1998/01/28 by Klaus M. Frahm, Thomas Guhr, Axel Müller-Groeling +2 · 20 citations
Chemistry · Physics and Astronomy · #Diagonal #Eigenvalues and eigenvectors #Lambda #Limit (mathematics) #Mathematical analysis #Mathematical physics #Matrix (chemical analysis) #Molecular spectroscopy and chirality #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Random matrix #Spectroscopy and Quantum Chemical Studies #Statistical physics #Statistics #cond-mat
paper · pdf · doi:10.1006/aphy.1998.5853
published in Annals of Physics 270(2), 292-327 (Elsevier BV) · 25 pages, revtex, 5 figures, Postscript file also available at http://germania.ups-tlse.fr/frahm
arxiv created 1998/01/28 · openalex publication_date 1998/12/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 X2(r) and the number variance Σ2(r). The graded eigenvalue approach leads to an expression for X2(r) which is valid for all values of the parameter λ 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 λ≫ 1 the Breit-Wigner width Γ1 measured in units of the mean level spacing D is much larger than unity. In this limit, closed analytical expression for X2(r) and Σ2(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 Γ1 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 √(Γ1). This is rigorously shown and discussed in great detail. The Breit-Wigner Γ1 width itself governs the approach to the Poisson limit for r→∞. Our analytical findings are confirmed by numerical simulations of an ensemble of 500× 500 matrices, which demonstrate the universal validity of our results after proper unfolding.