Effective operator treatment of the anharmonic oscillator

We analyze the one-dimensional anharmonic oscillator using effective operator methods in both the strong and weak coupling limits. We show that in the case of a one-dimensional model space, the similarity transformation needed to define the effective Hamiltonian is related to the coefficients in the expansion of the wave function in the unperturbed harmonic oscillator basis. We obtain an infinite system of equations which is equivalent to those obtained from the Hill determinant solution of the anharmonic oscillator. The analytic properties of the resulting equations reveal the nonperturbative features of the underlying problem. Thus, we demonstrate the utility of the effective operator method for solving a nonanalytic strong coupling problem.