Uncommonly accurate energies for the general quartic oscillator. Issue 7 (19th December 2020) - Okun, Pavel; Burke, Kieron
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 two terms in the asymptotic expansion. We give analytic formulas for expansion in up to three 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. Abstract : The eigenvalues of different quartic oscillator potentials are computed to higher accuracy then previously available in the literature for future tests of asymptotic density functional theory. Various approximations (including WKB and perturbation theory) with pedagogical and research applications are analyzed for the quartic oscillator. In the graphic various quartic potentials (dashed) and corresponding ground‐state densities (solid) are shown.