Christiansen, Jacob S. and Eichinger, Benjamin and Vandenboom, Tom (2021) Finite-Gap CMV Matrices : Periodic Coordinates and a Magic Formula. International Mathematics Research Notices, 2021 (18). pp. 14016-14085. ISSN 1073-7928
Full text not available from this repository.Abstract
We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum E and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as E-dependent operator Möbius transforms of certain generating CMV matrices that 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.