PARAMETRIC DYNAMICS OF QUANTUM-SYSTEMS AND TRANSITIONS BETWEEN ENSEMBLES OF RANDOM MATRICES

We analyze ensembles of random matrices capable of describing the tran sitions between orthogonal, unitary and Poisson ensembles. Scaling law s found in complex Hermitian band random matrices and in additive rand om matrices allow us to apply them to represent the changes of the sta tistical properties of quantum systems under a variation of external p arameters. The properties of spectrum and eigenvectors of an illustrat ive dynamical system are compared with the properties of ensembles of random matrices. To describe the motion of the eigenvectors of the mat rix representing a dynamical system under a change of external paramet ers we define the relative localization length of the eigenvectors and analyze its properties. We propose a criterion for selection of gener ic basis, in which statistics of eigenvector components might be descr ibed by random matrices. The properties of products of unitary matrice s, representing composed quantum systems, are investigated.