Titel
The AGM of Gauss, Ramanujan’s corresponding theory, and spectral bounds of self-adjoint operators
Abstract
We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic–geometric mean iterations. This follows from connections to Jacobi theta functions and Ramanujan’s corresponding theories. As a consequence, we rediscover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau’s constant is obtained as half the cubic arithmetic–geometric mean of ∛2 and 1, which we believe to be a new result.
Stichwort
Arithmetic–geometric meanGabor systemTheta functionLatticeSpectral bounds
Objekt-Typ
Sprache
Englisch [eng]
Persistent identifier
Erschienen in
Titel
Monatshefte für Mathematik
Band
206
Ausgabe
3
ISSN
0026-9255
Erscheinungsdatum
2025
Seitenanfang
551
Seitenende
582
Publication
Springer Science and Business Media LLC
Erscheinungsdatum
2025
Zugänglichkeit
Rechte
Rechteangabe
© The Author(s) 2025
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