The perturbed two-dimensional oscillator: eigenvalues and infinite-field limits via continued fractions, renormalised perturbation theory and moment methods

Abstract

The author employs two independent methods to obtain consistent and accurate values of eigenvalues E( lambda ) of the perturbed two-dimensional oscillator Hamiltonian, H=H(0)+ lambda x2y2: (i) continued-fraction representations of the Rayleigh-Schrodinger perturbation series to large order, as well as a renormalised version of perturbation theory, and (ii) a method of moments based on the positivity properties of the factorised wavefunction. The latter generates converging upper and lower bounds to E( lambda ). The two methods are also used to obtain estimates for the eigenvalues F(0) of the infinite-field limit Hamiltonian, Hinfinity =px2+py2+x2y2.