is a speed of sound. The requirement for the speed of sound to be real leads to the stability condition $U(0)+(4/3)J(0)>0$ obtained above.

Two other spectra $\omega_{{\bf p}z}^{\rm ({\rm I})}$ and $\omega_{{\bf p}x}^{\rm ({\rm II})}$ describe the excitations related to the internal degrees of freedom, or “spin-quadrupole” waves. When the applied magnetic field is zero ($h=0$), both spectra become identical so that the system is described by two types of single-particle excitations. Note that the quadratic form determined by Eq. (\[eq:4.10\]) must be positive definite. This requirement implies $J({\bf p})-J(0)>0$ at any momentum ${\bf p}$, moreover $J(0)<0$ for ferromagnetic ordering. It is worth stressing that in contrast to the previous studies of spin-1 Bose-Einstein condensates [@Ohmi; @Ho; @Ueda; @Kurn], both spectra depends on the interaction parameter that is absolutely clear for the system of interacting atoms. This is due to the fact that we do not parameterize the interaction by the corresponding scattering lengths. Indeed, in doing so, $J({\bf p})=J(0)={4\pi\hbar^{2}a_{s}/m}$, where $a_{s}$ is the corresponding scattering length and the spectra become independent of the interaction parameters. Therefore, the description of the interaction effects in ultracold gases by the scattering length represents a sufficiently rough approximation that does not take into 