Abstract
The asymptotic functional form preservation method is developed for the perturbation theory to obtain the energy eigenvalues of anharmonic oscillators. The conventional energy perturbative series expansion for the anharmonic oscillator is strongly divergent even if the anharmonicity is small. Employing a transformation containing an unphysical parameter, we analytically continue this series expansion into a new series expansion applicable to all the range of the perturbation parameter. The unphysical parameter is determined by the principle of minimal sensitivity. This new series expansion is reduced to the conventional energy perturbative series expansion for small anharmonicity, and it preserves the correct asymptotic functional form when the perturbation parameter tends to infinity. Then, we use the full-range energy series expansion to calculate the energy eigenvalues of the anharmonic oscillator. In addition to excellent energy eigenvalues obtained for the oscillator with small and strong anharmonicity, accurate energy eigenvalues can be obtained using the full-range energy series expansion when the perturbation parameter tends to infinity.