{
  "abstract": "Let $B_3=B(0,3)\\subset\\mathbb{R}^3$. For every sufficiently large integer $k$, we construct a nonzero real function $u_k\\in C^2(\\overline{B_3})$ and a real potential $V_k\\in L^\\infty(B_3)$ such that $Δu_k=V_ku_k$, $u_k|_{\\partial B_3}=0$, $\\mathrm{ord}_0u_k=k$, and $\\|V_k\\|_\\infty\\le Ck^{3/2}$. Consequently, for every sufficiently large $N$, there is such a Dirichlet solution with potential norm smaller than $N$ and vanishing order at least $cN^{2/3}$. The construction extends directly to higher dimensions and to spheres. Together with previous results, this example indicates that $C(1+\\|V\\|_\\infty^{2/3})$ is the sharp bound for the vanishing order in the real-valued case. It also indicates that the bound conjectured independently by Kukavica \\cite{Kukavica1998} and Kenig \\cite{Kenig2006} is not attainable in general.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01615",
  "author_total": 4,
  "authors": [
    {
      "name": "Shuowen Feng"
    },
    {
      "name": "Zaihui Gan"
    },
    {
      "name": "Renjin Jiang"
    },
    {
      "name": "Fanghua Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cwvr46rc2z310.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01615",
  "fulltext_url": null,
  "id": "cwvr46rc2z310",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01615",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7216011424",
  "pdf_url": "https://arxiv.org/pdf/2610.01615",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cwvr46rc2z310.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01615",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01615"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Vanishing orders of Dirichlet solutions to the Schrödinger equation in dimensions three and higher",
  "title_text": "Vanishing orders of Dirichlet solutions to the Schrödinger equation in dimensions three and higher",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/cwvr46rc2z310",
  "venue": "arXiv",
  "versions": []
}