{
  "abstract": "In this paper, we study a finite-data inverse nodal optimization problem for radial Schrödinger operators in an arbitrary fixed angular-momentum sector. The analysis is built directly at the Friedrichs endpoint and in the physical weighted space $L_d^p$, so that the singular radial geometry is retained rather than replaced by a regular one-dimensional model. The main purpose of this paper is to provide \\emph{a singular Friedrichs finite-data variational framework} valid in every angular-momentum sector, thereby extending the existing finite-data variational theories concerning either regular one-dimensional operators or the radial sector $\\ell=0$. By means of a Volterra representation of the Friedrichs branch, we prove weak continuity and continuous Fréchet differentiability of nodal radii, exact realization of compatible same-mode nodal data, existence of optimal potentials, and finite-codimensional constraint geometry. The same framework also incorporates mixed angular-momentum and spectral--nodal observations through finite-dimensional transversality. Remarkably, \\emph{a global uniqueness theorem} is established for inward displacements of the unique interior node of the second mode in the radial sector \\(\\ell=0\\). For a constant reference potential and \\(p>(d+2)/2\\), every such displacement admits a unique global optimizer. Unlike local inverse-mapping or one-dimensional integrability arguments, the proof first selects the admissible critical sign globally and then reduces every minimizer to a scalar mass-balance equation between a focusing ball branch and a logistic annulus branch. The strict opposite monotonicity of the two weighted masses makes the balance parameter unique, providing a global rigidity mechanism over the entire inward-displacement regime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23502",
  "author_total": 3,
  "authors": [
    {
      "name": "Xijun Deng"
    },
    {
      "name": "Zhisu Liu"
    },
    {
      "name": "Yonghui Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5m2hcts5pq04s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23502",
  "fulltext_url": null,
  "id": "5m2hcts5pq04s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23502",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214037724",
  "pdf_url": "https://arxiv.org/pdf/2609.23502",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5m2hcts5pq04s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.23502",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23502"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Finite-data inverse nodal optimization in angular-momentum sectors of Schrödinger operators",
  "title_text": "Finite-data inverse nodal optimization in angular-momentum sectors of Schrödinger operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/5m2hcts5pq04s",
  "venue": "arXiv",
  "versions": []
}