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

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 Vu₀. 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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BibTeX RIS

BibTeX

@misc{zhang2026directional,
  title = {{Directional Tangency and Anomalous Far-Field Decay for Schrödinger Equations}},
  author = {Ruming Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23472},
  eprint = {2609.23472},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23472},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - Directional Tangency and Anomalous Far-Field Decay for Schr{ö}dinger Equations
AU  - Ruming Zhang
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23472
UR  - https://arxiv.org/abs/2609.23472
AB  - 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.
ER  -

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