A Novel Technique for ”Complex” Spectra Analysis

The choice of a suitable random matrix model of a complex system is very sensitive to the nature of its complexity. The statistical spectral analysis of various complex systems requires, therefore, a thorough probing of a wide range of random matrix ensembles which is not an easy task. It is highly desirable, if possible, to identify a common mathematcal structure among all the ensembles and analyze it to gain information about the ensembleproperties. Our successful search in this direction leads to Calogero Hamiltonian, a one-dimensional quantum hamiltonian with inverse-square interaction, as the common base. This is because both, the eigenvalues of the ensembles, and, a general state of Calogero Hamiltonian, evolve in an analogous way for arbitrary initial conditions. The varying nature of the complexity is reflected in the different form of the evolution parameter in each case. A complete investigation of Calogero Hamiltonian can then help us in the spectral analysis of complex systems. PACS numbers: 05.45+b, 03.65 sq, 05.40+j Typeset using REVTEX 1 Recent statistical studies in various branches of theoretical physics, ranging from Calogero model of 1-d fermionic system [1], random matrix (RM) model of disordered systems, matrix models of random surfaces to non-linear sigma model of quantum chaotic systems have revealed the presence of a common mathematical structure [2–4]. The connectingweb of these various models with each other is well-described in [3]. However, so far, the connection of RM model with other models was established only for standard Gaussian ensembles (SGE), that is, Gaussain ensembles invariant under unitary transformation. This was achieved by showing that distribution of the eigenvalues of the ensemble is governed by a Fokker-Planck (F-P) equation [6,7] similar to that of Dyson’s ”Brownian” motion model [5]. Through the reduction of F-P equation to the Schrodinger equation, the latter model is already known to be connected to Calogero Hamiltonian and thereby to various other models [6–8]. In this paper, we explore RM models with non-invariant distributions, and, following the same route as in the case of SGE, connect them to Calogero Hamiltonian. This gives us a new technique to analyze the spectral behaviour of the quantum operators of complex systems. The connection between Complex systems and Calogero hamiltonian seems to be widerannging. The eigen value dynamics of Hermitian operators, for example, Hamiltonian of complex quantum systems e.g. chaotic systems, disordered systems seems to have an intimate connection with the particle-dynamics of Calogero-Moser (CM) Hamiltonian. The latter describes the dynamics of particles interacting via pairwise inverse square interaction and confined to move along a real line [1], Ĥ = ∑