Distribution of points on the real line under a class of repulsive potentials

In a 1979 paper, Ventevogel and Nijboer showed that classical point particles interacting via the pair potential $\phi(x)=\left(1+x^4\right)^{-1}$ are not equally spaced in their ground states in one dimension when the particle density is high, in contrast with many other potentials such as inverse power laws or Gaussians. In this paper, we explore a broad class of potentials for which this property holds; we prove that under the potentials $f_\alpha(x)=\left(1+x^\alpha\right)^{-1}$, when $\alpha>2$ is an even integer, there is a corresponding $s_\alpha>0$ such that under density $\rho={n}/{s_\alpha}$, the configuration that places $n$ particles at each point of $s_\alpha\mathbb{Z}$ minimises the average potential energy per particle and is therefore the exact ground state. In other words, the particles form clusters, while the clusters do not approach each other as the density increases; instead they maintain a fixed spacing. This is, to the best of our knowledge, the first rigorous analysis of such a ground state for a naturally occurring class of potential functions.