论文标题
关于爱泼斯坦Zeta功能差异的最小值和Lennard-Jones晶格能的精确解决方案
On Minima of Difference of Epstein Zeta Functions and Exact Solutions to Lennard-Jones Lattice Energy
论文作者
论文摘要
令$ζ（s，z）= \ sum _ {（m，n）\ in \ mathbb {z}^2 \ backslash \ {0 \}} \ frac {（\ im（z））^s} {| mz+n |^{2s}} $ eisensteinsein/eSENSTEINSEATAINACTAINAUCTANICTAINCTAINCAL。受到广泛使用的Lennard-Jones潜力的动机 \ begin {equination} \对齐\ nonumber \ nonumber \ Mathcal {v}（| \ cdot |^2）：= 4 \ varepsilon \ big（（（（（（\fracσ{| \ cdot |}）） \ endaligned \ end {equation} 在物理学中，在本文中，我们考虑以下晶格最小化问题\ begin {qore} \对齐\ nonumber \ min_ {z {z \ in \ mathbb {h}} \ big（ζ（ζ（ζ（6，z）） \ endaligned \ end {equation} 并完全对所有$ b \ in \ r $ in \ r $中的最小化器进行分类。我们的结果解决了Blanc-Lewin \ cite {Bla2015}的开放问题，以及Bétermin\ cite {Bet2018}的猜想。此外，我们的证明方法适用于一般最小化问题\ begin {equination} \对齐\ nonumber \ min_ {z {z \ in \ mathbb {h}} \ big（ζ（ζ（ζ（s_1，z）-bζ（s_1，z）-Bζ（s_2，z），s_2，z）对应于一般的Lennard-Jones潜力 \ begin {equination} \ aligned \ nonumber \ mathcal {v}（| \ cdot |^2）：= 4 \ varepsilon \ big（（（\fracσ{| \ cdot |}） \ big），\; \; s_1> s_2> 1。 \ endaligned \ end {equation}
Let $ζ(s,z)=\sum_{(m,n)\in\mathbb{Z}^2\backslash\{0\}}\frac{(\Im(z))^s}{|mz+n|^{2s}}$ be the Eisenstein series/Epstein Zeta function. Motivated by widely used Lennard-Jones potential \begin{equation}\aligned\nonumber \mathcal{V}(|\cdot|^2):=4\varepsilon\Big( (\fracσ{|\cdot|})^{12}-(\fracσ{|\cdot|})^{6} \Big), \endaligned\end{equation} in physics, in this paper, we consider the following lattice minimization problem \begin{equation}\aligned\nonumber \min_{z\in\mathbb{H}}\Big(ζ(6,z)-bζ(3,z)\Big), \;\;b=\frac{1}{σ^6} \endaligned\end{equation} and completely classify the minimizers for all $b\in \R$. Our results resolve an open problem in Blanc-Lewin \cite{Bla2015}, and a conjecture by Bétermin \cite{Bet2018}. Furthermore, our method of proofs works for general minimization problem \begin{equation}\aligned\nonumber \min_{z\in\mathbb{H}}\Big(ζ(s_1,z)-bζ(s_2,z)\Big), \;\;s_1>s_2>1 \endaligned\end{equation} which corresponds to general Lennard-Jones potential \begin{equation}\aligned\nonumber \mathcal{V}(|\cdot|^2):=4\varepsilon\Big( (\fracσ{|\cdot|})^{2s_1}-(\fracσ{|\cdot|})^{2s_2} \Big),\;\;s_1>s_2>1. \endaligned\end{equation}
