Lattice Energy Minimization for the Difference of Theta and Epstein Zeta Functions

Let , and be the lattice in . Let be the Theta function, and be the Epstein zeta function. Motivated by the widely used Buckingham potential in physics, in this paper, we explore the lattice minimization problem of for any . A key finding of this work is that the coefficient significantly influences the optimal lattice configuration, leading to three distinct phase transition patterns as varies from to : For , the optimal lattice undergoes a hexagonal wide rhombic square rectangular transition. For , the transition follows hexagonal narrow rhombic . For , an extended transition sequence appears: hexagonal wide rhombic square rectangular narrow rhombic . Notably, we identify a previously overlooked crystallization phenomenon: the narrow rhombic lattice, a special type of rhombic structure distinct from those reported by Luo and Wei, and by Bétermin. Our results provide partial answers to conjectures and open problems. Moreover, we establish complete minimization results for potentials formed by the difference of two quite different functions. We also derive a necessary condition for minimizers, given by (see (1.1)), which has broader applicability to noncompletely monotonic functions.