{
  "abstract": "We establish sharp endpoint mapping properties for the wave operators $W_\\pm(H,-Δ)$ 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 $p=\\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\\mathbb{R}^2)$ and on $L^\\infty(\\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calderón--Zygmund type: $$ L^1(\\mathbb{R}^2)\\longrightarrow L^{1,\\infty}(\\mathbb{R}^2),\\ \\ \\ \\mathcal{H}^1(\\mathbb{R}^2)\\longrightarrow L^1(\\mathbb{R}^2),\\ \\ \\ L^\\infty(\\mathbb{R}^2)\\longrightarrow \\mathrm{BMO}(\\mathbb{R}^2). $$ 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 $L^1(\\mathbb{R}^2)$ and $L^\\infty(\\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\\ge3$, 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 $L^\\infty$-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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30602",
  "author_total": 3,
  "authors": [
    {
      "name": "Han Cheng"
    },
    {
      "name": "Changxing Miao"
    },
    {
      "name": "Xiaohua Yao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07br3cvr254rm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30602",
  "fulltext_url": null,
  "id": "07br3cvr254rm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30602",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204941617",
  "pdf_url": "https://arxiv.org/pdf/2608.30602",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07br3cvr254rm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:25Z",
      "native_id": "2608.30602",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30602"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators",
  "title_text": "Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/07br3cvr254rm",
  "venue": "arXiv",
  "versions": []
}