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

We consider the spectral statistics of the superposition of a random diagonal matrix and a GUE matrix. By means of two alternative superanalytic approaches, 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 valid for all values of the parameter lambda governing the strength of the GUE admixture on the unfolded scale. A new twofold integration representation is 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 expressions for X-2(r) and Sigma(2)(r) can be derived by (i) evaluating the double integral perturbatively or (ii) an ab initio perturbative calculation employing the coset method. The instructive comparison between both approaches reveals that random fluctuations of Gamma(1) manifest themselves in modifications 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 simulations of an ensemble of 500 x 500 matrices. which demonstrate the universal validity of our results after proper unfolding. (C) 1998 Academic Press.