AI 中文总结
本文发现方向相切机制导致Schrödinger方程异常远场衰减，通过Fermi曲面几何分解解析源与奇点贡献，并构造势验证Born修正中的慢衰减现象。
AI 中文摘要
我们识别出一种方向相切机制，该机制导致Schrödinger方程中出现异常远场衰减。对于常系数算子，我们基于Fermi曲面的几何形状和观测方向，引入了极限预解式的方向分解。在二维情形下，我们证明解析源产生超代数阶小的相切贡献，而位于方向相切集上的横向非解析性则产生阶为\\(R^{-(2\alpha+1)}\\)的代数贡献。将奇点移离相切集可恢复超代数衰减，这表明几何对齐对异常行为至关重要。我们进一步构造了一个有界、衰减的实值势，使得相同机制通过有效源\\(Vu_0\\)出现在第一Born修正中。在合适的长程范围内，由此产生的相切贡献比常规二维远场项衰减得更慢。
英文摘要
We identify a directional tangency mechanism responsible for anomalous far-field decay in Schrödinger equations. For constant-coefficient operators, we introduce a directional decomposition of the limiting resolvent based on the geometry of the Fermi surface and the observation direction. In two dimensions, we show that an analytic source produces a super-algebraically small tangency contribution, whereas a transverse non-analyticity located on the directional tangency set generates an algebraic contribution of order \(R^{-(2α+1)}\). Shifting the singularity away from the tangency set restores super-algebraic decay, showing that geometric alignment is essential to the anomalous behavior. We further construct a bounded, decaying real-valued potential for which the same mechanism occurs in the first Born correction through the effective source \(Vu_0\). In a suitable long-range regime, the resulting tangency contribution decays more slowly than the conventional two-dimensional far-field term.
