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

Uncommonly accurate energies for the general quartic oscillator