Minimal Soft Lattice Theta Functions

We study the minimality properties of a new type of “soft” theta functions. For a lattice $L\subset {\mathbb{R}}^d$, a $L$-periodic distribution of mass ${\mu }_L$ and an other mass ${\nu }_z$ centred at $z\in {\mathbb{R}}^d$, we define, for all scaling parameter $\alpha >0$, 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-centred-cubic and face-centred-cubic lattices.