Sum Rules for a Binary Solution and Effective Interactions Between He-3 Quasiparticles in Superfluid He-4

Sum rules are derived for dynamic structure factors for a binary solution and applied to the Woo-Tan-Massey theory of He-3-He-4 solutions. We demonstrate that in the long-wavelength region the effective He-3 quasiparticle interaction V-k contains no linear k dependence. In other words, V-k -> A + Bk-2 as k -> 0 This result is exact, and requires no empirical information. It is consistent with expressions of Vk assumed by Eckstein, Eckstein, Kuper, and Ron, and by Ebner, based on fitting experimental data such as osmotic pressure measurements.