Lyapunov exponents, one-dimensional Anderson localization and products of random matrices

摘要

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localization; its interest in this particular context is that it provides a reasonable measure of the localization 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 localization properties. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Lyapunov analysis: from dynamical systems theory to applications'.