Asymptotic Analysis and Phase Transition of the Bessel-Kuznetsov Transform with an Oscillatory Phase

Abstract

The spectral side of the Kuznetsov trace formula for $GL(2)$ is governed by the Bessel-Kuznetsov integral transform $\check{\phi}(t)$. 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 $e(\alpha x)$. In this paper, we provide a rigorous and explicit asymptotic analysis of $\check{\phi}(t)$ 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 $\alpha$. We prove that in the sub-critical regime ($\alpha \le 1/2\pi$), the transform decays rapidly. Conversely, in the super-critical regime ($\alpha > 1/2\pi$), the geometric oscillations resonate with the spectral kernel, yielding a localized main term of order $O(t^{-1})$ with a remarkably simplified arithmetic phase.