Dimension reduction techniques for the minimization of theta functions on lattices

• Published in: Journal of mathematical physics 2017-07, Vol.58 (7), p.1
• Main Authors: Bétermin, Laurent, Petrache, Mircea
• Format: Article
• Language: English
• Subjects: Body centered cubic lattice; Bose-Einstein condensates; Condensates; Crystal lattices; Crystals; Face centered cubic lattice; Lattice theory; Mathematics; Optimization; Physics
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.4995401

We consider the minimization of theta functions Λ ( α ) = ∑ p ∈ Λ e − π α | p | 2 amongst periodic configurations Λ ⊂ R d , by reducing the dimension of the problem, following as a motivation the case d = 3, where minimizers are supposed to be either the body-centered cubic or the face-centered cubic lattices. A first way to reduce dimension is by considering layered lattices, and minimize either among competitors presenting different sequences of repetitions of the layers, or among competitors presenting different shifts of the layers with respect to each other. The second case presents the problem of minimizing theta functions also on translated lattices, namely, minimizing ( Λ , u ) ↦ Λ + u ( α ) , relevant to the study of two-component Bose-Einstein condensates, Wigner bilayers and of general crystals. Another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers. The first direction leads to the question of minimization amongst orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we study in detail in two dimensions.