Studies of Bogomolny's semiclassical quantization of integrable and nonintegrable systems

The semiclassical quantization scheme formulated by Bogomolny, employing a suitably chosen Poincare surface of section (PSS), has been used to calculate approximate energy eigenvalues for the quantum analogues of several Hamiltonian systems. Using a finite approximation to the transfer operator in coordinate space, we have carried out calculations of the energy eigenvalues for the circle billiard and the 45 degrees wedge billiard, both of which are integrable systems. Calculations have also been performed for the 49 degrees and 60 degrees wedges, which classically exhibit hard chaos, and for the 41 degrees and 30 degrees wedges which classically exhibit soft chaos or mixed behaviour. In all cases, the low-lying energy eigenvalues are in excellent agreement with the exact quantum energy eigenvalues and, in the case of the 49 degrees wedge, are better than the results obtained using any other semiclassical quantization scheme. We have also studied a variant of Bogomolny's approach which employs the symbolic dynamics to construct a representation of the transfer operator. We show explicitly for the 49 degrees wedge billiard how the accessible part of the PSS in phase space is divided into cells labelled by sequences of n symbols. This leads to a systematic way of finding the pruning rules for this system. Results are presented for the 49 degrees wedge based on schemes employing 2-, 3-, and 4-symbol cells. Because of the extensive pruning in this system, this approach cannot be easily implemented for the general case of n-symbol cells. Furthermore, the numerical results are not as good as those obtained from finite approximation to the transfer operator in coordinate space.