周期集的局部能量最优性
Local Energy Optimality of Periodic Sets
AI总结：
本文研究了在高斯核心模型中周期点集的能量最小化局部最优性，通过分析参数空间和周期集的点密度，推导出周期点集在不同参数下的最优条件，特别证明了对于奇数整数n≥9，D^n_+集在足够大的c值下是局部最优的。
摘要：
我们研究了在R^n中周期点集在高斯核心模型下的局部最优性，即对于径向对势函数f_c(r)=e^{-c r} (c>0)。通过考虑适合的参数空间对于m-周期集，我们可以局部严格分析点集的能量，在具有相同点密度的周期集家族内。我们推导出周期点集对所有c而言为f_c-临界点的特征，这涉及到包含在集合中的加权球面2设计。特别是对于2-周期集如D^n_+家族，我们得到能量函数的海森矩阵表达式，允许在某些情况下认证f_c-最优性。对于奇数整数n≥9，我们特别证明D^n_+在足够大的c值下在周期集内是局部最优的。
英文摘要：
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~$c$.
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