TY - JOUR

T1 - Optimal and non-optimal lattices for non-completely monotone interaction potentials

AU - Bétermin, Laurent

AU - Petrache, Mircea

PY - 2019

Y1 - 2019

N2 - We investigate the minimization of the energy per point E f 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 E f 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 E f 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 E f than the triangular one. Many open questions are also presented.

AB - We investigate the minimization of the energy per point E f 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 E f 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 E f 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 E f than the triangular one. Many open questions are also presented.

KW - Completely monotone functions

KW - Laplace transform

KW - Lattice energies

KW - Lennard-Jones potentials

KW - Theta functions

KW - Triangular lattice

UR - http://www.scopus.com/inward/record.url?scp=85064686267&partnerID=8YFLogxK

U2 - 10.1007/s13324-019-00299-6

DO - 10.1007/s13324-019-00299-6

M3 - Journal article

AN - SCOPUS:85064686267

VL - 9

SP - 2033

EP - 2073

JO - Analysis and Mathematical Physics

JF - Analysis and Mathematical Physics

SN - 1664-2368

IS - 4

ER -