Discrete symmetries and spectral statistics

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Abstract
We calculate the 2-point spectral statistics associated with a given irreducible representation (i.e. symmetry class) for time-reversal invariant systems possessing discrete symmetries using semiclassical periodic orbit theory. When the representation in question is real or pseudo-real, our results conform to those of the Gaussian orthogonal ensemble (GOE) of random matrices. When it is complex, we find instead Gaussian unitary ensemble (GUE) behaviour. This provides a direct semiclassical explanation for the recent observation by Leyvraz et al (1996) of GUE correlations in the desymmetrized spectra of certain symmetric billiards in the absence of any time-reversal invariance breaking (e.g. magnetic) fields.

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