Energy levels of an anharmonic oscillator in both weak and strong coupling limit using convergency of Morse-Feshbach non-linear perturbation series

We make an extensive rigorous study on convergent behaviour of Morse-Feshbach nonlinear perturbation series (MFNPS) to find out energy levels of the anharmonic oscillator (AHO) in both weak and strong coupling limit. We develop a new method of multi step optimal splitting in order to get convergency in MFNPS for ground state of AHO and found that two step optimal splitting is sufficient to provide convergency in MFNPS. Unlike the ground state the optimal splitting parameters for excited states is modified according to their dependency on state in order to achieve convergency in MFNPS.