Optimal and non-optimal lattices for non-completely monotone interaction potentials
No Thumbnail Available
Date
2019
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
We investigate the minimization of the energy per point Ef among d-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function f(|x|2). We formulate criteria for minimality and non-minimality of some lattices for Ef at fixed scale based on the sign of the inverse Laplace transform of f when f is a superposition of exponentials, beyond the class of completely monotone functions. We also construct a family of non-completely monotone functions having the triangular lattice as the unique minimizer of Ef at any scale. For Lennard-Jones type potentials, we reduce the minimization problem among all Bravais lattices to a minimization over the smaller space of unit-density lattices and we establish a link to the maximum kissing problem. New numerical evidence for the optimality of particular lattices for all the exponents are also given. We finally design one-well potentials f such that the square lattice has lower energy Ef than the triangular one. Many open questions are also presented.
Description
Keywords
Lattice energies, Theta functions, Lennard-Jones potentials, Triangular lattice, Completely monotone functions, Laplace transform, Primary 74G65, Secondary 82B20, 11F27