# Some curious results related to a conjecture of Strohmer and Beaver

## TL;DR

This paper investigates a specific case of a conjecture on Gaussian Gabor frames with rectangular lattices, revealing new and intriguing results using advanced mathematical tools, and connecting to energy minimization and heat distribution problems.

## Contribution

It presents novel findings on the optimality of square lattices for Gaussian Gabor frames at density 2, employing determinants of Laplace--Beltrami operators and special functions.

## Key findings

- Square lattice optimality for the Tolimieri and Orr bound implies optimality for the sharp lower frame bound.
- Results extend to energy minimization problems over lattices.
- Findings involve determinants of Laplace--Beltrami operators and special functions like Eisenstein series.

## Abstract

We study results related to a conjecture formulated by Strohmer and Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of density 2. Although this case has been fully treated by Faulhuber and Steinerberger, the results in this work are new and quite curious. Indeed, the optimality of the square lattice for the Tolimieri and Orr bound already implies the optimality of the square lattice for the sharp lower frame bound. Our main tools include determinants of Laplace--Beltrami operators on tori as well as special functions from analytic number theory, in particular Eisenstein series, zeta functions, theta functions and Kronecker's limit formula. We note that our results also carry over to energy minimization problems over lattices and a heat distribution problem over flat tori.