  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.
