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International Conference on Squeezed States

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. Many complex N -level quantum systems exhibiting universal behaviour depending only on symmetry of Hamiltonian matrix of the system are divided into: Gaussian orthogonal ensemble GOE(N), or Gaussian unitary ensemble GUE(N), or Gaussian symplectic ensemble GSE(N). The Gaussian ensembles are used in study of quantum systems whose classicallimit analogs are chaotic. The Poisson ensemble PE (Poisson random-sequence spectrum) is composed of uncorrelated and randomly distributed energy levels and it describes quantum systems whose classical-limit analogs are integrable. The standard statistical measure is Wigner’s distribution of the ith nearest neighbour spacing: si = ∆ 1Ei = Ei+1 − Ei, i=1,...,N-1. (1)