Triangular-lattice crystallization for the 2D Lennard-Jones potential
In plain words
In the plane, particles with Lennard-Jones forces should arrange into the triangular lattice. Existing proofs work only for forces with a much narrower attractive well.
Precise statement
For the Lennard-Jones pair potential in $R^2$, prove that $N$-particle minimizers of $E_N$ converge (energy per particle and structure away from an $o(N)$ boundary layer) to the triangular lattice with the spacing that minimizes the lattice energy. Theil's 2006 proof and its extensions (E, Li 2009) need a narrow potential well and steep core, which excludes Lennard-Jones.
What would settle it
An extension of the 2D crystallization proof to the Lennard-Jones potential itself.
Status in the literature
Optimality of the triangular lattice for Lennard-Jones is known only within the restricted class of Bravais lattices and for some densities.