ferromagnetic phase of a weakly interacting Bose gas of spin-1 atoms with Bose-Einstein condensate. To this end, we employ the Bogoliubov model [@Bogoliubov] based on $c$-number treatment of creation and annihilation operators for condensate particles. Our starting point is the following Hamiltonian consisting of the kinetic energy term $H_{0}$ and the terms corresponding to the potential interaction $V_{p}$ as well as the interaction between the internal degrees of freedom $V_{\lambda}$: $$\label{eq:3.1}
H=H_{0}+V_{\rm p}+V_{\lambda},$$ where $$\begin{gathered}
H_{0}=\sum_{\bf p}a^{\dagger}_{{\bf p}\alpha}\left[\varepsilon_{\bf p}\delta_{\alpha\beta}- hS^{z}_{\alpha\beta}\right]a_{{\bf p}\beta}, \quad S^{z}_{\alpha\beta}\equiv\lambda^{2}_{\alpha\beta}, \label{eq:3.2} \\
V_{\rm p}={1\over 2\cal{V}}\sum_{{\bf p}_{1},\ldots{\bf p}_{4}}U({\bf p}_{1}-{\bf p}_{3})a^{\dagger}_{{\bf p}_{1}\alpha}a^{\dagger}_{{\bf p}_{2}\beta}a_{{\bf p}_{3}\alpha}a_{{\bf p}_{4}\beta}\,\delta_{{\bf p}_{1}+{\bf p}_{2},\,{\bf p}_{3}+{\bf p}_{4}}, \label{eq:3.3}\end{gathered}$$ and $V_{\lambda}$ is given by Eq. (\[eq:2.11\]). Here $\varepsilon_{\bf p}=p^{2}/2m$ is the kinetic energy of a particle, $a^{\dagger}_{{\bf p}\alpha}$ and $a_{{\bf p}\alpha}$ are the bosonic creation and annihilation operators satisfying Eqs. (\[eq:2.1\]), $h=g\mu_{B}H$ with $g$, $\mu_{B}$, and $H$ being, respectively, the Landé hyperfine factor [@Ueda], the Bohr magneton, and external magnetic field directed along $z$-axis. Note that usually the interaction Hamiltonian is written in terms of the corresponding scattering lengths describing the low energy collisions of atoms at ultra low temperature [@Ohmi; @Ho; @Ueda; @Kurn]. However, such parametrization of interaction does not take into account the local character of 