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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2024_2051_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\root 3 \of {2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mroot> <mn>2</mn> <mn>3</mn> </mroot> </math></EquationSource> </InlineEquation> and 1, which we believe to be a new result.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年1月22日

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  • Markus Faulhuber,
  • Anupam Gumber,
  • Irina Shafkulovska

摘要

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 \(\root 3 \of {2}\) 2 3 and 1, which we believe to be a new result.