We study the minimality properties of a new type of “soft” theta functions. For a lattice $$L\subset {\mathbb {R}}^d$$ L ⊂ R d , an L -periodic distribution of mass $$\mu _L$$ μ L , and another mass $$\nu _z$$ ν z centered at $$z\in {\mathbb {R}}^d$$ z ∈ R d , we define, for all scaling parameters $$\alpha >0$$ α > 0 , the translated lattice theta function $$\theta _{\mu _L+\nu _z}(\alpha )$$ θ μ L + ν z ( α ) as the Gaussian interaction energy between $$\nu _z$$ ν z and $$\mu _L$$ μ L . We show that any strict local or global minimality result that is true in the point case $$\mu =\nu =\delta _0$$ μ = ν = δ 0 also holds for $$L\mapsto \theta _{\mu _L+\nu _0}(\alpha )$$ L ↦ θ μ L + ν 0 ( α ) and $$z\mapsto \theta _{\mu _L+\nu _z}(\alpha )$$ z ↦ θ μ L + ν z ( α ) when the measures are radially symmetric with respect to the points of $$L\cup \{z\}$$ L ∪ { z } and sufficiently rescaled around them (i.e., at a low scale). The minimality at all scales is also proved when the radially symmetric measures are generated by a completely monotone kernel. The method is based on a generalized Jacobi transformation formula, some standard integral representations for lattice energies, and an approximation argument. Furthermore, for the honeycomb lattice $${\mathsf {H}}$$ H , the center of any primitive honeycomb is shown to minimize $$z\mapsto \theta _{\mu _{{\mathsf {H}}}+\nu _z}(\alpha )$$ z ↦ θ μ H + ν z ( α ) , and many applications are stated for other particular physically relevant lattices including the triangular, square, cubic, orthorhombic, body-centered-cubic, and face-centered-cubic lattices.

中文翻译：

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最小软晶格 Theta 函数

我们研究了一种新型“软”θ 函数的极小特性。对于格 $$L\subset {\mathbb {R}}^d$$ L ⊂ R d ，质量 $$\mu _L$$ μ L 的 L 周期分布​​和另一个质量 $$\nu _z$ $ ν z 以 $$z\in {\mathbb {R}}^d$$ z ∈ R d 为中心，我们定义，对于所有缩放参数 $$\alpha >0$$ α > 0 ，平移的点阵 theta 函数$$\theta _{\mu _L+\nu _z}(\alpha )$$ θ μ L + ν z ( α ) 作为 $$\nu _z$$ ν z 和 $$\mu _L$ 之间的高斯相互作用能$μL。我们表明，在 $$\mu =\nu =\delta _0$$ μ = ν = δ 0 点情况下为真的任何严格的局部或全局极小结果也适用于 $$L\mapsto \theta _{\mu _L+\nu _0}(\alpha )$$ L ↦ θ μ L + ν 0 ( α ) 和 $$z\mapsto \theta _{\mu _L+\nu _z}(\alpha )$$ z ↦ θ μ L + ν z ( α ) 当测量相对于 $$L\cup \{z\}$$ L ∪ { z } 的点径向对称并在它们周围充分重新缩放（即，在低比例下）。当径向对称度量由完全单调的内核生成时，也证明了所有尺度的最小值。该方法基于广义雅可比变换公式、格能的一些标准积分表示和近似参数。此外，对于蜂窝晶格 $${\mathsf {H}}$$ H ，