  In a gas of $N$ weakly interacting bosons \cite{Bogo1, Bogo2}, a truncated
canonic Hamiltonian $\widetilde{h}_c$ follows from dropping all the interaction
terms between free bosons with momentum $\hbar\mathbf{k}\ne\mathbf{0}$.
Bogoliubov Canonic Approximation (BCA) is a further manipulation, replacing the
number \emph{operator} $\widetilde{N}_{in}$ of free particles in
$\mathbf{k}=\mathbf{0}$, with the total number $N$ of bosons. BCA transforms
$\widetilde{h}_c$ into a different Hamiltonian
$H_{BCA}=\sum_{\mathbf{k}\ne\mathbf{0}}\epsilon(k)B^\dagger_\mathbf{k}B_\mathbf{k}+const$,
where $B^\dagger_\mathbf{k}$ and $B_\mathbf{k}$ create/annihilate non
interacting pseudoparticles. The problem of the \emph{exact} eigenstates of the
truncated Hamiltonian is completely solved in the thermodynamic limit (TL) for
a special class of eigensolutions $|\:S,\:\mathbf{k}\:\rangle_{c}$, denoted as
\textquoteleft s-pseudobosons\textquoteright, with energies
$\mathcal{E}_{S}(k)$ and \emph{zero} total momentum. Some preliminary results
are given for the exact eigenstates (denoted as \textquoteleft
$\eta$-pseudobosons\textquoteright), carrying a total momentum
$\eta\hbar\mathbf{k}$ ($\eta=\:1,\:2,\: \dots$). A comparison is done with
$H_{BCA}$ and with the Gross-Pitaevskii theory (GPT), showing that some
differences between exact and BCA/GPT results persist even in the TL. Finally,
it is argued that the emission of $\eta$-pseudobosons, which is responsible for
the dissipation $\acute{a}$ \emph{la} Landau \cite{L}, could be significantly
different from the usual picture, based on BCA pseudobosons.

Quantum Gases;Quantum Physics