Absence of absolutely continuous spectrum for random scattering zippers

TL;DR

This paper proves that random scattering zippers, modeled as unitary operators similar to matrix Jacobi matrices, lack absolutely continuous spectrum due to positive Lyapunov exponents in the presence of randomness.

Contribution

It establishes the absence of absolutely continuous spectrum for random scattering zippers by analyzing Lyapunov exponents, extending spectral theory in random unitary systems.

Findings

Lyapunov exponents are positive for the system

Absence of absolutely continuous spectrum is proven

Results apply to infinite identical scattering events

Abstract

A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries. For infinite identical events and random phases, Lyapunov exponents positivity is proved and yields to the absence of absolutely continuous spectrum.