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We study the spectral bounds of self-adjoint operators on the Hilbert spaceof square-integrable functions, arising from the representation theory of theHeisenberg group. Interestingly, starting either with the von Neumann latticeor the hexagonal lattice of density 2, the spectral bounds obey well-knownarithmetic-geometric mean iterations. This follows from connections to Jacobitheta functions and Ramanujan's corresponding theories. As a consequence were-discover that these operators resemble the identity operator as the densityof the lattice grows. We also prove that the conjectural value of Landau'sconstant is obtained as the cubic arithmetic-geometric mean of $\sqrt[3]{2}$and 1, which we believe to be a new result.

The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators