前向散射的渐近结构
The asymptotic structure of forward scattering
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中文总结 AI 辅助
本文在渐近欧几里得情形下重新研究前向散射中受扰平面波的结构，通过双面紧化给出物理空间渐近的初等描述，证明了完全多齐次性，并得到两类模型问题。
中文摘要 AI 辅助
受扰平面波是欧几里得空间及渐近欧几里得空间散射理论中的基本对象。本文研究受扰平面波在前向方向的结构，该方向中出射球面波通常是奇异的，且与入射平面波相连。Melrose与Zworski利用与相交拉格朗日子流形对相关的拉格朗日分布概念，在渐近锥形流形的更一般框架下给出了微局部描述，并在此基础上证明了S矩阵是傅里叶积分算子（FIO）。本文在渐近欧几里得情形下重新研究该问题，在此情形下Melrose-Zworski所用的振荡积分达到了其相对于更一般渐近锥形情形的最复杂形式。我们寻求基于物理空间渐近的更初等描述，该描述通过双面紧化X ← ℝᵈ来确定，其中每个面对应一个渐近 regime。我们证明了完全多齐次性，在主面（标记为'bf'）处出现一个输运方程作为模型问题，在前表面（标记为'ff'）处出现量子反谐振子作为模型问题。
英文摘要
Perturbed plane waves are fundamental objects in scattering theory on Euclidean space and asymptotically Euclidean spaces. In this paper, we investigate the structure of perturbed plane waves in the $\textit{forward}$ direction, in which the outgoing spherical wave is typically singular and conjoined to the incoming plane wave. Melrose & Zworski provided a microlocal description (in the more general setting of asymptotically conic manifolds) using their notion of Lagrangian distributions associated to pairs of intersecting Legendrian submanifolds, on the way to proving that the S-matrix is an FIO. Here, we revisit the problem in the asymptotically Euclidean case, for which the oscillatory integrals used by Melrose--Zworski attain their most complicated form (relative to the more general asymptotically conic case). We seek a more elementary description in terms of physical-space asymptotics. These are specified using a two-faced compactification $X\hookleftarrow \mathbb{R}^d$, with one face for each asymptotic regime. We prove full polyhomogeneity. A transport equation arises as a model problem at the main face (`bf'). The quantum inverted harmonic oscillator arises as a model problem at the front face (`ff').
