带振荡相位的贝塞尔-库兹涅佐夫变换的渐近分析与相变
Asymptotic Analysis and Phase Transition of the Bessel-Kuznetsov Transform with an Oscillatory Phase
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中文总结 AI 辅助
针对解析数论中带高度振荡线性相位的检验函数场景，对GL(2)的Kuznetsov迹公式谱侧的贝塞尔-库兹涅佐夫变换做半经典极限下的渐近分析，借助WKB近似发现依赖扭参数的尖锐相变并证明不同区域的衰减/共振特性。
中文摘要 AI 辅助
GL(2)的Kuznetsov迹公式的谱侧由贝塞尔-库兹涅佐夫积分变换$\breve{\u03d5}(t)$支配。经典界保证了该变换对光滑、非振荡的检验函数具有快速衰减性，但解析数论中的现代应用——尤其是涉及扭移位卷积和的应用——经常遇到具有高度振荡线性相位$e(\u03b1 x)$的检验函数。本文中，我们在此类振荡条件下，对半经典极限$t \to \u221e$中的$\breve{\u03d5}(t)$给出了严格且显式的渐近分析。通过将WKB近似应用于虚阶贝塞尔核，我们识别出一种依赖于扭参数$\u03b1$的尖锐相变。我们证明，在亚临界区域（$\u03b1 \u2264 1/2\u03c0$），该变换快速衰减。相反，在超临界区域（$\u03b1 > 1/2\u03c0$），几何振荡与谱核发生共振，产生阶为$O(t^{-1})$的局部化主项，其算术相位显著简化。
英文摘要
The spectral side of the Kuznetsov trace formula for $GL(2)$ is governed by the Bessel-Kuznetsov integral transform $\checkϕ(t)$. While classical bounds guarantee rapid decay of this transform for smooth, non-oscillatory test functions, modern applications in analytic number theory---particularly those involving twisted shifted convolution sums---frequently encounter test functions exhibiting a highly oscillatory linear phase $e(αx)$. In this paper, we provide a rigorous and explicit asymptotic analysis of $\checkϕ(t)$ in the semiclassical limit $t \to \infty$ under such oscillatory conditions. By applying the WKB approximation to the imaginary-order Bessel kernel, we identify a sharp phase transition dependent on the twist parameter $α$. We prove that in the sub-critical regime ($α\le 1/2π$), the transform decays rapidly. Conversely, in the super-critical regime ($α> 1/2π$), the geometric oscillations resonate with the spectral kernel, yielding a localized main term of order $O(t^{-1})$ with a remarkably simplified arithmetic phase.
