Products of random matrices and generalised quantum point scatterers

Details Products of random matrices and generalised quantum point scatterers Products of random matrices and generalised quantum point scatterers

2010-04-14

arxiv.org/abs/1004.2415v2

J. Stat. Phys. 140(3), 427-466 (2010)

Physics

Condensed Matter

Disordered Systems and Neural Networks

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

Scientific paper

10.1007/s10955-010-0005-x

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.

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