Quasiclassical Surface of Section Perturbation Theory

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Perturbation theory, the quasiclassical approximation and the quantum surface of section method are combined for the first time. This gives a new solution of the the long standing problem of quantizing the resonances generically appearing in classical perturbation theory. Our method is restricted to two dimensions. In that case, however, the results are simpler, more explicit and more easily expressed visually than the results of earlier techniques. The method involves expanding the `phase' of the wavefunction in powers of the square root of the small parameter. It gives explicit WKB-like wavefunctions and energies even for certain systems which classically show hard chaos.