The excitation spectrum for Bose fluids with weak interactions (Q2249414)

The paper summarizes recent progress made towards a rigorous justification of the Bogoliubov approximation introduced to explain the superfluid behavior of weakly interacting Bose gases. First, for bosons interacting via a pair-interaction potential, the Hamiltonian \(H_N\) is given acting on the Hilbert space of permutation-symmetric wave functions. Different quantities derived from the Hamiltonian are considered. Then the problem is considered in the limit of infinite number of particles (\(N \to\infty\)). For the ideal Bose gas (non-interacting bosons), the ground energy, the ground state wave function, the spectrum of \(N\) bosons, and the eigenstate of \(H_N\) corresponding to an eigenvalue are stated. Then, the second quantization on the bosonic Fock space is considered by introducing the creation and annihilation operators which satisfy the canonical commutation relations. As a result, the Hamiltonian \(H_N\) is equal to the restriction of the Hamiltonian on the Fock space being a basis for the Bogoliubov approximation discussed in the paper. Then the resulting Hamiltonian is obtained which is quadratic in the creation and annihilation operators, and explicitly diagonalized via a Bogoliubov transformation. This Bogoliubov approximation allows one to compute the complete excitation spectrum. Then, the Hartree limit (HL) is discussed, analyzing the validity of the Bogoliubov approximation beyond the ground state energy. The HL is an extreme form of a mean-field limit where the interaction potential extends over the whole size of the system but the interaction is sufficiently weak (of order \(1/N\)) in order for the interaction energy to be of the same order as the kinetic energy. The main result of the paper is the theorem on the ground state energy \(E_0\) of \(H_N\) and the excitation spectrum \({H_N}-{E_0}\). It is shown that the Bose-Einstein condensation (BEC) is only guaranteed for excitation energies small compared to \(N\), and the existence of BEC is one of the key assumptions entering the Bogoliubov approximation. Moreover, the results of the paper are applied also to the parts of the spectrum \(H_N\) with excitation energies corresponding to collective excitations. So, the results reviewed in the paper discuss the excitation spectrum of an interacting Bose gas in a limit of weak, long-range interactions. The results verify the Bogoliubov prediction that the spectrum consists of sums of elementary excitations. In the translation invariant case, the excitation energy turns out to be linear in the momentum for small momentum and Landau's criterion for superfluidity is verified. The presented methods can be generalized to inhomogeneous systems without translation invariance. Finally, a list of open problems is given, related to the Bogoliubov approximation: (i) the existence of BEC in more general case than the Hartree limit, (ii) the study of the excitation spectrum of cold atomic gases in the Gross-Pitaevskii limit, (iii) the study of the low energy excitation spectrum in the thermodynamic limit and its relation to the phenomenon of superfluidity, (iv) the study of some problems in the Hartree limit, in particular the existence of collective excitations described by solutions of the Hartree equation.