立方三维环面上点散射体的间距统计
Spacing statistics for a point scatterer on the cubic three-torus
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中文总结 AI 辅助
研究立方三维环面上点散射体的新特征值间距统计，证明其收敛于独立于参数的分布，并刻画了小间隙渐近形式。
中文摘要 AI 辅助
我们研究了立方三维环面上固定点相互作用的新的特征值。它们的平均一连续间距收敛到一个与相互作用参数无关的概率分布。我们通过由三平方同余密度构造的随机亚纯函数的连续零点来刻画该分布。其小间隙分布具有形式$s^5\Psi(\log_2(s^{-2}))+o(s^5)$，其中$\Psi$是连续、正且以1为周期的函数。证明结合了周期均方逼近、加权希尔伯特变换估计以及保留素数之间相互作用的傅里叶估计。
英文摘要
We study the new eigenvalues of a fixed point interaction on the cubic three-torus. Their mean-one consecutive spacings converge to a probability law independent of the interaction parameter. We characterize this law by the consecutive zeros of a random meromorphic function built from three-squares congruence densities. Its small-gap distribution has the form $s^5Ψ(\log_2(s^{-2}))+o(s^5)$, where $Ψ$ is continuous, positive, and one-periodic. The proof combines periodic mean-square approximation, a weighted Hilbert-transform estimate, and Fourier estimates that retain the interactions between primes.
