Lennard-Jones型晶格能的三角晶格优化：一种计算机辅助方法
Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method
摘要：
众所周知，任何Lennard-Jones型势能都必须具有由二维三角晶格给出的周期性基态。在本文中，我们描述了一个计算机辅助的方法，严格显示这种全球极小的结果之间的$2$维晶格一旦指数的潜力已被固定。将该方法应用于广泛使用的经典$(12,6)$ Lennard-Jones势，这是本文的主要结果。此外，本文还导出了三角格点最小时的逆密度（即余体积）的一个新的界，改进了[L. Bétermin和P. Zhang，\textit{Commun. Contemp. Math.}，17（2015），1450049]和[L. Bétermin，\textit{SIAM J. Math. Anal.}，48（2016），3236--3269]。同样的结果也被证明持有其他指数作为额外的例子和一个新的猜想暗示的任何参数的三角形格的全局最优性。
英文摘要：
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. Bétermin and P. Zhang, \textit{Commun. Contemp. Math.}, 17 (2015), 1450049] and [L. Bétermin, \textit{SIAM J. Math. 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.
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