Abstract
Exact eigenfunctions, which simultaneously diagonalise the Hamiltonian of a 2:1 resonant, two-dimensional harmonic oscillator and an additional constant of the motion, cubic in the cartesian displacement coordinates and momenta, are found by direct solution of the Schrodinger equation in parabolic coordinates. The connection with the usual harmonic-oscillator cartesian basis is established and used in the formulation of a second-order perturbation theory for the oscillator with a particular form of nonlinear coupling. Uniform semiclassical quantisation of the unperturbed oscillator is discussed.