Large-omega Asymptotic Expansions and Perturbation Theory for the Harmonic Oscillator

In Part I, a large parameter asymptotic expansion of a perturbed simple harmonic oscillator is presented. Perturbations of the form lambda (x sup alpha) are studied where lambda is the coupling constant. The method is demonstrated by a general derivation of the wave functions to third order for any alpha and quantum number n. Additional 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 Schrodinger 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 wave function in terms of Hermite polynomials, perturbation theory results can be readily obtained, for any state, to arbitrarily high order. (Author)