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Ariznabarreta, Gerardo and Mañas Baena, Manuel
(2014)
*Matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems.*
Advances in mathematics, 264
.
pp. 396-463.
ISSN 0001-8708

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Official URL: http://dx.doi.org/10.1016/j.aim.2014.06.019

## Abstract

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.

Item Type: | Article |
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Additional Information: | ©Elsevier. |

Uncontrolled Keywords: | Matrix orthogonal Laurent; Polynomials; Borel–Gauss factorization; Christoffel–Darboux kernels; Toda type integrable hierarchies |

Subjects: | Sciences > Physics > Physics-Mathematical models Sciences > Physics > Mathematical physics |

ID Code: | 31456 |

Deposited On: | 16 Jul 2015 11:47 |

Last Modified: | 10 Dec 2018 15:09 |

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