Endpoint Mapping Properties of Wave Operators for Schrödinger Operators in Dimensions n≥3
Abstract
We study endpoint mapping properties of low-energy wave operators for Schrödinger operators H=−Δ+V on ℝn, n≥3. In dimension four, we prove that a zero-energy resonance prevents L1 boundedness, whether or not zero is also an eigenvalue, under |V(x)|≲⟨ x⟩−β with β>10. For a zero-energy eigenvalue without a resonance, we obtain a complete low-energy Lp classification in every dimension n≥3 under β>n+4. In particular, L∞ boundedness is equivalent to the vanishing of the zeroth, first, and harmonic second moments of Vψ for every zero-energy eigenfunction ψ. The proof identifies the finite-rank obstruction and shows that the remaining eigenvalue correction cannot cancel its critical asymptotic profiles. The same conclusions hold for the full wave operators when the corresponding high-energy bounds are available.
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BibTeX
@misc{cheng2026endpoint,
title = {{Endpoint Mapping Properties of Wave Operators for Schrödinger Operators in Dimensions $n\ge3$}},
author = {Han Cheng and Changxing Miao and Xiaohua Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00869},
eprint = {2610.00869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00869},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Endpoint Mapping Properties of Wave Operators for Schrödinger Operators in Dimensions n≥3
AU - Han Cheng
AU - Changxing Miao
AU - Xiaohua Yao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00869
UR - https://arxiv.org/abs/2610.00869
AB - We study endpoint mapping properties of low-energy wave operators for Schrödinger operators H=−Δ+V on ℝ^n, n≥3. In dimension four, we prove that a zero-energy resonance prevents L^1 boundedness, whether or not zero is also an eigenvalue, under |V(x)|≲⟨ x⟩^−β with β>10. For a zero-energy eigenvalue without a resonance, we obtain a complete low-energy L^p classification in every dimension n≥3 under β>n+4. In particular, L^∞ boundedness is equivalent to the vanishing of the zeroth, first, and harmonic second moments of Vψ for every zero-energy eigenfunction ψ. The proof identifies the finite-rank obstruction and shows that the remaining eigenvalue correction cannot cancel its critical asymptotic profiles. The same conclusions hold for the full wave operators when the corresponding high-energy bounds are available.
ER -