Level spacing for band random matrices

Anderson localization of flexural waves in disordered elastic beams

We study, both experimentally and numerically, the Anderson localization phenomenon in flexural waves of a disordered elastic beam, which consists of a beam with randomly spaced notches. We found that the effect of the disorder on the system is stronger above a crossover frequency f_c than below it. For a chosen value of disorder, we show that above f_c the normal-mode wave functions are localized as occurs in disordered solids, while below f_c the wave functions are partially and fully extended, but their dependence on the frequency is not governed by a monotonous relationship, as occurs in other classical and quantum systems. These findings were corroborated with the calculation of the participation ratio, the localization length and a level statistics. In particular, the nearest spacing distribution is obtained and analyzed with a suitable phenomenological expression, related to the level repulsion.

From closed to open one-dimensional Anderson model: transport versus spectral statistics

Using the phenomenological expression for the level spacing distribution with only one parameter 0 ≤ β ≤ ∞ covering all regimes of chaos and complexity in a quantum system, we show that transport properties of the one-dimensional Anderson model of finite size can be expressed in terms of this parameter. Specifically, we demonstrate a strictly linear relation between β and the normalized localization length for the whole transition from strongly localized to extended states. This result allows one to describe all transport properties in the open system entirely in terms of the parameter β and the strength of the coupling to the continuum. For nonperfect coupling, our data show a quite unusual interplay between the degree of internal chaos defined by β and the degree of openness of the model. The results can be experimentally tested in single-mode waveguides with either bulk or surface disorder.