Quasiclassical surface of section perturbation theory

Abstract

Perturbation theory, the quasiclassical approximation, and the quantum surface of section method are combined. This gives a new solution of 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 wave function in powers of the square root of the small parameter. It gives explicit WKB-like wave functions and energies even for certain systems which classically show hard chaos. It also gives certain classes of states in some nonintegrable systems. The relationship of the method to earlier techniques is discussed.