Semiclassical surface of section perturbation theory

We derive analytic expressions for the wavefunctions and energy levels in the semiclassical approximation for perturbed integrable systems. We find that some eigenstates of such systems are substantially different from any of the unperturbed states, which requires some sort of a resonant perturbation theory. We utilize the semiclassical surface of section method by Bogomolny that reduces the spatial dimensions of the problem by one. Among the systems considered are the circular billiard with a perturbed boundary, including the short stadium; the perturbed rectangular billiard, including the tilted square and the square in magnetic field; the bouncing ball states in the stadium and slanted stadium; and the whispering gallery modes. The surface of section perturbation theory is compared with the Born-Oppenheimer approximation, which is an alternative way to describe some classes of states in these systems. We discuss the derivation of the trace formulas from Bogomolny's transfer operator for the chaotic, integrable, and almost integrable systems.