A Complete Proof of Mueller--Ho Conjecture

Abstract

For $s>0$, and $z\in\mathbb H=\{z=x+iy:y>0\}$, define $\theta(s;z)=\sum_{m,n\in\mathbb Z}e^{-\pi s\big(m^2y+\frac{(mx-n)^2}{y}\big)}$ and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-\pi \big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2\pi(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respectively. We prove that, up to the modular group, there exist three thresholds $0<\alpha_a<\alpha_b<\alpha_c<1$ such that \begin{equation}\aligned\nonumber \Minima_{z\in\mathbb H,\,(a,b)\in\mathbb R^2}\Big(\theta(1;z)+\alpha J(z;a,b)\Big)= \begin{cases} \;\big(e^{i{\pi}/{3}};1/3,1/3\big),&\hbox{if}\;\; \alpha\in[0,\alpha_a),\\ \;\big(e^{i{\pi}/{3}};1/3,1/3\big)\;\hbox{or}\;\big(e^{i{\varphi_{\alpha_a}}};1/2,1/2\big),&\hbox{if}\;\; \alpha=\alpha_a,\\ \;\big(e^{i{\varphi_\alpha}};1/2,1/2\big),\;\varphi_\alpha\in(\varphi_{\alpha_a},\frac{\pi}{2}),\;\;&\hbox{if}\;\; \alpha\in(\alpha_a,\alpha_b),\\ \;\big(i;1/2,1/2\big),\;&\hbox{if}\;\; \alpha\in[\alpha_b,\alpha_c],\\ \;\big(iy_\alpha;1/2,1/2\big),\; y_\alpha\in(1,\sqrt3],\;\;y_1=\sqrt3,\;\;&\hbox{if}\;\; \alpha\in(\alpha_c,1], \end{cases} \endaligned\end{equation} where the complex variable $z$ and vector $(a,b)$, represent the total vortex shapes and relative positions of the two-component Bose gas, respectively. Consequently, this completely proves the conjecture of Mueller and Ho~\cite{Mue2002} (2002) arising in Bose--Einstein condensates.