On minima of difference of theta functions and application to hexagonal crystallization
On minima of difference of theta functions and application to hexagonal crystallization
复制标题
θ函数差极小值及其在六方晶系结晶中的应用
作者：
S. Luo;Juncheng Wei
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(α;z)=∑(m,n)∈Z2e-απy|mz+n|2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ heta (alpha ;z)=sum _{(m,n)in mathbb {Z}^2} e^{-alpha frac{pi }{y }|mz+n|^2}$$end{document} be the theta function associated with the lattice L=1Im(z)Z⊕zZdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$L =sqrt{frac{1}{{ ext {Im}}(z)}}left( {mathbb Z}oplus z{mathbb Z}
ight) $$end{document}. In this paper we consider the following minimization problem of difference of two theta functions minHθ(α;z)-βθ(2α;z)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} min _{ mathbb {H} } left( heta (alpha ; z)-eta heta (2alpha ; z)
ight) end{aligned}$$end{document}where α≥1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$alpha ge 1$$end{document} and β∈(-∞,+∞)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ eta in (-infty , +infty )$$end{document}. We prove that there is a critical value βc=2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$eta _c=sqrt{2}$$end{document} (independent of αdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$alpha $$end{document}) such that if β≤βcdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$eta le eta _c$$end{document}, the minimizer is 12+i32documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$frac{1}{2}+ifrac{sqrt{3}}{2}$$end{document} (up to translation and rotation) which corresponds to the hexagonal lattice, and if β>βcdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$eta >eta _c$$end{document}, the minimizer does not exist. Our result partially answers some questions raised in Bétermin (SIAM J Math Anal 48(5):3236–269, 2016), Bétermin (Nonlinearity 31(9):3973–4005, 2018), Bétermin et al. (Models Methods Appl Sci 31(2):293–325, 2021) and Bétermin and Petrache (Anal Math Phys 9(4):2033–2073, 2019) and gives a new proof in the hexagonal crystallization among lattices under Yukawa potential.
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(α;z)=∑(m,n)∈Z2e-απy|mz+n|2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ heta (alpha ;z)=sum _{(m,n)in mathbb {Z}^2} e^{-alpha frac{pi }{y }|mz+n|^2}$$end{document} be the theta function associated with the lattice L=1Im(z)Z⊕zZdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$L =sqrt{frac{1}{{ ext {Im}}(z)}}left( {mathbb Z}oplus z{mathbb Z}
ight) $$end{document}. In this paper we consider the following minimization problem of difference of two theta functions minHθ(α;z)-βθ(2α;z)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} min _{ mathbb {H} } left( heta (alpha ; z)-eta heta (2alpha ; z)
ight) end{aligned}$$end{document}where α≥1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$alpha ge 1$$end{document} and β∈(-∞,+∞)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ eta in (-infty , +infty )$$end{document}. We prove that there is a critical value βc=2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$eta _c=sqrt{2}$$end{document} (independent of αdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$alpha $$end{document}) such that if β≤βcdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$eta le eta _c$$end{document}, the minimizer is 12+i32documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$frac{1}{2}+ifrac{sqrt{3}}{2}$$end{document} (up to translation and rotation) which corresponds to the hexagonal lattice, and if β>βcdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$eta >eta _c$$end{document}, the minimizer does not exist. Our result partially answers some questions raised in Bétermin (SIAM J Math Anal 48(5):3236–269, 2016), Bétermin (Nonlinearity 31(9):3973–4005, 2018), Bétermin et al. (Models Methods Appl Sci 31(2):293–325, 2021) and Bétermin and Petrache (Anal Math Phys 9(4):2033–2073, 2019) and gives a new proof in the hexagonal crystallization among lattices under Yukawa potential.
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