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arXiv · 2110.06008

A variational principle for Gaussian lattice sums

Laurent Bétermin·Markus Faulhuber·Stefan Steinerberger
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Abstract

We consider a two-dimensional analogue of Jacobi theta functions and prove that, among all lattices $\Lambda \subset \mathbb{R}^2$ with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be interpreted as the dual of a 1988 result of Montgomery who proved that the hexagonal lattice minimizes the maximal values. Our inequality resolves a conjecture of Strohmer and Beaver about the operator norm of a certain type of frame in $L^2(\mathbb{R})$. It has implications for minimal energies of ionic crystals studied by Born, the geometry of completely monotone functions and a connection to the elusive Landau constant.

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Laurent Bétermin, Markus Faulhuber, Stefan Steinerberger. 2021-10-12. A variational principle for Gaussian lattice sums. https://arxiv.org/abs/2110.06008

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