2002/04/30 by Antonio M. Garcı́a-Garcı́a, A. M. Garcia-Garcia, J. J. M. Verbaarschot · 2 citations
Chemistry · Mathematics · Physics and Astronomy · #Eigenvalues and eigenvectors #Gaussian #Hamiltonian (control theory) #Lyapunov exponent #Mathematical analysis #Mathematics #Molecular spectroscopy and chirality #Nonlinear system #Physics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos #Quantum chaos and dynamical systems #Quantum dynamics #Quantum mechanics #Random matrix #Statistical physics #Statistics #cond-mat.dis-nn #cond-mat.mes-hall #hep-th
paper · pdf · doi:10.1103/physreve.67.046104
published as Phys.Rev. E67 (2003) 046104 · 27 pages, 4 figures, modified conclusions in section four
arxiv created 2002/10/07 · openalex publication_date 2003/04/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We investigate the spectral properties of a generalized Gaussian orthogonal ensemble capable of describing critical statistics. The joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the Calogero-Sutherland (CS) Hamiltonian. Taking advantage of the correspondence between CS particles and eigenvalues, and utilizing a recently conjectured expression by Kravtsov and Tsvelik for the finite temperature density-density correlations of the CS model, we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model. Such linear behavior is a signature of critical statistics. This random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self-similar or fractal Poincaré section of its classical counterpart. This is shown in detail for two examples: the anisotropic Kepler problem and a kicked particle in a well potential. In both cases the number variance and the Delta(3) statistic are accurately described by our analytical results.