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

Research output: Contribution to journalJournal articlepeer-review

  • Laurent Bétermin
  • Mircea Petrache

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.

Original languageEnglish
JournalAnalysis and Mathematical Physics
Volume9
Issue number4
Pages (from-to)2033–2073
ISSN1664-2368
DOIs
Publication statusPublished - 2019

    Research areas

  • Completely monotone functions, Laplace transform, Lattice energies, Lennard-Jones potentials, Theta functions, Triangular lattice

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