Between Poisson and GUE statistics: Role of the Breit-Wigner width

We consider the spectral statistics of the superposition of a random diagon al matrix and a GUE matrix. By means of two alternative superanalytic appro aches, the coset method and the graded eigenvalue method, we derive the two -level correlation function X-2(r) and the number variance Sigma(2)(r). The graded eigenvalue approach leads to an expression for X-2(r) which is vali d for all values of the parameter lambda governing the strength of the GUE admixture on the unfolded scale. A new twofold integration representation i s found which can be easily evaluated numerically. For lambda much greater than 1 the Breit-Wigner width Gamma(1) measured in units of the mean level spacing D is much larger than unity. In this limit, closed analytical expre ssions for X-2(r) and Sigma(2)(r) can be derived by (i) evaluating the doub le integral perturbatively or (ii) an ab initio perturbative calculation em ploying the coset method. The instructive comparison between both approache s reveals that random fluctuations of Gamma(1) manifest themselves in modif ications of the spectral statistics. The energy scale which determines the deviation of the statistical properties from GUE behavior is given by root Gamma(1). This is rigorously shown and discussed in great detail. The Breit -Wigner Gamma(1) width itself governs the approach to the Poisson limit for r --> infinity. Our analytical findings are confirmed by numerical simulat ions of an ensemble of 500 x 500 matrices. which demonstrate the universal validity of our results after proper unfolding. (C) 1998 Academic Press.