Local energy optimality of periodic sets

We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r) = e^{-cr}$ with $c \gt 0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$‑designs contained in the set. Especially for $2$‑periodic sets like the family $\mathsf{D}^{+}_n$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$-optimality in certain cases. For odd integers $n \geq 9$ we can hereby in particular show that $\mathsf{D}^{+}_n$ is locally $f_c$-optimal among $2$‑periodic sets for all sufficiently large $c$.

2010 Mathematics Subject Classification

11Hxx, 52Cxx, 82Bxx

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Both authors were supported by the Erwin-Schrödinger-Institute (ESI) during a stay in fall 2014 for the program on Minimal Energy Point Sets, Lattices and Designs.

The second author gratefully acknowledges support by DFG grant SCHU 1503/7-1.

Received 18 October 2018

Accepted 23 February 2021

Published 15 July 2021