Semiclassical calculation of eigenvalues for higher order resonances in nonseparable oscillator systems

It is shown how the "trajectory-close method" introduced in earlier papers of this series can be used to treat other resonant systems semiclassically. The method, which does not involve the use of any curvilinear coordinate system, is illustrated for two coupled oscillators which have 3:1, 4:1, 5:1, 3:2, and 5:2 internal resonances. It is readily executed and it is shown how it can be extended to the three-oscillator case. This work supplements our earlier studies of 1:1, 2:1, and 3:1 resonant systems using this technique. Shapes of eigentrajectories and of corresponding quantum mechanical wave functions are compared for each of these systems. The paper also contains a survey of and comparison with other semiclassical methods which have been applied to systems with internal resonance