Richard L. Weaver
Dept. of Theor. and Appl. Mech., Univ. of Illinois, 104 S. Wright St., Urbana, IL 61801
Conventional wisdom holds that a finite reverberant system with chaotic ray trajectories will have, at high frequencies, eigenvalue statistics identical to those of the Gaussian orthogonal ensemble of random matrices (GOE). It also holds that a nonchaotic system will have simple Poissonian statistics. Recent experiments on the eigenvalues of elastic blocks with angled cuts, recent calculations of the eigenfrequencies of membranes with staircases like jagged boundaries and the eigenfrequencies of a rectangular domain with a single isotropic point scatterer have, however, found GOE statistics even in these quasi-integrable systems---even though all rays in such systems are nonchaotic. In this work the rectangular domain with isotropic point scatterers is studied further. It is shown that the long-range level repulsion in this system is not in precise accord with the predictions of the GOE, nor is the long range spectral rigidity. GOE does, though, correctly describe the short range statistics. A quantitative prediction for the range in which GOE applies is advanced based upon the lifetime of a ray against mixing---i.e., based upon the cross section of the scatterer. This prediction is corroborated by numerical calculations of the eigenfrequencies. [Work supported by NSF.]