Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method

It is well-known that any Lennard-Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among $2$-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical $(12,6)$ Lennard-Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in [L. B'etermin and P. Zhang, $\mathit{\text{Commun.}}\mathit{\text{Contemp.}}\mathit{\text{Math.}}$, 17 (2015), 1450049] and [L. B'etermin, $\mathit{\text{SIAM}}\mathit{\text{J.}}\mathit{\text{Math.}}\mathit{\text{Anal.}}$, 48 (2016), 3236–3269]. The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated.