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: { \it 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 $N \geq 3$.} The notions of { \it level homogenization with level clustering} and { \it level homogenization with level repulsion} are introduced.
Categories: cond-mat.stat-mech