On the Infimum of the Excitation Spectrum of a Homogeneous Bose Gas

We consider a second quantized homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature, and in the grand canonical setting (the chemical potential μ is fixed, the number of particles can vary). The interaction potential has a positive Fourier transform. We define the infimum of the excitation spectrum as the infimum of the grandcanonical Hamiltonian restricted to the subspace of total momentum k. We investigate its upper bound given by the expectation value at squeezed states. Our method can be viewed as a rigorous version of the usual Bogoliubov approach.