Approximation Methods Applied to the Pullen-Edmonds Hamiltonian

In this work we have studied the Hamburger theorem sequence which uses the moments of the Hamiltonian evaluated for a particular state, as well as a variety of approximation schemes derivable from the \emph{t}-expansion and also a Lanczos tridiagonalization scheme. Each of these calculational schemes has been applied to the well-studied Pullen-Edmonds Hamiltonian for the representation of a $2D$ isotropic harmonic oscillator with an interaction potential of the form $x^{2}y^{2}$. We further investigate truncated approximations from moments, matrix truncations relative to the natural $2D$ simple harmonic oscillator states $|n_{x}n_{y}>$, and a class of analytic truncations in the spirit of Feenberg perturbation theory. Each of these different approximation schemes will be compared with respect to effort, accuracy, and calculational problems.