Description of Quantum Systems by Random Matrix Ensembles of High Dimensions

Abstract

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 $N\ge 3$.} The notions of {\it level homogenization with level clustering} and {\it level homogenization with level repulsion} are introduced.

arXiv comment3 pages; to appear in: "Proceedings of ICSSUR'6, Sixth International Conference on Squeezed States and Uncertainty Relations, 24th May 1999 - 29th May 1999, Universita` degli Studi di Napoli 'Federico II', Naples, Italy", NASA, Greenbelt, Maryland, USA (1999)