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We investigate analytically and numerically the eigenstate thermalizationhypothesis (ETH) in terms of a Hermitian operator in the eigenkets selectedfrom the elements of a real (complex) symmetric random matrix of the Gaussianorthogonal ensemble (Gaussian unitary ensemble) according to the theory ofrandom matrices. We propose a method to construting the commensurateeigenvector basis that are appropriate for the numerical calculation as well asthe diagonal or off-diagonal ETH diagnostics. We also investigate theintegrablity-chaos transition with independent perturbations in terms of theBerry autocorrelation in semicalssical limit, where there is a phase spacespanned by the momentum-like projection and the range of local wave function.In such a semiclassical framework, we further develop a method which is base onthe correlations of adjacent local points (CALP), to investigate theintegrablity-chaos transition In the presence of uncorrelated peturbations andcorrelated integrable eigenstate fluctuations, which is applicable for both theergodic and nonergodic systems (where the Berry autocorrelation is vanishinglysmall and be nearly one, respectively, similar to the characteristic of inverseparticipation ratio (IPR)).More importantly,this work reveals and illustratesto a certain extent the essential role of the Golden ratio and the constant 31in the quantum chaos physics.

Eigenstate thermalization and integrability-chaos transition base on the commensurate eigenvector basis and the correlations of adjacent local points