On lattice hexagonal crystallization for non-monotone potentials

Let $L =\sqrt{\frac{1}{\Im(z)}}\Big({\mathbb Z}\oplus z{\mathbb Z}\Big)$ where $z \in \mathbb{H}=\{z= x+ i y\;\hbox{or}\;(x,y)\in\mathbb{C}: y>0\}$ be the two dimensional lattices with unit density. Assuming that $\alpha\geq1$, we prove that \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}|\mathbb{P}|^2 e^{- \pi\alpha|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice. More generally we prove that for $\alpha \geq 1$ \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}(|\mathbb{P}|^2-\frac{b}{\alpha}) e^{- \pi\alpha|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice for $b\leq\frac{1}{2\pi}$ and does not exist for $b>\frac{1}{2\pi}$. 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 \cite{Oreport, Bet2016, Bet2018, Bet2019AMP} and extend the main results in \cite{LW2022} on minima of difference of two theta functions.