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arXiv · 2609.23472

Directional Tangency and Anomalous Far-Field Decay for Schr{ö}dinger Equations

Ruming Zhang
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Abstract

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.

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Ruming Zhang. 2026-09-20. Directional Tangency and Anomalous Far-Field Decay for Schr{ö}dinger Equations. https://arxiv.org/abs/2609.23472

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