Perturbation approximation for higher modes in nearly regular two-dimensional cavities

A perturbation theory for weakly distorted regular cavity which has classical ray trajectories lying on invariant tori, is constructed to a higher perturbation order, than for the general case. This is possible because of a special structure of semi-classical eigenvalues for integrable Hamiltonians. The perturbation magnitude here has an order of a characteristic wavelength of a mode instead of usual wavelength square. The results are expressed in solutions of the Hill equation. The set includes modes localized along stable periodic ray trajectories; scar modes, corresponding to unstable periodic trajectories; weakly distorted modes of regular cavity, and intermediate cases. The application of the method to square, circular and elliptical cavities is outlined.