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

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We investigate the minimization of the energy per point $$E_f$$Ef among d-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function $$f(|x|^2)$$f(|x|2). We formulate… Click to show full abstract

We investigate the minimization of the energy per point $$E_f$$Ef among d-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function $$f(|x|^2)$$f(|x|2). We formulate criteria for minimality and non-minimality of some lattices for $$E_f$$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 $$E_f$$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 $$E_f$$Ef than the triangular one. Many open questions are also presented.

Keywords: lattices non; optimal non; non optimal; optimal lattices; non completely; completely monotone

Journal Title: Analysis and Mathematical Physics
Year Published: 2019

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