Modified wave operators are unbounded on L^p for p≠ 2

Abstract

We study the L^p-boundedness of Isozaki--Kitada type modified wave operators associated with Schrödinger operators P=−∂_x² + V in one dimension for potentials V including long-range potentials such as negative Coulomb-like ones V(x) = −(1+|x|²)^−μ/2 for μ>0. For such potentials, if $ μ\in (1, 2)$, we prove that the middle and the high energy parts are bounded on L^p (ℝ) for any p ∈ [1, ∞] but the low energy part is unbounded except for p=2. Moreover, if μ∈ (0, 1], we show that even the middle energy part is unbounded except for p=2. Actually, regarding the L^p-boundedness in the middle and the high energy regimes, we give a complete classification of slowly decaying potentials without imposing any sign condition.

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BibTeX RIS

BibTeX

@misc{hoshiya2026modified,
  title = {{Modified wave operators are unbounded on $L^p$ for $p\neq 2$}},
  author = {Akitoshi Hoshiya and Kouichi Taira},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29729},
  eprint = {2609.29729},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29729},
}

RIS

TY  - UNPB
TI  - Modified wave operators are unbounded on L^p for p≠ 2
AU  - Akitoshi Hoshiya
AU  - Kouichi Taira
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29729
UR  - https://arxiv.org/abs/2609.29729
AB  - We study the L^p-boundedness of Isozaki--Kitada type modified wave operators associated with Schrödinger operators P=−∂_x^2 + V in one dimension for potentials V including long-range potentials such as negative Coulomb-like ones V(x) = −(1+|x|^2)^−μ/2 for μ>0. For such potentials, if $ μ\in (1, 2)$, we prove that the middle and the high energy parts are bounded on L^p (ℝ) for any p ∈ [1, ∞] but the low energy part is unbounded except for p=2. Moreover, if μ∈ (0, 1], we show that even the middle energy part is unbounded except for p=2. Actually, regarding the L^p-boundedness in the middle and the high energy regimes, we give a complete classification of slowly decaying potentials without imposing any sign condition.
ER  -

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