On non-Gaussian potentials having triangular lattice as minimizer at any fixed density

A fundamental problem in the crystallization conjecture and lattice energy is whether the triangular lattice can be a minimizer of lattice energy at any fixed density for non-Gaussian potentials. By numerical computations, Bétermin-Petrache observed that there exists a positive, decreasing, and convex function such that the answer is positive. In this paper, we rigorously confirm this observation and provide a mathematical justification by constructing two global minimality results on classical theta functions and their derivatives. Furthermore we provide a broad class of functions with this property.