Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. II. Quartic and Sextic Anharmonicity Cases

TL;DR

This paper extends a formalism for radial anharmonic oscillators to quartic and sextic cases, developing highly accurate approximants for eigenfunctions and energies across dimensions and coupling regimes.

Contribution

It introduces multi-parameter approximants for quartic and sextic oscillators, achieving unprecedented accuracy in eigenvalues and eigenfunctions for various dimensions.

Findings

Approximants achieve 8-12 digit accuracy in energies.

Eigenfunction deviations are less than 10^{-6}.

Method applies across dimensions and coupling strengths.

Abstract

In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials $V(r)= r^2 + g^{2(m-1)}\, r^{2m}, m=2,3$, respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8-12 figures for any $D=1,2,3\ldots $ and $g \geq 0$, while the relative deviation of the Approximant from the exact eigenfunction is less than $10^{-6}$ for any $r \geq 0$.