On Minima of Sum of Theta Functions and Application to Mueller–Ho Conjecture
On Minima of Sum of Theta Functions and Application to Mueller–Ho Conjecture
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论Theta函数和的极小值及其在Mueller-Ho猜想中的应用
DOI：
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发表时间：
2020
影响因子：
2.5
通讯作者：
Juncheng Wei
中科院分区：
文献类型：
--
作者：
S. Luo;Juncheng Wei
让z=x+iy∈H：={z=x+iy∈C:y>；0}documentclass[12pt]{minimal}usepackage{amsath}usepackage{wa ysym}usepackage{amssymb}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}egin{Document}$$z=x+iy in{mathbb{H}：={z=x+i in{mathbb{C}}：y>；0}$$end{Document}和θ(S;z)=∑(m，n)∈Z2e-Sπy|mz+n|2DocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$heta(S；Z)=Sum_{(m，n)in{mathbb{Z}^2}e^{-S Frac{pi}{y}|mz+n|^2}$$End{Document}是与格Λ=Z⊕zZ DocumentClass[12pt]{Minimum}UsPackage{amsath}UsPack{waysym}UsPack{amsFonts}UsPack{amssymb}Usepackage{upsbsy}UsPack{upgreek}设置长度{oddsidemargin}{-69pt}egin$Lambda={{mathbb{Z}Oplus z{mathbb{Z}$$End{Document}相关联的theta函数。在本文中，我们考虑最小化问题0.1minHθ2;z+12+ρθ(1;z)，ρ∈[0，∞]，Minhθ1;z+12+ρθ(2;z)，ρ∈[0，∞)，DocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{amssymts}usepackage{amssymts}usepackage{amssymb}usepackage{mathrsfs}usepackage{oddsidemargin}{-69pt}在{Document}$egin中{;}ρ∈_{{mathbb{H}heta Left(2；c{fraz+1}{2})
夜)+
Ho Heta(1；z)，；；
HO in[0，inty)，\min_{{mathbb{H}Heta Left(1；frac{z+1}{2}
夜)+
Ho Heta(2；z)，；；
Ho in[0，inty)，end{aligned}end{aligned}$$end{Document}，其中参数ρ∈[0，∞)DocumentClass[12pt]{Minimal}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$
Ho in[0，inty)$$end{Document}表示两个交织在一起的格的竞争，而参数S=1，2Documentclass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{upgreek}setlong{oddsidemargin}{-69pt}的具体选择是由我们提出的物理模型决定的，它可以推广到本文提出的策略和方法。我们发现，作为ρ文档类[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsFonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$
Ho$$end{Document}变化，最优网格允许一种新的模式：它们从矩形移动(长边和短边的比例从3DocumentClass[12pt]{Minimum}UsPack{amsath}UsPack{wa ysym}UsPack{amsFonts}UsPackage{amsbsy}UsPackage{amsbsy}UsPackage{mathsfs}UsPackage{upgreek}setLength{oddsidemargin}{-69pt}egin{DocumentClass$$Sqrt{3}$end{Document}连续变化到1)，正方形和菱形(角度从π/2Documentclass[12pt]{Minimum}usepackage{amsath}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$pi/2$$end{Document}中的$$pi/2$$end{Document}到π/3DocumentClass[12pt]{最小}usepackage{amsmax}usepackage{amssymb}usepackage{amssymb}usepackage{mathsbsy}usepackage{upgresek}{setododemin in{Document}$$69pi/$end}；几何上，直到一个不变群(经典模群的一个子群)，它们在一条特殊的曲线上连续移动；此外，存在ρ文档类[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amssymb}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$
Ho$$end{Document}使得最优格始终是正方形格。这是关于参数为ρ的theta函数的极小化问题的第一个新颖的、也是完整的结果文档类[12pt]{Minimum}usepackage{amsath}usepackage{waysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{文档}$
Ho$$end{文档}。这与单一theta函数(ρ=∞文档类[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$
HO=INTY$$END{DOCUMENT}案例)，其中以六角点阵为主。因此，我们对Mueller和Ho(Phys Rev Lett 88：180403,2002年)猜想(并得到数值和实验验证)的玻色-爱因斯坦凝聚竞争系统中涡旋的最优晶格排列给出了部分和肯定的回答；这是Mueller-Ho猜想的第一个进展。最后，我们指出我们提出的策略和方法是一般的，并且可以用于更一般的格上的极小化问题。
Let z=x+iy∈H:={z=x+iy∈C:y>0}documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$z=x+iy in {mathbb {H}}:={z= x+ i yin {mathbb {C}}: y>0}$$end{document} and θ(s;z)=∑(m,n)∈Z2e-sπy|mz+n|2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ heta (s;z)=sum _{(m,n)in {mathbb {Z}}^2 } e^{-s frac{pi }{y }|mz+n|^2}$$end{document} be the theta function associated with the lattice Λ=Z⊕zZdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Lambda ={{mathbb {Z}}}oplus z{{mathbb {Z}}}$$end{document}. In this paper we consider minimization problems 0.1minHθ2;z+12+ρθ(1;z),ρ∈[0,∞),minHθ1;z+12+ρθ(2;z),ρ∈[0,∞),documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} egin{aligned} min _{ {mathbb {H}} } heta left( 2;frac{z+1}{2}
ight) +
ho heta (1;z),;;
ho in [0,infty ), \ min _{ {mathbb {H}} } heta left( 1; frac{z+1}{2}
ight) +
ho heta (2; z),;;
ho in [0,infty ), end{aligned} end{aligned}$$end{document}where the parameter ρ∈[0,∞)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$
ho in [0,infty )$$end{document} represents the competition of two intertwining lattices, and the particular selection of the parameters s=1,2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$s=1,2$$end{document} is determined by the physical model, which can be generalized by our strategy and method proposed here. We find that as ρdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$
ho $$end{document} varies, the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short sides changes from 3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$sqrt{3}$$end{document} to 1 continuously), square and rhombus (the angle changes from π/2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$pi /2$$end{document} to π/3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$pi /3$$end{document} continuously) to hexagonal continuously; geometrically, up to an invariant group (a subgroup of the classical modular group), they move continuously on a special curve; furthermore, there exists a closed interval of ρdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$
ho $$end{document} such that the optimal lattices is always a square lattice. This is the first, novel and also the complete result on the minimizer problem for theta functions with parameter ρdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$
ho $$end{document}. This is in sharp contrast to optimal lattice shapes for a single theta function (ρ=∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$
ho =infty $$end{document} case), for which the hexagonal lattice prevails. As a consequence, we give a partial and positive answer to optimal lattice arrangements of vortices in competing systems of Bose–Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller and Ho (Phys Rev Lett 88:180403, 2002); this is the first progress on the Mueller–Ho conjecture. Lastly, we mention that the strategy and method we propose here is general, and can be used in much more general minimization problems on the lattices.
来自两种相互作用系统的非六方晶格
影响因子：
2
作者：
Luo, Senping;Ren, Xiaofeng;Wei, Juncheng
通讯作者：
Wei, Juncheng