Polygonal billiards: Correspondence between classical trajectories and quantum eigenstates

Correspondence between classical trajectories and quantum eigenstates in polygonal billiards is studied. Level statistics of polygonal billiards are close to GOE-type fluctuation but they depend weakly on the energy. Among the eigenfunctions with quite irregular patterns which cannot be distinguished from typical patterns of classical chaotic billiards, there are regular eigenfunctions each of which is associated with a certain classical counterpart. Moreover, some of regular eigenstates do not seem to correspond to periodic orbits which typically form families in the polygonal boundary. The analysis using periodic-orbit quantization and also Fourier transform of the density of states indeed yields evidence that there are certainly contributions from the corner orbits which connect vertices of the boundary.