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 $c$.

中文翻译：

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周期集的局部能量优化

我们研究了 $\mathbb{R}^n$ 中周期性点集的局部最优性，用于高斯核心模型中的能量最小化，即径向对势函数 $f_c(r) = e^{-cr}$ 与$c \gt 0$。通过为 $m$-周期集考虑合适的参数空间，我们可以在具有相同点密度的周期集族中局部严格地分析点集的能量。我们根据包含在集合中的加权球形 $2$-designs 推导出周期点集的特征是 $f_c$-critical 对于所有 $c$。特别是对于像 $\mathsf{D}^{+}_n$ 族这样的 $2$-周期集，我们获得了能量函数的 Hessian 表达式，允许在某些情况下证明 $f_c$-最优性。