interaction and the divergencies appear when computing the ground state energy or chemical potential so that it is necessary to use the renormalization of the coupling constant [@Pethick; @Stringari; @JPhys2017]. Moreover, as we see below, it may lead to an incomplete structure of the spectrum of single-particle excitations. Therefore, the interaction given by Eqs. (\[eq:2.11\]), (\[eq:3.3\]) is characterized by the corresponding functions $U({\bf p})$ and $J({\bf p})$.

Since the number of Bose condensed atoms is a macroscopic value proportional to the volume of the system ${\cal V}$, the next step, according to the Bogoliubov model [@Bogoliubov], is to replace the creation and annihilation operators of condensed atoms with zero momentum by $c$-numbers, $a_{0}^{\dagger}\to\sqrt{\cal V}\Psi^{*}_{\alpha}$ and $a_{0}\to\sqrt{\cal V}\Psi_{\alpha}$, in all operators of relevant physical quantities, where $\Psi_{\alpha}$ represents the condensate wave function. This procedure has been proved to be exact in the thermodynamic limit [@Ginibre]. The $c$-number terms in the Hamiltonian and those that are quadratic in creation and annihilation operators allow to define the ground state and the corresponding spectrum of single-particle excitations (quasiparticles), while the higher order terms in creation and annihilation operators are relevant when describing the interaction effects between the quasiparticles themselves. Therefore, performing the above replacement 