Structure Functions in an Interacting Boson System*

3m (k) = dw d~S(k ,.,) for a system of Bosons of mass m at zero temperature and density n to the first approximation beyond the Bogoliubov (zeroth) approximation. The two-body interaction u is characterized by the s-wave scatteringlength and is taken to be a constant in k-space. The small dimensionless parameter g = 4irams 0, where s0 = (4iran) 1/2/m is the phonon speed in the zeroth approximation. We choose units such that h = m = s 0 1. We write S(k, ~) as a coherent one-phonon part, gS1(k, ~,) = Z(k) 6(o — ~), and a multi-phonon part [2, 3]. In order to isolate the effects of backflow, S1 is constructed to exhaust the f-sum rule, gS1(k) = ~_-k 2 thus, Z(k) = k2/2t~kand ~ is the energy of the one-phonon excitations. The form of the multiphonon term and the k-dependence of Sm(k) is of interest. The method of calculation of S(k, c...) is based on the generalized dielectric formulation [4] of Bose systems and can be outlined as follows. We relate S(k, o.,) to the density-density response function [4] F(k, w); S(k, w) = —Im F(k, t~.,)/ir.The effects of backflow are conveniently taken into account by the identity [4] gF(k, w) = (k2/~.~2)[l +gF33(k, s,)] based on the continuity equation, where F33 is the longitudinal current response function 141. Aperturbation expansion for S(k, ~,) is developed by expanding F33, F, S, ~ and u in powers of g, e.g., gS(k, ~) = S(°)(k,cu)+gSU)(k, cu)+ .,~ = w~(o)+g~(1)+...; t)/g= 1+gu~1~+ ... We find that to 0(g°) 5(k, ~.i)has only a coherent one-phonon part S~0kk, w) = Z(°)(k) ~ — ~o)), where Z(°)(k)= k2/2o.~°~ and 4o) = k(1+k2/4)’/2. To 0(g) we obtain