• Corpus ID: 238634561

A variational principle for Gaussian lattice sums

  title={A variational principle for Gaussian lattice sums},
  author={Laurent B'etermin and Markus Faulhuber and Stefan Steinerberger},
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… 
Szeg\H{o} type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials
Consider the subspace Wn of L2(C, dA) consisting of all weighted polynomials W (z) = P (z)·e− 1 2 , where P (z) is a holomorphic polynomial of degree at most n−1, Q(z) = Q(z, z̄) is a fixed,
Gabor frame bound optimizations
In some cases optimality results for the square lattice are obtained, while in other cases the lattices optimizing the frame bounds and the condition number are different.
Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere
We consider the problem of finding an N -point configuration on the sphere S d ⊂ R d +1 with the smallest absolute maximum value over S d of its total potential. The potential induced by each point y


Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies
We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H.~Montgomery. Minimal theta functions. \textit{Glasgow Mathematical
Two-Dimensional Theta Functions and Crystallization among Bravais Lattices
It is proved that if a function is completely monotonic, then the triangular lattice minimizes its energy per particle among Bravais lattices for any given density, and the global minimality is deduced, i.e., without a density constraint, of a triangular lattICE for some Lennard-Jones-type potentials and attractive-repulsive Yukawa potentials.
On Born’s Conjecture about Optimal Distribution of Charges for an Infinite Ionic Crystal
Born’s conjecture about the optimality of the rock salt alternate distribution of charges on a cubic lattice (and more generally on a d-dimensional orthorhombic lattice) is proved and the results hold for a class of completely monotone interaction potentials which includes Coulomb-type interactions.
Universal optimality of the $E_8$ and Leech lattices and interpolation formulas
We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they
Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture
The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the $E_8$ lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they
Some curious results related to a conjecture of Strohmer and Beaver
We study results related to a conjecture formulated by Thomas Strohmer and Scott Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with
Quantum Harmonic Analysis on Lattices and Gabor Multipliers
We develop a theory of quantum harmonic analysis on lattices in $${\mathbb {R}}^{2d}$$ R 2 d . Convolutions of a sequence with an operator and of two operators are defined over a lattice, and using
On the hyperbolic metric of the complement of a rectangular lattice
AbstractThe density of the hyperbolic metric on the complement of a rect-angular lattice is investigated. The question is related to conformalmapping of symmetric circular quadrilaterals with all
Optimal and non-optimal lattices for non-completely monotone interaction potentials
We investigate the minimization of the energy per point $$E_f$$Ef among d-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function
A Minimum Problem for the Epstein Zeta-Function
  • R. Rankin
  • Mathematics
    Proceedings of the Glasgow Mathematical Association
  • 1953
In some recent work by D. G. Kendall and the author † on the number of points of a lattice which lie in a random circle the mean value of the variance emerged as a constant multiple of the value of