Riemann–Hilbert problems, matrix orthogonal polynomials
and discrete matrix equations with singularity confinement
| dc.contributor.author | Cassatella-Contra, Giovanni A. | |
| dc.contributor.author | Mañas Baena, Manuel Enrique | |
| dc.date.accessioned | 2023-06-20T03:56:19Z | |
| dc.date.available | 2023-06-20T03:56:19Z | |
| dc.date.issued | 2012-04 | |
| dc.description | Wiley-Blackwell. The authors thanks economical support from the Spanish Ministerio de Ciencia e Innovación, research project FIS2008-00200. GAC acknowledges the support of the grant Universidad Complutense de Madrid. Finally, MM reckons illuminating discussions with Dr. Mattia Cafasso in relation with orthogonality and singularity confinement, and both authors are grateful to Prof. Gabriel Álvarez Galindo for several discussions and for the experimental confirmation, via Mathematica, of the existence of the confinement of singularities in the 2 × 2 case. | |
| dc.description.abstract | n this paper, matrix orthogonal polynomials in the real line are described in terms of a RiemannHilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not. | |
| dc.description.department | Depto. de Física Teórica | |
| dc.description.faculty | Fac. de Ciencias Físicas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Ministerio de Ciencia e Innovación, Spain | |
| dc.description.sponsorship | Universidad Complutense de Madrid | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/31478 | |
| dc.identifier.doi | 10.1111/j.1467-9590.2011.00541.x | |
| dc.identifier.issn | 0022-2526 | |
| dc.identifier.officialurl | http://dx.doi.org/10.1111/j.1467-9590.2011.00541.x | |
| dc.identifier.relatedurl | http://onlinelibrary.wiley.com | |
| dc.identifier.relatedurl | http://arxiv.org/abs/1106.0036 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/44696 | |
| dc.issue.number | 3 | |
| dc.journal.title | Studies in applied mathematics | |
| dc.language.iso | eng | |
| dc.page.final | 274 | |
| dc.page.initial | 252 | |
| dc.publisher | Wiley-Blackwell | |
| dc.relation.projectID | FIS2008-00200 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 51-73 | |
| dc.subject.keyword | 2nd-Order differential-equations | |
| dc.subject.keyword | Painleve equations | |
| dc.subject.keyword | Formulas | |
| dc.subject.ucm | Física-Modelos matemáticos | |
| dc.subject.ucm | Física matemática | |
| dc.title | Riemann–Hilbert problems, matrix orthogonal polynomials and discrete matrix equations with singularity confinement | |
| dc.type | journal article | |
| dc.volume.number | 128 | |
| dcterms.references | [1] P. Painlevé, Le¸cons sur la théorie analytique des equations différentielles (Leçons de Stockholm, delivered in 1895), Hermann, Paris (1897). Reprinted in Œvres de Paul Painlevé, vol. I, Éditions du CNRS, Paris, (1973). [2] R. Conte (Editor), The Painlevé Property, One Century Later, Springer Verlag, New York, (1999). [3] B. Grammaticos, A. Ramani and V. Papageorgiou, Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67, 1825-1828 1991). [4] A. Ramani, B. Grammaticos, J. Hietarinta, Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67, 1829-1832 (1991). [5] A. Ramani, D. Takahashi, B. Grammaticos, Y. Ohta, The ultimate discretisation of the Painlevé equations, Physica D, 114, Issues 3-4, 185-196 (1998). [6] J. Hietarinta and C. Viallet, Discrete Painlevé I and singularity confinement in projective space, Chaos Solitons and Fractals 11 (2000), 29-32. [7] S. Lafortune and A. Goriely, Singularity confinement and algebraic integrability, J. Math.l Phys. 45 (2004), 1191-1208. [8] G. Freud, On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Royal Irish Acad. A76 (1976), 1-6. [9] W. Van Assche, Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials, Proceedings of the International Conference on Difference Equations, special Functions and Orthogonal Polynomials, World Scientific (2007), 687-725. [10] A. P. Magnus, Freud’s equations for orthogonal polynomials as discrete Painlevé equations, Symmetries and Integrability of Difference Equations (Canterbury, 1996), London Math. Soc. Lecture Note Ser., 255, Cambridge University Press, 1999, pp. 228-243. [11] David Damanik, Alexander Pushnitski, and Barry Simon, The Analytic Theory of Matrix Orthogonal Polynomials, Surveys in Approximation Theory 4, 2008. pp. 1-85 and also arXiv:0711.2703. [12] A. S. Fokas, A. R. Its and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Commun. Math. Phys. 147 (1992), 395-430. [13] E. Daems and A. B. J. Kuijlaars, Multiple orthogonal polynomials of mixed type and nonintersecting Brownian motions, J. Approx. Theory 146 (2007), 91-114. [14] M. G. Krein, Infinite J-matrices and a matrix moment problem, Dokl. Akad. Nauk. SSSR 69 (2) (1949), 125-128. [15] M. G. Krein, Fundamental aspects of the representation theory of hermitian operators with de- ficiency index (m,m), AMS Translations, Series 2, vol. 97, Providence, Rhode Island, 1971, pp. 75-143. [16] Yu. M. Berezanskii, Expansions in eigenfunctions of self-adjoint operators, Transl. Math. Monographs 17, Amer. Math. Soc., (1968). [17] J. S. Geronimo, Scattering theory and matrix orthogonal polynomials on the real line, Circuits Systems Signal Process 1 (1982), 471-495. [18] A. I. Aptekarev, E. M. Nikishin,The scattering problem for a discrete Sturm–Liouville operator, Math. USSR-Sb. 49 (1984), 325-355. [19] A. J. Durán, F. J. Grünbaum, Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations, Journal of Approximation Theory 134 (2005), 267-280. [20] A. J. Durán, F. J. Grünbaum, Structural formulas for orthogonal matrix polynomials satisfying second order differential equations, I, Constr. Approx. 22 (2005), 255-271. [21] Rodica D. Constin, Matrix valued polynomials generated by the scalar-type Rodrigues’ formulas, Journal of Approximation Theory 161 (2009), 693- 705. [22] A. J. Durán, Matrix inner product having a matrix symmetric second order differential operator, Rocky Mountain Journal of Mathematics, 27 (1997), 585-600. [23] A. J. Durán, F. J. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Res. Notices 10 (2004), 461-484. [24] J. Borrego, M. Castro, A. J. Durán, Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits , arXiv:1102.1578v1. [25] A. J. Durán and M. D. de la Iglesia, Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions, Int. Math. Research Notices, Vol. 2008, Article ID rnn084, 24 pages. [26] Cantero, M.J., Moral, L. and Velázquez, L., Differential properties of matrix orthogonal polynomials, J. Concrete Appl. Math. v3 i3. 313-334. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 0d5b5872-7553-4b33-b0e5-085ced5d8f42 | |
| relation.isAuthorOfPublication.latestForDiscovery | 0d5b5872-7553-4b33-b0e5-085ced5d8f42 |
Download
Original bundle
1 - 1 of 1
