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

In a gas of N weakly interacting bosons [1,2], a truncated canonic Hamiltonian H-c follows from dropping all the interaction terms between free bosons with momentum hk not equal 0. Bogoliubov Canonic Approximation (BCA) is a further manipulation, replacing the number operator (N) over tilde (in) of free particles in k = 0, with the total number N of bosons. BCA transforms into a different Hamiltonian H-BCA = Sigma k not equal 0, epsilon(k)(BkBk)-B-dagger + const, where B-k(dagger), and B-k 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 vertical bar S, k >(c) denoted as 's-pseudobosons', with energies epsilon(S)(k) and zero total momentum. Some preliminary results are given for the exact eigenstates (denoted as 'eta-ippseudobosons'), carrying a total momentum eta hk (eta = 1, 2,...). 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 thermodynamic limit (TL). Finally, it is argued that the emission of eta-pseudobosons, which is responsible for the dissipation a la Landau [3], could be significantly different from the usual picture, based on BCA pseudobosons. (C) 2016 Elsevier B.V. All rights reserved.