Description of Quantum Systems by Random Matrix Ensembles of High Dimensions

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The new Theorem on location of maximum of probability density functions of dimensionless second difference of the three adjacent energy levels for N-dimensional Gaussian orthogonal ensemble GOE(N), N-dimensional Gaussian unitary ensemble GUE(N), N-dimensional Gaussian symplectic ensemble GSE(N), and Poisson ensemble PE, is formulated: The probability density functions of the dimensionless second difference of the three adjacent energy levels take on maximum at the origin for the following ensembles: GOE(N), GUE(N), GSE(N), and PE, where N ≥ 3. The notions of level homogenization with level clustering and level homogenization with level repulsion are introduced.