On the configuration of systems of interacting particles with minimum potential energy per particle

Previous work on infinite one-dimensional systems of interacting particles is continued. In the case of two-body potentials φ(x) = φ(-x), whose Fourier transform ĝf(k) eicsts, it is shown that a necessary condition that the equidistant configuration has for a certain range of densities minimum potential energy per particle among all configurations of the same density, is that ĝf(k)⩾0 for all k. An analogous theorem is proved for systems of particles in two and three dimensions.