Linear Programming Bounds for Periodic Energy Problems

Abstract

We develop linear programming bounds for the energy of configurations in Euclidean space which are periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial interpolation problem. We construct sequences of such problems whose solutions would complete the Cohn-Kumar universal optimality conjecture for the hexagonal lattice. We solve the base cases and obtain novel results about the optimality of the hexagonal lattice among certain periodic configurations for a wide range of interactions, including inverse power laws and Gaussians.