A new representation of the Hamiltonian operator for bosons and fermions. Quantization of free energy and dependence of the Landau criterion on temperature (Q5959471)

Three pages of short notes on three subjects covered by the title and by the three recent papers by the author (plus his two older books). This means that the text (the majority of which are formulae) forms an extended abstract by itself. Its main theorem states that in terms of the (either bosonic or fermionic) creation and annihilation operators one can introduce a density-like functional and an ``'averaged'' operator \(H\) in such a way that the projection of \(H\) on the pertaining Fock space coincides with the usual partial differential Hamiltonian of the system. The subsequent commentary (adding, in the similar spirit, an entropy and free energies) is split in the weak- and strong-interaction parts. Curiously enough, the paper was originally presented in Russian (I was told by my Russian friends that such a practice has also some pecuniary benefits) but the text concerning the latter case (including the operator representation of the free energies and related discussion) appears only in its ``'translated'' English update where one misses (for compensation?) the explicit half-page form of the equation which determines the \(k=1\) solutions and, via the loss of reality of its eigenvalues, describes Landau's boundary of the domain of the superfluidity (hence, interested people and the specialists should rather read both the language mutations).