Fine structure of a one-dimensional discrete point system

Abstract

We consider a system of N points x 1 < ... < x N on a segment of the real line. An ideal system (crystal) is a system where all distances between neighbors are the same. Deviation from idealness is characterized by a system of finite differences ∇ 1 = x x+1 − x i , ∇ +1 = ∇ +1 − ∇ , for all possible i and k. We find asymptotic estimates as N → ∞, k→∞, for a system of points minimizing the potential energy of a Coulomb system in an external field.