On minima of sum of theta functions and Mueller-Ho Conjecture

Abstract

Let $z=x+iy\in \mathbb{H}:=\{z=x+iy\in \mathbb{C}:y>0\}$ and $\theta (s;z)={\sum }_{(m,n)\in {\mathbb{Z}}^2}{e}^{-s\frac{\pi }{y}|mz+n{|}^2}$ be the theta function associated with the lattice $\Lambda =\mathbb{Z}\oplus z\mathbb{Z}$. In this paper we consider the following pair of minimization problems $$\underset{\mathbb{H}}{\min }\theta (2;\frac{z+1}{2})+\rho \theta (1;z),\rho \in [0,\infty ),$$ $$\underset{\mathbb{H}}{\min }\theta (1;\frac{z+1}{2})+\rho \theta (2;z),\rho \in [0,\infty ),$$ where the parameter $\rho \in [0,\infty )$ represents the competition of two intertwining lattices. We find that as $\rho$ varies the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short side changes from $\sqrt{3}$ to 1), square, rhombus (the angle changes from $\pi /2$ to $\pi /3$) to hexagonal; furthermore, there exists a closed interval of $\rho$ such that the optimal lattices is always square lattice. This is in sharp contrast to optimal lattice shapes for single theta function ($\rho =\infty$ case), for which the hexagonal lattice prevails. As a consequence, we give a partial answer to optimal lattice arrangements of vortices in competing systems of Bose-Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller-Ho \cite{Mue2002}.

42 pages; comments welcome

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