Proof of a Conjecture, for Large Wave Numbers, in Mihara and Puff's Theory of the Structure Factor of Liquid Helium

To establish their theory of the structure factor for liquid helium, Mihara and Puff argue the validity of the relation ${〈{\hat{H}}^3〉}_F\sim {E_F}^3\left(\overrightarrow{\mathrm{k}}\right)+4\mathrm{}〈K〉\epsilon \left(\overrightarrow{\mathrm{k}}\right)E_F\left(\overrightarrow{\mathrm{k}}\right)$ which expresses the ${\omega }^3$ moment ${〈H^3〉}_F$ in terms of the Bijl-Feynman excitation spectrum $E_F\left(\overrightarrow{\mathrm{k}}\right)$, the free boson excitation spectrum $\epsilon \left(\overrightarrow{\mathrm{k}}\right)$, and the kinetic energy per particle of the liquid in its groundstate, $〈K〉$. In this paper we show that ~ means "is equal to for wave numbers greater than about $2A^{-1\mathrm{}}$" for the case of liquid helium and that it means "is exactly equal to" for the case of a low-density Bose fluid. The latter result has the interesting effect of providing a means of deriving the Bogoliubov theory of a weakly coupled Bose system without the well-known canonical transformation of the Hamiltonian.