Radial power-like potentials: from the Bohr-Sommerfeld ℓ ≠ 0 state energies to the exact ones
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=sign(m) rm} 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 nr ∈ [0,100] and for angular momenta ℓ=1,2,3,4,5 in dimensions d=2,3. It is shown that for large nr the B-S energies EBS 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 EBS 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=r4. 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 ≥ 0 and sufficiently small for non-positive ones m < 0 for all nr, ℓ studied.
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BibTeX
@misc{valle2026radial,
title = {{Radial power-like potentials: from the Bohr-Sommerfeld $\ell \neq 0$ state energies to the exact ones}},
author = {J. C. del Valle and A. V. Turbiner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36421},
eprint = {2610.36421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36421},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Radial power-like potentials: from the Bohr-Sommerfeld ℓ ≠ 0 state energies to the exact ones
AU - J. C. del Valle
AU - A. V. Turbiner
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36421
UR - https://arxiv.org/abs/2610.36421
AB - 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=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 ∈ [0,100] and for angular momenta ℓ=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 ≥ 0 and sufficiently small for non-positive ones m < 0 for all n_r, ℓ studied.
ER -