Scattering zippers and their spectral theory

TL;DR

This paper introduces scattering zippers, a class of unitary operators generalizing several models, and develops their spectral theory using Weyl discs and Sturm-Liouville oscillation methods.

Contribution

It defines scattering zippers, analyzes their spectral properties, and establishes a bijection with matrix-valued measures, extending existing models and theories.

Findings

Weyl discs characterize the spectral properties of scattering zippers.

A bijection between semi-infinite operators and matrix measures is proven.

Oscillation theory is adapted to compute spectra of finite and periodic operators.

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 and generalizes Blatter-Browne and Chalker-Coddington models and CMV matrices. Weyl discs are analyzed and used to prove a bijection between the set of semi-infinite scattering zipper operators and matrix valued probability measures on the unit circle. Sturm-Liouville oscillation theory is developed as a tool to calculate the spectra of finite and periodic scattering zipper operators.