2022/03/01 by Senping Luo, Juncheng Wei, Luo, Senping +1
Mathematics · Physics and Astronomy · #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #math-ph #math.CA #math.MP
paper · pdf · doi:10.48550/arxiv.2203.00264
29 pages; comments welcome
arxiv created 2022/03/01 · openalex publication_date 2022/03/01 · arxiv updated 2022/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let z=x+iy ∈ ℍ:=\z= x+ i y∈ℂ: y>0\ and θ(α;z)=∑(m,n)∈ℤ2 e-α(π)/(y )|mz+n|2 be the theta function associated with the lattice L =\mathbb Z⊕ z\mathbb Z. In this paper we consider the following minimization problem of difference of two theta functions \alignedmin ℍ (θ(α; z)-βθ(2α; z)) \endaligned where α≥ 1 and β∈ (-∞, +∞). We prove that there is a critical value βc=√2 (independent of α) such that if β≤βc, the minimizer is (1)/(2)+i(√3)/(2) (up to translation and rotation) which corresponds to the hexagonal lattice, and if β>βc, the minimizer does not exist. Our result partially answers some questions raised in \citeBet2016, Bet2018, Bet2020, Bet2019AMP and gives a new proof in the crystallization of hexagonal lattice under Yukawa potential.