Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. I. Generalities, cubic anharmonicity case

Abstract

For the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text] the perturbation theory (PT) in powers of coupling constant [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) is developed constructively in [Formula: see text]-space and in [Formula: see text]-space, respectively. The Riccati–Bloch (RB) equation and generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate [Formula: see text]-space and in [Formula: see text]-space, respectively, exploring the logarithmic derivative of wave function [Formula: see text]. It is shown that PT in powers of [Formula: see text] developed in RB equation leads to Taylor expansion of [Formula: see text] at small [Formula: see text] while being developed in GB equation leads to a new form of semiclassical expansion at large [Formula: see text]: it coincides with loop expansion in path integral formalism. In complementary way PT for large [Formula: see text] developed in RB equation leads to an expansion of [Formula: see text] at large [Formula: see text] and developed in GB equation leads to an expansion at small [Formula: see text]. Interpolating all four expansions for [Formula: see text] leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at [Formula: see text] for any coupling constant [Formula: see text] and dimension [Formula: see text]. As a concrete application, the low-lying states of the cubic anharmonic oscillator [Formula: see text] are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is [Formula: see text] for [Formula: see text] for coupling constant [Formula: see text] and dimension [Formula: see text] In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7–8 s.d. for [Formula: see text] and [Formula: see text]