Abstract
The asymptotic iteration method is used to calculate the eigenenergies for the asymmetrical quantum anharmonic oscillator potentials V(x) = Sigma(2 alpha)(j=2)A(j)x(j), with (alpha = 2) for quartic, and (alpha = 3) for sextic asymmetrical quantum anharmonic oscillators. An adjustable parameter beta is introduced in the method to improve its rate of convergence. Comparing the present results with the exact numerical values, and with the numerical results of the earlier works, it is found that asymptotically, this method gives accurate results over the full range of parameter values A(j).