Local Energy Optimality of Periodic Sets - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2018

Local Energy Optimality of Periodic Sets

Résumé

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 $f_c(r)=e^{-c r}$ with $c>0$. 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 periodic sets for all sufficiently large~$c$.

Dates et versions

hal-01943098 , version 1 (03-12-2018)

Identifiants

Citer

Renaud Coulangeon, Achill Schürmann. Local Energy Optimality of Periodic Sets. 2018. ⟨hal-01943098⟩

Collections

CNRS IMB INSMI
12 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More