2023/02/10 by Senping Luo, Juncheng Wei, Luo, Senping +1
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2302.05042
openalex publication_date 2023/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let L =√((1)/(\Im(z)))(\mathbb Z⊕ z\mathbb Z) where z ∈ ℍ=\z= x+ i y \hboxor (x,y)∈ℂ: y>0\ be the two dimensional lattices with unit density. Assuming that α≥1, we prove that \alignedminL∑ℙ∈ L, |L|=1|ℙ|2 e- πα|ℙ|2 \endaligned is achieved at hexagonal lattice. More generally we prove that for α≥ 1 \alignedminL∑ℙ∈ L, |L|=1(|ℙ|2-\fracbα) e- πα|ℙ|2 \endaligned is achieved at hexagonal lattice for b≤(1)/(2π) and does not exist for b>(1)/(2π). As a consequence, we provide two classes of non-monotone potentials which lead to hexagonal crystallization among lattices. Our results partially answer some questions raised in \citeOreport, Bet2016, Bet2018, Bet2019AMP and extend the main results in \citeLW2022 on minima of difference of two theta functions.