On Minima of Difference of Epstein Zeta Functions and Exact Solutions to Lennard-Jones Lattice Energy

Let $\zeta (s,z)={\sum }_{(m,n)\in {\mathbb{Z}}^2\mathrm{\backslash }\{0\}}\frac{(\mathrm{\Im }(z){)}^s}{\mathrm{\mid }mz+n{\mathrm{\mid }}^{2s}}$ be the Eisenstein series/Epstein Zeta function. Motivated by widely used Lennard-Jones potential \begin{equation}\aligned\nonumber $\mathcal{V}$(|\cdot|^2):=4\varepsilon\Big( (\fracσ{|\cdot|})^{12}-(\fracσ{|\cdot|})^{6} \Big), \endaligned\end{equation} in physics, in this paper, we consider the following lattice minimization problem \begin{equation}\aligned\nonumber \min_{z\in$\mathbb{H}$}\Big(ζ(6,z)-bζ(3,z)\Big), ;;b=\frac{1}{σ^6} \endaligned\end{equation} and completely classify the minimizers for all $b\in \mathbb{R}$. Our results resolve an open problem in Blanc-Lewin \cite{Bla2015}, and a conjecture by Bétermin \cite{Bet2018}. Furthermore, our method of proofs works for general minimization problem \begin{equation}\aligned\nonumber \min_{z\in$\mathbb{H}$}\Big(ζ(s_1,z)-bζ(s_2,z)\Big), ;;s_1>s_2>1 \endaligned\end{equation} which corresponds to general Lennard-Jones potential \begin{equation}\aligned\nonumber $\mathcal{V}$(|\cdot|^2):=4\varepsilon\Big( (\fracσ{|\cdot|})^{2s_1}-(\fracσ{|\cdot|})^{2s_2} \Big),;;s_1>s_2>1. \endaligned\end{equation}