{
  "abstract": "Discrete energy-level structures in Coulomb-like localized bound states are conventionally obtained by direct diagonalization of the Schrödinger equation or by semiclassical approximations that fail at low energies. Here we develop a geometric phase-space framework that recovers the exact discrete energy levels and shell degeneracy of three-dimensional Coulomb bound states from classical symplectic geometry and global boundary constraints, bypassing explicit operator diagonalization. Using Kustaanheimo--Stiefel regularization, we map negative-energy Kepler orbits to four-dimensional isotropic harmonic oscillators. The group composition law of the symplectic flow determines an integral kernel built from the classical two-point action and Van Vleck amplitude, whose infinitesimal generator satisfies an exact linear evolution equation---a property we term \\textit{quadratic closure}. Imposing three geometric constraints---decay at infinity, regularity at the origin, and fiber invariance under the Hopf fibration---we recover the exact $E_n/E_1=1/n^2$ scaling and $g_n=n^2$ degeneracy. The phase scale $α$ sets the action unit; all spectral structural features are independent of its value. This framework offers a symmetry-transparent route to spectral data: for quadratic Hamiltonians, the classical kernel incurs zero semiclassical error even at ground-state energies, and the geometric constraints directly identify symmetry-protected degeneracies without diagonalization. We apply the method to shallow impurity states, moiré superlattices, and semiclassical transport, demonstrating how geometric screening rules simplify spectral analysis in condensed-matter environments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19159",
  "author_total": 5,
  "authors": [
    {
      "name": "Gang Zheng"
    },
    {
      "name": "Peng Chen"
    },
    {
      "name": "Mengli Wang"
    },
    {
      "name": "Wenqi Xue"
    },
    {
      "name": "Benniu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6098e65jezrse.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19159",
  "fulltext_url": null,
  "id": "6098e65jezrse",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19159",
  "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",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3109|Quantum chaos and dynamical systems"
    ]
  ],
  "openalex": "W7213587068",
  "pdf_url": "https://arxiv.org/pdf/2609.19159",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6098e65jezrse.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.19159",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19159"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Three-Dimensional Coulomb Discrete Spectrum via Symplectic Geometry and Phase-Space Constraints: A Geometric Framework for Condensed-Matter Coulomb-Like Systems",
  "title_text": "Three-Dimensional Coulomb Discrete Spectrum via Symplectic Geometry and Phase-Space Constraints: A Geometric Framework for Condensed-Matter Coulomb-Like Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/6098e65jezrse",
  "venue": "arXiv",
  "versions": []
}