Radial power-like potentials: from the Bohr-Sommerfeld $S$-state energies to the exact ones

Abstract

Following our previous study of the Bohr-Sommerfeld (B-S) quantization condition for one-dimensional case (del Valle \& Turbiner (2021) \cite{First}), we extend it to $d$-dimensional power-like radial potentials. The B-S quantization condition for $S$-states of the $d$-dimensional radial Schr\"odinger equation is proposed. Based on numerical results obtained for the spectra of power-like potentials, $V(r)=r^m$ with $m \in [-1, \infty)$, the correctness of the proposed B-S quantization condition is established for various dimensions $d$. It is demonstrated that by introducing the {\it WKB correction} $\gamma$ (supposedly coming from the higher order WKB terms) into the r.h.s. of the B-S quantization condition leads to the so-called {\it exact WKB quantization condition}, which reproduces the exact energies, while $\gamma$ remains always very small. For $m=2$ (any integer $d$) and for $m=-1$ (at $d=2$) the WKB correction $\gamma=0$: for $S$ states the B-S spectra coincides with the exact ones. Concrete calculations for physically important cases of linear, cubic, quartic, and sextic oscillators, as well as Coulomb and logarithmic potentials in dimensions $d=2,3,6$ are presented. Radial quartic anharmonic oscillator is considered briefly.