# Finite-gap CMV matrices: Periodic coordinates and a Magic Formula

## TL;DR

This paper establishes a bijective correspondence between isospectral tori of finite-gap CMV matrices and special periodic block-CMV matrices satisfying a Magic Formula, advancing the understanding of spectral theory on the unit circle.

## Contribution

It introduces a new class of matrices called MCMV, linking almost-periodic CMV matrices with finite-gap spectra to periodic matrices via a functional model.

## Key findings

- Proves a bijective correspondence between isospectral tori and periodic matrices.
- Defines MCMV matrices as spectrally-dependent Möbius transforms.
- Resolves Simon's conjecture on Caratheodory functions as quadratic irrationalities.

## 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.