Application of Bogomolny’s transfer operator to semiclassical quantization of a chaotic system

The transfer operator developed recently by Bogomolny is used to calculate approximate semiclassical energy eigenvalues for a chaotic system, the wedge billiard. Only four classical trajectories, starting and ending on the Poincaré surface of section and crossing it once in between, are required to calculate an element of the T matrix. A 100×100T matrix is shown to give excellent results for the first 20 energy eigenvalues. It is also shown how the actions of the shortest periodic orbits of the 49° wedge can be obtained by calculating the Fourier transforms of Tr(Tm).