On minima of difference of theta functions and application to hexagonal crystallization

Abstract

Let L = 1 Im(z) Z ⊕ zZ where z ∈ H = {z = x + iy or (x, y) ∈ C : y > 0} be the two dimensional lattices with unit density. Assuming that α ≥ 1, we prove that min L P∈L,|L|=1|P| 2 e −πα|P| 2 is achieved at hexagonal lattice. More generally we prove that for α ≥ 1 minis 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 [1,7,10,14] and extend the main results in [34] on minima of difference of two theta functions.