Calculus of Variations and Geometric Measure Theory
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M. Petrache - L. B├ętermin

Dimension reduction techniques for theta functions on lattices

created by petrache on 01 Aug 2016
modified on 29 Aug 2016

[BibTeX]

Preprint

Inserted: 1 aug 2016
Last Updated: 29 aug 2016

Year: 2016
Links: arxiv link

Abstract:

We consider the minimization of theta functions $\theta_\Lambda(\alpha)=\sum_{p\in\Lambda}e^{-\pi\alpha
p
^2}$ amongst lattices $\Lambda\subset \mathbb R^d$, by reducing the dimension of the problem, following as a motivation the case $d=3$, where minimizers are supposed to be either the BCC or the FCC 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 $(L,u)\mapsto \theta_{L+u}(\alpha)$. 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.


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