2018/09/03 by Laurent Bétermin, Bétermin, Laurent
Mathematics · Physics and Astronomy · #74G65 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Optimization and Control (math.OC) #math-ph #math.MP #math.OC #msc:74G65
paper · pdf · doi:10.48550/arxiv.1809.00473
21 pages, 4 figures. Accepted manuscript. To appear in Constructive Approximation
arxiv created 2019/11/12 · arxiv updated 2019/11/13
We study the minimality properties of a new type of "soft" theta functions. For a lattice L⊂ ℝd, a L-periodic distribution of mass μL and an other mass νz centred at z∈ ℝd, we define, for all scaling parameter α>0, the translated lattice theta function θμL+νz(α) as the Gaussian interaction energy between νz and μL. We show that any strict local or global minimality result that is true in the point case μ=ν=δ0 also holds for L↦ θμL+ν0(α) and z↦ θμL+νz(α) when the measures are radially symmetric with respect to the points of 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 H, the center of any primitive honeycomb is shown to minimize z↦ θ_μH+νz(α) 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.