Products of random matrices and generalised quantum point scatterers

Abstract

To every product of $2\times 2$ matrices, there corresponds a one-dimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of {\em 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 $\text{SL}(2,\mathbb{R})$. We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.

38 pages, 13 pdf figures. V2 : conclusion added ; Definition of function $\Omega$ changed