Random matrix theory for systems with an approximate symmetry and widths of acoustic resonances in thin chaotic plates

摘要

We discuss a random matrix model of systems with an approximate symmetry andpresent the spectral fluctuation statistics and eigenvector characteristics forthe model. An acoustic resonator like, e.g., an aluminium plate may have anapproximate symmetry. We have measured the frequency spectrum and the widthsfor acoustic resonances in thin aluminium plates, cut in the shape of theso-called three-leaf clover. Due to the mirror symmetry through the middleplane of the plate, each resonance of the plate belongs to one of two modeclasses and we show how to separate the modes into these two classes usingtheir measured widths. We compare the spectral statistics of each mode classwith results for the Gaussian orthogonal ensemble. By cutting a slit ofincreasing depth on one face of the plate, we gradually break the mirrorsymmetry and study the transition that takes place as the two classes aremixed. Presenting the spectral fluctuation statistics and the distribution ofwidths for the resonances, we find that this transition is well described bythe random matrix model.