A variational principle for Gaussian lattice sums

Abstract

We consider a two-dimensional analog 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 seen as the inhomogeneous counterpart to Montgomery's 1988 result, which proved that the hexagonal lattice minimizes the maximum.

arXiv comment61 pages, 11 figures, 98 references; to appear in Forum of Mathematics, Pi

References in corpus (10)

• The sphere packing problem in dimension 24
• Two-component Bose-Einstein Condensates with Large Number of Vortices
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• Minimal Operator Norms via Minimal Theta Functions
• On Born's conjecture about optimal distribution of charges for an infinite ionic crystal
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• A Short Note on the Frame Set of Odd Functions
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Cited by in corpus (2)

• Maximal polarization for periodic configurations on the real line
• On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials