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The present short note is simply intended to communicate that I haveanalytically diagonalized the Bogoliubov truncated Hamiltonian$H_c$~\cite{Bogo1,Bogo2}, in an interacting bosonic gas. This is the naturalprosecution of my work~\cite{MS}, now denoted as (I), where the diagonalizationwas performed only in the subspace corresponding to zero momentum collectiveexcitations (CE).

Exact diagonalization of the truncated Bogoliubov Hamiltonian