论文标题
关于theta函数差异的最小值，并应用于六边形结晶
On minima of difference of theta functions and application to hexagonal crystallization
论文作者
论文摘要
令$ z = x+ iy \ in \ mathbb {h}：= \ {z = x+ i y \ in \ mathbb {c}：y> 0 \} $和$θ（α; z）= \ sum _ {（m，m，n）\ } | mz+n |^2} $是与晶格$ l = {\ mathbb z} \ oplus z {\ mathbb z} $关联的theta函数。在本文中，我们考虑以下最小化两个theta函数的差异问题\ begin {equination} \对齐\ nonumber \ min_ {\ mathbb {\ mathbb {h}} \ big（θ（θ（α; z; z）-βθ（z）-βθ（z）-βθ（2α; z）\ big）\ big）\ endAligned \ ene $ neque. （ - \ infty， +\ \ infty）$。 We prove that there is a critical value $β_c=\sqrt2$ (independent of $α$) such that if $β\leqβ_c$, the minimizer is $\frac{1}{2}+i\frac{\sqrt3}{2}$ (up to translation and rotation) which corresponds to the hexagonal lattice, and if $β>β_C$，最小化不存在。我们的结果部分回答了\ cite {Bet2016，Bet2018，Bet2020，Bet2019Amp}提出的一些问题，并在Yukawa的潜力下给出了六角形格的结晶方面的新证明。
Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ θ(α;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-α\frac{π}{y }|mz+n|^2}$ be the theta function associated with the lattice $L ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following minimization problem of difference of two theta functions \begin{equation}\aligned\nonumber \min_{ \mathbb{H} } \Big(θ(α; z)-βθ(2α; z)\Big) \endaligned\end{equation} where $α\geq 1$ and $ β\in (-\infty, +\infty)$. We prove that there is a critical value $β_c=\sqrt2$ (independent of $α$) such that if $β\leqβ_c$, the minimizer is $\frac{1}{2}+i\frac{\sqrt3}{2}$ (up to translation and rotation) which corresponds to the hexagonal lattice, and if $β>β_c$, the minimizer does not exist. Our result partially answers some questions raised in \cite{Bet2016, Bet2018, Bet2020, Bet2019AMP} and gives a new proof in the crystallization of hexagonal lattice under Yukawa potential.
