欧氏球的Pólya猜想
Pólya's conjecture for Euclidean balls
AI总结：
本文证明了圆盘、任意平面扇形及任意维度球（Dirichlet情形）的Pólya猜想，使其成为首个验证该猜想的非铺砌平面域，并发展了特征值与格点计数函数间的一致界限关系。
摘要：
谱几何中著名的Pólya猜想（1954）指出，有界欧氏域上Dirichlet和Neumann Laplacian的特征值计数函数可分别由Weyl渐近的主项从上方和下方进行估计。已知Pólya猜想对于可铺砌欧氏空间的域成立，此外对高维中的一些特殊域也成立。本文证明了圆盘的Pólya猜想，使其成为首个验证该猜想的非铺砌平面域。我们还对任意平面扇形确认了Pólya猜想，并在Dirichlet情形下对任意维度的球也予以确认。在此过程中，我们发展了圆盘谱问题与某些格点计数问题间的已知联系。一个关键的新颖要素是，正如最后一位署名作者在近期工作中所观察到的，相应的特征值与格点计数函数不仅在渐近意义上相关，而且实际上满足某些一致界限。我们的证明是纯解析的，除了在圆盘Neumann问题情形下，覆盖谱参数取值的短区间需要严格的计算机辅助论证。
英文摘要：
The celebrated Pólya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. Pólya's conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya's conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya's conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk.
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