Asymptotic series for the quantum quartic anharmonic oscillator

We propose a numerical method to find solutions of the one-dimensional Schrödinger equation when the potential is symmetric and can be expanded in a polynomial form. We used a non-perturbative method, in which we include explicitly the correct asymptotic behavior of the wave function computed by the WKB method. The numerical convergence is very fast and allows to compute the energy eigenvalues and eigenfunctions simultaneously. The method is applied to the quartic anharmonic oscillator with one and two wells, we compute the energy eigenvalues for the ground state and for the first six excited states, the results obtained are in agreement with those reported previously in the literature.