Classical and quantum chaos of the wedge billiard. II. Quantum mechanics and quantization rules.

Abstract

In this second of two papers on the classical-quantum correspondence of the wedge billiard, attention is focused on testing a number of quantization schemes based on the Gutzwiller periodic-orbit theory. To begin with, accurate values of the energy eigenvalues of the Schr\"odinger equation for the 49\ifmmode^\circ\else\textdegree\fi{} wedge and the 60\ifmmode^\circ\else\textdegree\fi{} wedge have been calculated by means of a large matrix diagonalization. These are used to judge the success of various approaches to the problem of determining approximate semiclassical energy eigenvalues, knowing only the characteristics of the periodic orbits of the classical system. First, it is shown that the periodic-orbit sum of the Gutzwiller trace formula is not absolutely convergent for either the 49\ifmmode^\circ\else\textdegree\fi{} wedge or the 60\ifmmode^\circ\else\textdegree\fi{} wedge. Nevertheless, the periodic-orbit sum may be conditionally convergent. For the 60\ifmmode^\circ\else\textdegree\fi{} wedge, a calculation including 1621 primitive periodic orbits yield peaks that are close to the lowest 20 eigenvalues of the Schr\"odinger equation. Results for the 49\ifmmode^\circ\else\textdegree\fi{} wedge are less successful, however. It is shown that the infinite families of primitive periodic orbits with nearly the same action, described in the preceding paper, cannot be treated in the usual way by the stationary-phase approximation. Finally, a number of quantization rules based on the staircase function and on the zeros of the dynamical \ensuremath{\zeta} function are studied. The Riemann-Siegel look-alike equation is found to give good results for the lowest 20 energy eigenvalues of the 49\ifmmode^\circ\else\textdegree\fi{} wedge, but misses several pairs of eigenvalues over the range of the next 30 eigenvalues. However, the smoothed version of this equation, formulated by Berry and Keating, gives good results for all the energy eigenvalues over the range of the lowest hundred eigenvalues. Even better results are found when the functional equation is combined with the dynamical \ensuremath{\zeta} function expressed as a simple product over about a thousand primitive periodic orbits. Surprisingly, the best results of all are obtained from the zeros of Bogomolny's functional determinant making use of only 16 irreducible orbits.