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

Some curious results related to a conjecture of Strohmer and Beaver