To every product of 2 × 2 matrices, there corresponds a onedimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of random matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in SL (2, R).

We study products of arbitrary random real 2 × 2 matrices that are close to the identity matrix. Using the Iwasawa decomposition of SL(2, R), we identify a continuum regime where the mean values and the covariances of the three Iwasawa parameters are simultaneously small. In this regime, the Lyapunov exponent of the product is shown to assume a scaling form. In the general case, the corresponding scaling function is expressed in terms of Gauss' hypergeometric function. A number of particular cases are also considered, where the scaling function of the Lyapunov exponent involves other special functions (Airy, Bessel, Whittaker, elliptic). The general solution thus obtained allows us, among other things, to recover in a unified framework many results known previously from exactly solvable models of one-dimensional disordered systems.

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties. AMS classification scheme numbers: 60B20 , 60G51 , 82B44 PACS numbers: 72.15.Rn , 02.50.-r arXiv:1207.0725v2 [cond-mat.dis-nn] 11 Sep 2012 1D Anderson localisation and products of random matrices

We study the one-dimensional Schrödinger equation with a disordered potential of the formwhere φ(x) is a Gaussian white noise with mean µg and variance g, and κ(x) is a random superposition of delta functions distributed uniformly on the real line with mean density ρ and mean strength v. Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers : φ(x) models the force field acting on the diffusing particle and κ(x) models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponentwhere N is the integrated density of states per unit length and γ the reciprocal of the localisation length. By using the continuous version of the Dyson-Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean v = 2g. Building on this result, we then solve the general case-in the low-energy limit-in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour

Abstract. We consider the Schrödinger equation with a random potential of the form2 where w is a Lévy noise. We focus on the problem of computing the so-called complex Lyapunov exponentwhere N is the integrated density of states of the system, and γ is the Lyapunov exponent. In the case where the Lévy process is non-decreasing, we show that the calculation of Ω reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Lévy process. We review the known solvable cases-where Ω can be expressed in terms of special functions-and discover a new one.