In Part I, a large parameter asymptotic expansion of a perturbed simple harmonic oscillator is presented. Perturbations of the form λx^∝ are studied were λ is the coupling constant. The method is demonstrated by a general derviation of the wavefunctions to third order for any ∝ and quantum number n. Additonal expressions are presented for the ground states to tenth order in the energy. It is also shown that the "matching condition" for the basic and perturbation theory solutions is automatically satisfied due to the non-singular nature of the perturbation.

In Part II, the solution to the Schrödinger equation for a one dimensional harmonic oscillator under the influence of two general classes of potentials is investigated using high-order perturbation theory. It is shown that by utilizing a finite expansion of the perturbation theory wavefunction in terms of Hermite polynomials, perturbation theory results can be readily obtained, for any state, to arbitrarily high order. Results are presented explicitly to second order for a general set of potentials; whereas, tenth order results are given for the even and odd anharmonic oscillators as well as a periodic cosine perturbation. It is shown that by rearranging the perturbation theory expansion as a series of rational fractions (Padé Table) very accurate values are obtained for eigenvalues, even for large values of the coupling constant. Comparison of results obtained for the even anharmonic oscillator with recently published work indicates excellent agreement.