Uncommonly accurate energies for the general quartic oscillator (bibtex)
by Okun, Pavel and 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 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.
Reference:
Uncommonly accurate energies for the general quartic oscillator Okun, Pavel and Burke, Kieron, International Journal of Quantum Chemistry n/a, e26554 (2020). [supplementary information]
Bibtex Entry:
@article{OB20,
	Pub-num 	   = {201},
	Title 		   = {Uncommonly accurate energies for the general quartic oscillator},
	Author 		   = {Okun, Pavel and 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 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.},
	Doi 		   = {https://doi.org/10.1002/qua.26554},
%%	Issn		   = {},
	Year 		   = {2020},
	Month 		   = {June},
	Journal		   = {International Journal of Quantum Chemistry},
	Volume 		   = {n/a},
%%	Issue 		   = {},
	Number 		   = {n/a},
	Pages 		   = {e26554},
%%	Publisher 	   = {},
	Url 		   = {https://onlinelibrary.wiley.com/doi/abs/10.1002/qua.26554},
	arXiv		   = {arXiv:2007.04762},
%%	keywords 	   = {},
    supp-info      = {OB20s}
}
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