Exact canonic eigenstates of the truncated Bogoliubov Hamiltonian in an interacting bosons gas

In a gas of N weakly interacting bosons [1] and [2], a truncated canonic Hamiltonian Hc   follows from dropping all the interaction terms between free bosons with momentum ℏk≠0ℏk≠0. Bogoliubov Canonic Approximation (BCA) is a further manipulation, replacing the number operator  N˜in of free particles in k=0k=0, with the total number N of bosons. BCA Hc   transforms into a different Hamiltonian HBCA=∑k≠0ϵ(k)Bk†Bk+const, where Bk† and BkBk create/annihilate non interacting pseudoparticles. The problem of the exact   eigenstates of the truncated Hamiltonian is completely solved in the thermodynamic limit (TL) for a special class of eigensolutions |S,k〉c|S,k〉c, denoted as ‘s-pseudobosons’, with energies ES(k)ES(k) and zero total momentum. Some preliminary results are given for the exact eigenstates (denoted as ‘η  -pseudobosons’), carrying a total momentum ηℏkηℏk(η=1,2,…)(η=1,2,…). A comparison is done with HBCA and with the Gross-Pitaevskii theory (GPT), showing that some differences between exact and BCA/GPT results persist even in the thermodynamic limit (TL). Finally, it is argued that the emission of η-pseudobosons, which is responsible for the dissipation á la Landau [3], could be significantly different from the usual picture, based on BCA pseudobosons.