TY - JOUR

T1 - Range of spectral correlations in pseudointegrable systems

T2 - Gaussian-orthogonal-ensemble statistics in a rectangular membrane with a point scatterer

AU - Weaver, Richard L.

AU - Sornette, Didier

N1 - Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 1995

Y1 - 1995

N2 - 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 (GOE) of random matrices. It also holds that a nonchaotic system will have simple Poissonian statistics. Recent experiments on the eigenvalues of elastic blocks with angled cuts and recent calculations of the eigenfrequencies of membranes with staircaselike jagged boundaries and the eigenfrequencies of a rectangular domain with a single isotropic point scatterer have, however, found GOE statistics even in these pseudointegrable systems-even though all rays in such systems are nonchaotic. In this work, the rectangular domain with a single isotropic point scatterer is studied further. In contrast to recent related work, the scatterer is characterized here by its t matrix and scattering cross section. 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 scattering cross section of the scatterer. This prediction is corroborated by numerical calculations of the eigenfrequencies.

AB - 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 (GOE) of random matrices. It also holds that a nonchaotic system will have simple Poissonian statistics. Recent experiments on the eigenvalues of elastic blocks with angled cuts and recent calculations of the eigenfrequencies of membranes with staircaselike jagged boundaries and the eigenfrequencies of a rectangular domain with a single isotropic point scatterer have, however, found GOE statistics even in these pseudointegrable systems-even though all rays in such systems are nonchaotic. In this work, the rectangular domain with a single isotropic point scatterer is studied further. In contrast to recent related work, the scatterer is characterized here by its t matrix and scattering cross section. 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 scattering cross section of the scatterer. This prediction is corroborated by numerical calculations of the eigenfrequencies.

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U2 - 10.1103/PhysRevE.52.3341

DO - 10.1103/PhysRevE.52.3341

M3 - Article

AN - SCOPUS:0000594319

SN - 2470-0045

VL - 52

SP - 3341

EP - 3350

JO - Physical Review E

JF - Physical Review E

IS - 4

ER -