The quantum three-dimensional Sinai billiard - A semiclassical analysis

We present a comprehensive semiclassical investigation of the three-dimensi onal Sinai billiard, addressing a few outstanding problems in "quantum chao s". We were mainly concerned with the accuracy of the semiclassical trace f ormula in two and higher dimensions and its ability to explain the universa l spectral statistics observed in quantized chaotic systems. For this purpo se we developed an efficient KKR algorithm to compute an extensive and accu rate set of quantal eigenvalues. We also constructed a systematic method to compute millions of periodic orbits in a reasonable time. Introducing a pr oper measure for the semiclassical error and using the quantum and the clas sical databases for the Sinai billiards in two and three dimensions, we con cluded that the semiclassical error (measured in units of the mean level sp acing) is independent of the dimensionality, and diverges at most as log (h ) over bar. This is in contrast with previous estimates. The classical spec trum of lengths of periodic orbits was studied and shown to be correlated i n a way which induces the expected (random matrix) correlations in the quan tal spectrum, corroborating previous results obtained in systems in two dim ensions. These and other subjects discussed in the report open the way to e xtending the semiclassical study to chaotic systems with more than two free doms. (C) 2000 Elsevier Science B.V. All rights reserved.