction. We observe renormalization of the magnon mass and the spin-wave velocity due to the Andreev-Bashkin entrainment effect. For a three-dimensional weakly interacting gas the spectrum can be obtained by applying the Bogoliubov transformation to a second-quantized Hamiltonian in which the microscopic two-body potentials in each channel are replaced by the corresponding off-shell scattering amplitudes. The superfluid drag density can be calculated by considering a mixture of phonons and magnons interacting via the effective potentials. We show that this problem is identical to the second-order perturbative treatment of a Bose polaron. In two dimensions the drag contributes to the magnon dispersion already in the first approximation. Our consideration provides a basis for systematic study of emergent phases in quantum degenerate Bose-Bose mixtures.