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Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula

Research output: Contribution to Journal/Magazine › Journal article › peer-review

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Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula. / Christiansen, Jacob S.; Eichinger, Benjamin; Vandenboom, Tom.
In: International Mathematics Research Notices, Vol. 2021, No. 18, 30.09.2021, p. 14016-14085.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Christiansen, JS, Eichinger, B & Vandenboom, T 2021, 'Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula', International Mathematics Research Notices, vol. 2021, no. 18, pp. 14016-14085. https://doi.org/10.1093/imrn/rnz213

APA

Christiansen, J. S., Eichinger, B., & Vandenboom, T. (2021). Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula. International Mathematics Research Notices, 2021(18), 14016-14085. https://doi.org/10.1093/imrn/rnz213

Vancouver

Christiansen JS, Eichinger B, Vandenboom T. Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula. International Mathematics Research Notices. 2021 Sept 30;2021(18):14016-14085. Epub 2020 Feb 13. doi: 10.1093/imrn/rnz213

Author

Christiansen, Jacob S. ; Eichinger, Benjamin ; Vandenboom, Tom. / Finite-Gap CMV Matrices : Periodic Coordinates and a Magic Formula. In: International Mathematics Research Notices. 2021 ; Vol. 2021, No. 18. pp. 14016-14085.

Bibtex

@article{3d1742c426c7461faadd4cba7b702e79,
title = "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 E and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as E-dependent operator M{\"o}bius 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.",
author = "Christiansen, {Jacob S.} and Benjamin Eichinger and Tom Vandenboom",
note = "Publisher Copyright: {\textcopyright} 2020 The Author(s). Published by Oxford University Press. All rights reserved.",
year = "2021",
month = sep,
day = "30",
doi = "10.1093/imrn/rnz213",
language = "English",
volume = "2021",
pages = "14016--14085",
journal = "International Mathematics Research Notices",
issn = "1073-7928",
publisher = "Oxford University Press",
number = "18",

}

RIS

TY - JOUR

T1 - Finite-Gap CMV Matrices

T2 - Periodic Coordinates and a Magic Formula

AU - Christiansen, Jacob S.

AU - Eichinger, Benjamin

AU - Vandenboom, Tom

N1 - Publisher Copyright: © 2020 The Author(s). Published by Oxford University Press. All rights reserved.

PY - 2021/9/30

Y1 - 2021/9/30

N2 - 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.

AB - 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.

U2 - 10.1093/imrn/rnz213

DO - 10.1093/imrn/rnz213

M3 - Journal article

AN - SCOPUS:85122299255

VL - 2021

SP - 14016

EP - 14085

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 18

ER -