Minimal Soft Lattice Theta Functions
Tóm tắt
We study the minimality properties of a new type of “soft” theta functions. For a lattice
$$L\subset {\mathbb {R}}^d$$
, an L-periodic distribution of mass
$$\mu _L$$
, and another mass
$$\nu _z$$
centered at
$$z\in {\mathbb {R}}^d$$
, we define, for all scaling parameters
$$\alpha >0$$
, the translated lattice theta function
$$\theta _{\mu _L+\nu _z}(\alpha )$$
as the Gaussian interaction energy between
$$\nu _z$$
and
$$\mu _L$$
. We show that any strict local or global minimality result that is true in the point case
$$\mu =\nu =\delta _0$$
also holds for
$$L\mapsto \theta _{\mu _L+\nu _0}(\alpha )$$
and
$$z\mapsto \theta _{\mu _L+\nu _z}(\alpha )$$
when the measures are radially symmetric with respect to the points of
$$L\cup \{z\}$$
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
$${\mathsf {H}}$$
, the center of any primitive honeycomb is shown to minimize
$$z\mapsto \theta _{\mu _{{\mathsf {H}}}+\nu _z}(\alpha )$$
, 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.
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