{
  "abstract": "Consider the two-dimensional semiclassical Schrödinger operator $P_\\hbar=-\\frac{\\hbar^2}{2}Δ+V$ on $\\mathbb R^2$, where $V$ has a nondegenerate well at the origin, its harmonic frequencies are rationally independent, and $V$ is $\\mathbb Z_2$-symmetric. We prove that, generically, the first three layers of the quantum Birkhoff normal form (QBNF) determine the full Taylor series of $V$ at the origin, up to the unavoidable spatial inversion $V(x)\\mapsto V(-x)$. The generic condition depends only on the jet of $V$ through order ten. Consequently, for real analytic potentials with a unique isolated global well at the origin, the low-lying semiclassical spectrum generically determines $V$ up to spatial inversion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.11111",
  "author_total": 1,
  "authors": [
    {
      "name": "Qiaoling Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f5epgs1c6rkrh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.11111",
  "fulltext_url": null,
  "id": "f5epgs1c6rkrh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.11111",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7202290806",
  "pdf_url": "https://arxiv.org/pdf/2608.11111",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f5epgs1c6rkrh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2608.11111",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.11111"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Generic Spectral Determination of Semiclassical Schrödinger Operators with $\\mathbb Z_2$-Symmetry",
  "title_text": "Generic Spectral Determination of Semiclassical Schrödinger Operators with ℤ_2-Symmetry",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/f5epgs1c6rkrh",
  "venue": "arXiv",
  "versions": []
}