Boson pairing in the generalized random-phase approximation

Abstract

We write out the equations of the generalized random-phase approximation for bosons in analogy with the corresponding calculation due to Anderson for fermions. This formalism has apparently not been published before. When we specialize to a $\delta$-function $s$-wave potential in real space and take $N_0\to 0$, we find a density-density response function with structure qualitatively like that assumed in a phenomenological model of the long-wavelength excitations in liquid helium, which was previously proposed by the authors. This result provides partial theoretical support for that phenomenological model although, as we discuss, the potential is unrealistic and the calculation is not fully self-consistent.