Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Ga...

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Authors: Ariznabarreta, Gerardo, Mañas Baena, Manuel Enrique
Format: article
Publication Date:2014
Country:España
Institution:Universidad Complutense de Madrid (UCM)
Repository:Docta Complutense
Language:English
OAI Identifier:oai:docta.ucm.es:20.500.14352/34793
Online Access:https://hdl.handle.net/20.500.14352/34793
Access Level:Open access
Keyword:51-73
Matrix orthogonal Laurent
Polynomials
Borel–Gauss factorization
Christoffel–Darboux kernels
Toda type integrable hierarchies
Física-Modelos matemáticos
Física matemática
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spelling Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systemsAriznabarreta, GerardoMañas Baena, Manuel Enrique51-73Matrix orthogonal LaurentPolynomialsBorel–Gauss factorizationChristoffel–Darboux kernelsToda type integrable hierarchiesFísica-Modelos matemáticosFísica matemáticaMatrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szegő polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasidefinite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows –connected with Darboux transformations–. We generalize the integrable flows of the Cafasso’s matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szegő polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.ElsevierUniversidad Complutense de Madrid20142014-09-2020142014-09-20journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/34793reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/347932026-06-02T12:44:21Z
dc.title.none.fl_str_mv Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems
title Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems
spellingShingle Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems
Ariznabarreta, Gerardo
51-73
Matrix orthogonal Laurent
Polynomials
Borel–Gauss factorization
Christoffel–Darboux kernels
Toda type integrable hierarchies
Física-Modelos matemáticos
Física matemática
title_short Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems
title_full Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems
title_fullStr Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems
title_full_unstemmed Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems
title_sort Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems
dc.creator.none.fl_str_mv Ariznabarreta, Gerardo
Mañas Baena, Manuel Enrique
author Ariznabarreta, Gerardo
author_facet Ariznabarreta, Gerardo
Mañas Baena, Manuel Enrique
author_role author
author2 Mañas Baena, Manuel Enrique
author2_role author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 51-73
Matrix orthogonal Laurent
Polynomials
Borel–Gauss factorization
Christoffel–Darboux kernels
Toda type integrable hierarchies
Física-Modelos matemáticos
Física matemática
topic 51-73
Matrix orthogonal Laurent
Polynomials
Borel–Gauss factorization
Christoffel–Darboux kernels
Toda type integrable hierarchies
Física-Modelos matemáticos
Física matemática
description Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szegő polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasidefinite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows –connected with Darboux transformations–. We generalize the integrable flows of the Cafasso’s matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szegő polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.
publishDate 2014
dc.date.none.fl_str_mv 2014
2014-09-20
2014
2014-09-20
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/34793
url https://hdl.handle.net/20.500.14352/34793
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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