Radial power-like potentials: From the Bohr–Sommerfeld S-state energies to the exact ones

Abstract

Following our previous study of the Bohr–Sommerfeld (BS) quantization condition for one-dimensional case [J. C. del Valle and A. V. Turbiner, Int. J. Mod. Phys. A 36, 2150221 (2021)], we extend it to [Formula: see text]-dimensional power-like radial potentials. The BS quantization condition for [Formula: see text]-states of the [Formula: see text]-dimensional radial Schrödinger equation is proposed. Based on numerical results obtained for the spectra of power-like potentials, [Formula: see text] with [Formula: see text], the correctness of the proposed BS quantization condition is established for various dimensions [Formula: see text]. It is demonstrated that by introducing the WKB correction [Formula: see text] into the right-hand side of the BS quantization condition leads to the so-called exact WKB quantization condition, which reproduces the exact energies, while [Formula: see text] remains always very small. For [Formula: see text] (any integer [Formula: see text]) and for [Formula: see text] (at [Formula: see text]) the WKB correction [Formula: see text]: for [Formula: see text] states the BS 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 [Formula: see text] are presented. Radial quartic anharmonic oscillator is considered briefly.