Random Oper, Stock. Eqs., Vol. 1, No. 3, pp. 279-286 (1993) © VSP 1993 G. CASATI1 and V. GIRKO2 Dip&rtimento di Fisica, Universit di Milano, Via Castelnuovo, 22100 Como, Italy Department of Cybernetics, Kyjiv University, 25201 7 Kyjiv, Ukraine Received for ROSE 20 December 1992 Abstract--We rigorously derive the limit distribution of the eigenvalues density for band random matrices whose elements have arbitrary distribution, non-zero mean and non-constant variance. The study of qualitative behaviour of quantum systems whose classical limit is chaotic has led to the consideration of random matrices. Indeed, analytical and numerical investigations indicate that the eigenfunctions of such quantum systems mimic a Gaussian random function. Then the matrix elements of an operator on the basis of such eigenfunctions can also be considered as random variables. It follows that random matrix ensembles are widely used to describe the statistical properties of spectra of complex systems in solid state, nuclear and atomic physics, liquid theory etc [1, 2]. The simplest case is the so-called Gaussian orthogonal ensemble (GOE) which was introduced by Wigner in the fifties to describe the properties of spectra of systems which are invariant under time-reversal. The GOE consists of real symmetric matrices with
Random Operators and Stochastic Equations – de Gruyter
Published: Jan 1, 1993
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