Finite-Gap CMV Matrices : Periodic Coordinates and a Magic Formula
(2021) In International Mathematics Research Notices 2021(18). p.14016-14085- 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... (More)
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.
(Less)
- author
- Christiansen, Jacob S. LU ; Eichinger, Benjamin LU and Vandenboom, Tom
- organization
- publishing date
- 2021-09-01
- type
- Contribution to journal
- publication status
- published
- subject
- in
- International Mathematics Research Notices
- volume
- 2021
- issue
- 18
- pages
- 70 pages
- publisher
- Oxford University Press
- external identifiers
-
- scopus:85122299255
- ISSN
- 1073-7928
- DOI
- 10.1093/imrn/rnz213
- language
- English
- LU publication?
- yes
- additional info
- Publisher Copyright: © 2020 The Author(s). Published by Oxford University Press. All rights reserved.
- id
- 109abbdf-97a5-426d-956a-16c92aa06402
- date added to LUP
- 2022-02-21 14:51:53
- date last changed
- 2022-04-24 06:13:56
@article{109abbdf-97a5-426d-956a-16c92aa06402, abstract = {{<p>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. </p>}}, author = {{Christiansen, Jacob S. and Eichinger, Benjamin and Vandenboom, Tom}}, issn = {{1073-7928}}, language = {{eng}}, month = {{09}}, number = {{18}}, pages = {{14016--14085}}, publisher = {{Oxford University Press}}, series = {{International Mathematics Research Notices}}, title = {{Finite-Gap CMV Matrices : Periodic Coordinates and a Magic Formula}}, url = {{http://dx.doi.org/10.1093/imrn/rnz213}}, doi = {{10.1093/imrn/rnz213}}, volume = {{2021}}, year = {{2021}}, }