Uncommonly accurate energies for the general quartic oscillator

TL;DR

This paper accurately computes energy levels of the general quartic oscillator using a large basis, confirming asymptotic expansions and providing benchmark results for testing approximation methods in physics and chemistry.

Contribution

It introduces a highly precise numerical method for solving the general quartic oscillator and extends asymptotic analysis with new terms and analytic formulas.

Findings

Confirmed asymptotic expansion for pure quartic oscillator levels

Extracted next two terms in the asymptotic expansion

Provided benchmark eigenvalues for various quartic potentials

Abstract

Recent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order $10^{-34}$, but few non-trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of $x^2$ and $x^4$ ) beyond this level of accuracy using a basis of several hundred oscillator states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the known asymptotic expansion for the levels of the pure quartic oscillator, and extract the next 2 terms in the asymptotic expansion. We give analytic formulas for expansion in up to 3 even basis states. We confirm the virial theorem for the various energy components to similar accuracy. The sextic oscillator levels are also given. These benchmark results should be useful for extreme tests of approximations in several areas of chemical physics and beyond.