On Minima of Difference of Epstein Zeta Functions and Exact Solutions to Lennard-Jones Lattice Energy

2022/12/21 by Luo, Senping, Wei, Juncheng
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2212.10727

Abstract

Let ζ(s,z)=∑_(m,n)∈ℤ²\backslash\0\\frac(\Im(z))^s|mz+n|^2s be the Eisenstein series/Epstein Zeta function. Motivated by widely used Lennard-Jones potential \alignedV(|⋅|²):=4ε( (\fracσ|⋅|)¹²-(\fracσ|⋅|)⁶ ), \endaligned in physics, in this paper, we consider the following lattice minimization problem \alignedmin_z∈ℍ(ζ(6,z)-bζ(3,z)), b=(1)/(σ⁶) \endaligned and completely classify the minimizers for all b∈ \R. Our results resolve an open problem in Blanc-Lewin \citeBla2015, and a conjecture by Bétermin \citeBet2018. Furthermore, our method of proofs works for general minimization problem \alignedmin_z∈ℍ(ζ(s₁,z)-bζ(s₂,z)), s₁gt;s₂gt;1 \endaligned which corresponds to general Lennard-Jones potential \alignedV(|⋅|²):=4ε( (\fracσ|⋅|)^2s₁-(\fracσ|⋅|)^2s₂ ), s₁gt;s₂gt;1. \endaligned

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