Pressure Dependence of the Roton Spectrum in Liquid Helium*

We suggest consistency checks on neutron scattering data on liquid helium under pressure. The theoretically calculated static structure function S(k[P) leads to pressure dependence of the roton gap, momentum, and curvature in general agreement with experiment. Recent experiments at Argonne, t Brookhaven, 2 and Chalk River a indicate that the excitation spectrum of liquid helium undergoes significant shifts when subjected to external pressure. We point out in this note that certain well-known relations exist between the measured properties, and that these relations can be used as consistency checks on the experiment. First, there are sum rules dealing with various moments of the dynamic structure function S(k, co). These sum rules have been used 4's in analyzing neutron and x-ray scattering properties at zero pressure. They are equally valid at higher pressures. However, as yet the available xperimental data are neither detailed nor accurate nough to justify similar analyses. Second, there exist perturbation formulas relating the excitation spectrum to the static structure function $(k). In particula r, the second-orde