Products of random matrices and generalised quantum point scatterers

TL;DR

This paper establishes a link between products of random 2x2 matrices and a quantum model with generalized point scatterers, enabling analytical solutions for certain invariant measures.

Contribution

It introduces a quantum model corresponding to general products of matrices in SL(2, R), providing new analytical insights into their invariant measures.

Findings

Derived explicit invariant measures for specific random matrix products

Established a quantum model correspondence for general matrix products

Provided analytical expressions for invariant measures in new cases

Abstract

To every product of $2\times2$ matrices, there corresponds a one-dimensional Schr\"{o}dinger 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.