Minimal Soft Lattice Theta Functions

Abstract

AbstractWe 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.