Superimposed particles in 1D ground states

TL;DR

This paper proves that for certain one-dimensional pair potentials, classical ground states at lower densities form a tower-lattice structure, characterized by towers of particles with heights differing by at most one, with a lattice constant of 1.

Contribution

It establishes the precise structure of ground states for a class of nonnegative, range-1 potentials in 1D, including cases with flat or cusp-like potentials.

Findings

Ground states are tower-lattices with heights differing by at most one.

The lattice constant of these ground states is exactly 1.

Results hold on finite intervals, rings, and the entire line.

Abstract

For a class of nonnegative, range-1 pair potentials in one dimensional continuous space we prove that any classical ground state of lower density >=1 is a tower-lattice, i.e., a lattice formed by towers of particles the heights of which can differ only by one, and the lattice constant is 1. The potential may be flat or may have a cusp at the origin, it can be continuous, but its derivative has a jump at 1. The result is valid on finite intervals or rings of integer length and on the whole line.