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Local Energy Optimality of Periodic Sets. (arXiv:1802.02072v1 [math.MG])

来源于:arXiv
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~$c$. 查看全文>>