{
  "abstract": "Two new Bohr-Sommerfeld (B-S) type quantization conditions for $d$-dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \\hbox{$V=\\operatorname{sign}(m)\\,r^m$} for $m=-3/2,-1,-1/2, 1,3,4,6$ and $V=\\log r$ (that corresponds to $m=0$) are studied in detail for radial quantum numbers $n_r \\in [0,100]$ and for angular momenta $\\ell=1,2,3,4,5$ in dimensions $d=2,3$. It is shown that for large $n_r$ the B-S energies $E_{BS}$ always tend to the exact ones. For $m=2$ (the radial harmonic oscillator) and $m=-1$ (the $d$-dimensional analogue of the $3D$ Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of $E_{BS}$ with the energies $E^{(L)}_{BS}$, obtained for the radial potentials modified by the Langer correction for $d=3$, are in good agreement in general: in detail, it was checked for the quartic oscillator $V=r^4$. It is demonstrated that introducing a dimensionless {\\it WKB correction} $γ$ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with $γ$ being very small for positive degrees $m \\geq 0$ and sufficiently small for non-positive ones $m < 0$ for all $n_r, \\ell$ studied.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36421",
  "author_total": 2,
  "authors": [
    {
      "name": "J. C. del Valle"
    },
    {
      "name": "A. V. Turbiner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7cv252p9625tn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36421",
  "fulltext_url": null,
  "id": "7cv252p9625tn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36421",
  "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.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3109|Quantum chaos and dynamical systems"
    ]
  ],
  "openalex": "W7224124142",
  "pdf_url": "https://arxiv.org/pdf/2610.36421",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7cv252p9625tn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:50Z",
      "native_id": "2610.36421",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36421"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Radial power-like potentials: from the Bohr-Sommerfeld $\\ell \\neq 0$ state energies to the exact ones",
  "title_text": "Radial power-like potentials: from the Bohr-Sommerfeld ℓ ≠ 0 state energies to the exact ones",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/7cv252p9625tn",
  "venue": "arXiv",
  "versions": []
}