### ASA 129th Meeting - Washington, DC - 1995 May 30 .. Jun 06

## 5pSA4. Eigenstatistics in rectangular membranes with point scatterers.

**Richard L. Weaver
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*Dept. of Theor. and Appl. Mech., Univ. of Illinois, 104 S. Wright St.,
Urbana, IL 61801
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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.]