Products of random matrices and generalised quantum point scatterers

Understand

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.