论文标题
Gauss的AGM，Ramanujan的相应理论，以及自伴操作员的光谱界限
The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators
论文作者
论文摘要
我们研究了由海森伯格组的代表理论引起的，研究了自相邻算子在正方形函数的希尔伯特空间上的光谱范围。有趣的是，从von Neumann晶格或密度2的六角形晶格开始，光谱界限遵守众所周知的算术几何平均迭代。这是从与Jacobi Theta函数和Ramanujan的相应理论的连接之后。结果，我们重新发现，这些操作员类似于身份操作员，随着晶格的密度的增长。我们还证明，Landau常数的猜想价值是作为$ \ sqrt [3] {2} $和1的立方算术几何平均值获得的，我们认为这是一个新的结果。
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 re-discover 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 the cubic arithmetic-geometric mean of $\sqrt[3]{2}$ and 1, which we believe to be a new result.
