On minima of differences of Epstein zeta functions and exact solutions to Lennard–Jones lattice energy

Abstract

Assume that $s_1\in [5,7]$ and $s_2,s\in [2,4]$. Let $\zeta$ and $\theta$ be the Epstein zeta and theta functions, respectively, associated with a two-dimensional lattice, and let $z$ be a general parameter associated with a 2d lattice. We completely classify the minimizers of $\zeta (s_1,z)-b\zeta (s_2,z)-a\zeta (s,z)$ and $\zeta (s_1,z)-b\zeta (s_2,z)-a\theta (1,z)$ for $a,b\ge 0$. A global picture of the geometric and algebraic aspects of energy minimization problems is presented, highlighting a distinct pattern that contrasts with the celebrated theorem by Montgomery (1988). As a consequence, we give a complete classification of the lattice minimization problem with the widely used Lennard–Jones potential $(\sigma /r{)}^{12}-(\sigma /r{)}^6$. Our results settle down several open problems/conjectures proposed in multiple directions. We completely resolve the conjecture by Bétermin (2018), providing the explicit and analytical thresholds in the classification. We provide positive answers to an open problem of Blanc–Lewin (2015). Our results also clarify the stability of the potential in crystallization among lattices (Bétermin–Petrache (2019), Cohn–Kumar (2009)) and provide answers to when a square lattice minimizes the lattice energy.