The quantum three-dimensional Sinai billiard – a semiclassical analysis

We present a comprehensive semiclassical investigation of the three-dimensional Sinai billiard, addressing a few outstanding problems in “quantum chaos”. We were mainly concerned with the accuracy of the semiclassical trace formula in two and higher dimensions and its ability to explain the universal spectral statistics observed in quantized chaotic systems. For this purpose we developed an efficient KKR algorithm to compute an extensive and accurate set of quantal eigenvalues. We also constructed a systematic method to compute millions of periodic orbits in a reasonable time. Introducing a proper measure for the semiclassical error and using the quantum and the classical databases for the Sinai billiards in two and three dimensions, we concluded that the semiclassical error (measured in units of the mean level spacing) is independent of the dimensionality, and diverges at most as log ℏ. This is in contrast with previous estimates. The classical spectrum of lengths of periodic orbits was studied and shown to be correlated in a way which induces the expected (random matrix) correlations in the quantal spectrum, corroborating previous results obtained in systems in two dimensions. These and other subjects discussed in the report open the way to extending the semiclassical study to chaotic systems with more than two freedoms.