论文标题
关于theta函数和穆勒的最小值
On minima of sum of theta functions and Mueller-Ho Conjecture
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（s; z; z）= \ sum _ {（m，m，n）\ } | mz+n |^2} $是与晶格$λ= {\ 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问题$$ \ min_ {\ mathbb {h}}θ（2; \ frac {z+1} {2} {2} {2} {2}）+ρθ（1; z），\; \; \;ρ\;ρ\;ρ; own [0，\ infty），\ infty），$ $ $ $ \ $ $ \ $ \ math \ math { \ frac {z+1} {2}）+ρθ（2; z），\; \;ρ\ in [0，\ infty），$$ 其中参数$ρ\在[0，\ infty）$中代表两个相互交织的晶格的竞争。我们发现，随着$ρ$的变化，最佳晶格会允许一种新颖的模式：它们从矩形（长边和短边的比例从$ \ \ sqrt3 $变化为1），正方形，菱形（角度从$π/2 $变为$π/3 $）到六边形;此外，存在$ρ$的闭合间隔，使得最佳晶格始终是正方形的晶格。这与单theta功能的最佳晶格形状形成鲜明对比（$ρ= \ infty $ case），六边形晶格占上风。结果，我们通过Mueller-Ho \ cite {Mue2002}猜想（以及数值和实验验证）的Bose-Einstein冷凝物竞争系统中的最佳晶格布置。
Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ θ(s;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-s \frac{π}{y }|mz+n|^2}$ be the theta function associated with the lattice $Λ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following pair of minimization problems $$ \min_{ \mathbb{H} } θ(2;\frac{z+1}{2})+ρθ(1;z),\;\;ρ\in[0,\infty),$$ $$ \min_{ \mathbb{H} } θ(1; \frac{z+1}{2})+ρθ(2; z),\;\;ρ\in[0,\infty),$$ where the parameter $ρ\in[0,\infty)$ represents the competition of two intertwining lattices. We find that as $ρ$ varies the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short side changes from $\sqrt3$ to 1), square, rhombus (the angle changes from $π/2$ to $π/3$) to hexagonal; furthermore, there exists a closed interval of $ρ$ such that the optimal lattices is always square lattice. This is in sharp contrast to optimal lattice shapes for single theta function ($ρ=\infty$ case), for which the hexagonal lattice prevails. As a consequence, we give a partial answer to optimal lattice arrangements of vortices in competing systems of Bose-Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller-Ho \cite{Mue2002}.
