Finite-gap CMV matrices: Periodic coordinates and a Magic Formula
Abstract
We prove a bijective unitary correspondence between 1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as spectrally-dependent operator M\"obius transforms of certain generating CMV matrices which are periodic up to a rotational phase; for this reason we call them "MCMV". Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities.
Keywords
Cite
@article{arxiv.1902.05850,
title = {Finite-gap CMV matrices: Periodic coordinates and a Magic Formula},
author = {Jacob S. Christiansen and Benjamin Eichinger and Tom VandenBoom},
journal= {arXiv preprint arXiv:1902.05850},
year = {2019}
}
Comments
40 pages; v2 incorporates uniqueness Proposition 4.10