Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators

Abstract

We establish sharp endpoint mapping properties for the wave operators W_±(H,−Δ) of two-dimensional Schrödinger operators H=−Δ+V with real-valued decaying potentials V. Together with the known non-endpoint L^p theory, our results give a complete classification of the L^p mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints p=1 and p=∞. When zero is a regular point of H, the wave operators fail to be bounded on L¹(ℝ²) and on L^∞(ℝ²), but they satisfy the atural substitute estimates of Calderón--Zygmund type: L¹(ℝ²)⟶ L^1,∞(ℝ²), ℋ¹(ℝ²)⟶ L¹(ℝ²), L^∞(ℝ²)⟶ BMO(ℝ²). When zero is instead a threshold singularity of the first kind---an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces L¹(ℝ²) and L^∞(ℝ²). Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions n≥3, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of H. A p-wave resonance obstructs both the L¹- and the L^∞-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

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Cite

BibTeX RIS

BibTeX

@misc{cheng2026endpoint,
  title = {{Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators}},
  author = {Han Cheng and Changxing Miao and Xiaohua Yao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.30602},
  eprint = {2608.30602},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30602},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators
AU  - Han Cheng
AU  - Changxing Miao
AU  - Xiaohua Yao
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2608.30602
UR  - https://arxiv.org/abs/2608.30602
AB  - We establish sharp endpoint mapping properties for the wave operators W_±(H,−Δ) of two-dimensional Schrödinger operators H=−Δ+V with real-valued decaying potentials V. Together with the known non-endpoint L^p theory, our results give a complete classification of the L^p mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints p=1 and p=∞. When zero is a regular point of H, the wave operators fail to be bounded on L^1(ℝ^2) and on L^∞(ℝ^2), but they satisfy the atural substitute estimates of Calderón--Zygmund type: L^1(ℝ^2)⟶ L^1,∞(ℝ^2), ℋ^1(ℝ^2)⟶ L^1(ℝ^2), L^∞(ℝ^2)⟶ BMO(ℝ^2). When zero is instead a threshold singularity of the first kind---an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces L^1(ℝ^2) and L^∞(ℝ^2). Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions n≥3, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of H. A p-wave resonance obstructs both the L^1- and the L^∞-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.
ER  -

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