The paper introduces a framework for studying unitary one-channel operators, which generalize CMV matrices and one-dimensional quantum walks. The key aspects are:
Definition of one-channel unitary operators: These are constructed as a product of two unitary operators, V and W, where V is a direct sum of finite-dimensional unitary matrices and W has a specific structure connecting the "shells" or "slices" of the underlying graph.
Transfer matrix formalism: The authors develop a transfer matrix approach to analyze the spectral properties of these one-channel unitary operators. They define transfer matrices T^♯ and T^♭ that relate the wave functions across the shells.
Spectral average formula and criteria for absolute continuity: The authors establish an analog of Carmona's formula, which expresses the spectral measure in terms of the transfer matrices. They then derive a criterion for absolute continuity of the spectrum based on the transfer matrices.
Application to random perturbations: The authors consider random Hilbert-Schmidt perturbations of periodic one-channel scattering zippers and prove that the spectrum remains purely absolutely continuous in a certain set Σ, with probability one.
The results generalize previous work on Hermitian one-channel operators and provide new tools for analyzing the spectral properties of unitary one-channel models, including higher-dimensional quantum walks and Chalker-Coddington networks.
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by Olivier Bour... at arxiv.org 10-01-2024
https://arxiv.org/pdf/2405.08898.pdfDeeper Inquiries