Gaussian Orthogonal Ensemble

It is possible to define a time-reversal operator T as follows [556]  

Preservation of normalisation follows from the transformation n = U n so that m = m U where W is the hermitian transpose of U. Thus, for m n = m WU n to be the same as m n , we require that UU7 = 1, i.e. U is unitary. Complex conjugation, of course, does not affect normalisation either. 

Operating twice with T must bring the system back onto itself and preserve normalisation, so 

For either of the situations just considered the appropriate level distribution is the Gaussian orthogonal ensemble or GOE distribution,4 defined 

3 In quantum mechanics, the operation t — —t (time-reversal) is to be accompanied by complex conjugation (i — —i) so that the Schrodinger equation remains invariant. This operation is called Wigner time-reversal. 

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Very accurate results were obtained for the classically chaotic Sinai billiard by Bohigas, Giannoni, and Schmit (see Fig. 2) which led them to the important conclusion (Bohigas, Giannoni and Schmit, 1984) Spectra of time-reversal invariant systems whose classical analogues are K systems show the same fluctuation properties as predicted by the Gaussian orthogonal ensemble (GOE) of random-matrix theory... 

One distinguishes between three different types depending on space-time symmetry classified by the Dyson parameter (3d = 1,2,4 (Guhr, Muller-Groeling and Weidenmuller, 1998). The Gaussian orthogonal ensemble (GOE, (3d = 1) holds for time-reversal invariance and rotational symmetry of the Hamiltonian... 

For Hamiltonians invariant under rotational and time-reversal transformations the corresponding ensemble of matrices is called the Gaussian orthogonal ensemble (GOE). It was established that GOE describes the statistical fluctuation properties of a quantum system whose classical analog is completely chaotic. 

For a space of eigenvectors of matrices of the gaussian orthogonal ensemble (k = N) the distribution of values of matrix elements of electromagnetic transition operators is gaussian, as follows from the central limit theorem. The ensemble averaging of hamiltonians guarantees that no correlations exist between the hamiltonian structure and the particular transition operator that is considered. 

Universality holds if a distribution applies not only to the Gaussian ensembles but also to the other ensembles based on the different orthogonal polynomials, such as the Legendre ensembles, within each of the three Dyson universality classes OE, UE, and SE [73],... 

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