An optimized second-order perturbative study of the asymmetrical quantum anharmonic oscillator

We study the energy eigenvalues of a cubic-quartic anharmonic oscillator with the operator method introduced by Feranchuk and Komarov (Ann. Phys. (N.Y.), 238 (1995) 370). Based on the minimization of the second-order perturbative correction and the energy-variance, a simple optimized second-order perturbative calculation is found to yield very accurate energies especially for the excited states: the asymptotic error is shown to be - 0.002 %. When the same method is applied to an oscillator with the sextic anharmonic potential x(2)/2 (j=3)Sigma(6) x(j), similar accuracy is obtained for the excited states with an asymptotic error of about -0.003 %.