Asymptotic Correlations of the Hamiltonian Matrix Elements

Abstract

The statistical correlations of the matrix elements of real symmetric Hamiltonians are studied using the assumption of representation invariance and the limit of large dimension N. The diagonal-diagonal correlation coefficient is expressed in terms of a parameter which gives the ratio of the dispersion of off-diagonal element to that of the diagonal element. It is shown that for a certain class of real-symmetric Hamiltonian ensembles in the limit N → ∞, the diagonal-diagonal correlation coefficient goes as λN−1, where λ is some positive constant independent of N and the correlation coefficient of two different eigenvalues is the same as the one obtained using the weak assumption of independent probabilities.