2026-10-28 · arXiv Quantum Physics
arXiv:2610.36421v1 Announce Type: new
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} $\gamma$ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with $\gamma$ being very small for positive degrees $m \geq 0$ and sufficiently small for non-positive ones $m
原文链接：https://arxiv.org/abs/2610.36421
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