Non-Abelian Toda-type equations and matrix valued orthogonal polynomials

Deaño, Alfredo; Morey, Lucía; Román, Pablo

doi:10.1090/proc/16637

Non-Abelian Toda-type equations and matrix valued orthogonal polynomials
HTML articles powered by AMS MathViewer

by Alfredo Deaño, Lucía Morey and Pablo Román;

Proc. Amer. Math. Soc. 152 (2024), 1613-1632

DOI: https://doi.org/10.1090/proc/16637

Published electronically: January 26, 2024

HTML | PDF

Abstract:

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail.

References

M. Adler and P. van Moerbeke, Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems, Comm. Math. Phys. 207 (1999), no. 3, 589–620. MR 1727239, DOI 10.1007/s002200050738
Gerardo Ariznabarreta and Manuel Mañas, Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems, Adv. Math. 264 (2014), 396–463. MR 3250290, DOI 10.1016/j.aim.2014.06.019
C. Berg, The matrix moment problem, Coimbra Lecture Notes on Orthogonal Polynomials, Nova Publishers, New York, 2008.
Amílcar Branquinho, Ana Foulquié-Moreno, and Juan C. García-Ardila, Matrix Toda and Volterra lattices, Appl. Math. Comput. 365 (2020), 124722, 16. MR 4008027, DOI 10.1016/j.amc.2019.124722
Amílcar Branquinho, Ana Foulquié Moreno, and Manuel Mañas, Matrix biorthogonal polynomials: eigenvalue problems and non-Abelian discrete Painlevé equations: a Riemann-Hilbert problem perspective, J. Math. Anal. Appl. 494 (2021), no. 2, Paper No. 124605, 36. MR 4153864, DOI 10.1016/j.jmaa.2020.124605
M. Bruschi, S. V. Manakov, O. Ragnisco, and D. Levi, The nonabelian Toda lattice-discrete analogue of the matrix Schrödinger spectral problem, J. Math. Phys. 21 (1980), no. 12, 2749–2753. MR 597590, DOI 10.1063/1.524393
Mattia Cafasso and Manuel D. de la Iglesia, Non-commutative Painlevé equations and Hermite-type matrix orthogonal polynomials, Comm. Math. Phys. 326 (2014), no. 2, 559–583. MR 3165468, DOI 10.1007/s00220-013-1853-4
Mattia Cafasso and Manuel D. de la Iglesia, The Toda and Painlevé systems associated with semiclassical matrix-valued orthogonal polynomials of Laguerre type, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 076, 17. MR 3830211, DOI 10.3842/SIGMA.2018.076
W. Riley Casper and Milen Yakimov, The matrix Bochner problem, Amer. J. Math. 144 (2022), no. 4, 1009–1065. MR 4461954, DOI 10.1353/ajm.2022.0022
Giovanni A. Cassatella-Contra and Manuel Mañas, Riemann-Hilbert problem and matrix discrete Painlevé II systems, Stud. Appl. Math. 143 (2019), no. 3, 272–314. MR 4019585, DOI 10.1111/sapm.12277
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 481884
David Damanik, Alexander Pushnitski, and Barry Simon, The analytic theory of matrix orthogonal polynomials, Surv. Approx. Theory 4 (2008), 1–85. MR 2379691
Alfredo Deaño, Bruno Eijsvoogel, and Pablo Román, Ladder relations for a class of matrix valued orthogonal polynomials, Stud. Appl. Math. 146 (2021), no. 2, 463–497. MR 4206387, DOI 10.1111/sapm.12351
Antonio J. Durán and F. Alberto Grünbaum, Structural formulas for orthogonal matrix polynomials satisfying second-order differential equations. I, Constr. Approx. 22 (2005), no. 2, 255–271. MR 2148533, DOI 10.1007/s00365-004-0577-2
B. Eijsvoogel, L. Morey, and P. Román, Duality and difference operators for matrix valued discrete polynomials on the nonnegative integers, Constr. Approx. (2023).
H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. B (3) 9 (1974), 1924–1925. MR 408647, DOI 10.1103/PhysRevB.9.1924
M. Gekhtman, Hamiltonian structure of non-abelian Toda lattice, Lett. Math. Phys. 46 (1998), no. 3, 189–205. MR 1661233, DOI 10.1023/A:1007579806383
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
Mourad E. H. Ismail, Erik Koelink, and Pablo Román, Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice, Adv. in Appl. Math. 110 (2019), 235–269. MR 3988721, DOI 10.1016/j.aam.2019.07.002
Mourad E. H. Ismail, Erik Koelink, and Pablo Román, Generalized Burchnall-type identities for orthogonal polynomials and expansions, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 072, 24. MR 3828871, DOI 10.3842/SIGMA.2018.072
Erik Koelink, Ana M. de los Ríos, and Pablo Román, Matrix-valued Gegenbauer-type polynomials, Constr. Approx. 46 (2017), no. 3, 459–487. MR 3735699, DOI 10.1007/s00365-017-9384-4
Erik Koelink and Pablo Román, Orthogonal vs. non-orthogonal reducibility of matrix-valued measures, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 008, 9. MR 3451358, DOI 10.3842/SIGMA.2016.008
Erik Koelink and Pablo Román, Matrix valued Laguerre polynomials, Positivity and noncommutative analysis, Trends Math., Birkhäuser/Springer, Cham, [2019] ©2019, pp. 295–320. MR 4042279, DOI 10.1007/978-3-030-10850-2_{1}6
S. V. Manakov, Complete integrability and stochastization of discrete dynamical systems, Ž. Èksper. Teoret. Fiz. 67 (1974), no. 2, 543–555 (Russian, with English summary); English transl., Soviet Physics JETP 40 (1974), no. 2, 269–274 (1975). MR 389107
L. Miranian, Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory, J. Phys. A 38 (2005), no. 25, 5731–5749. MR 2168386, DOI 10.1088/0305-4470/38/25/009
Juan Alfredo Tirao and Ignacio Nahuel Zurrián, Spherical functions of fundamental $K$-types associated with the $n$-dimensional sphere, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 071, 41. MR 3261873, DOI 10.3842/SIGMA.2014.071
M. Toda, Vibration of a chain with nonlinear interaction, J Phys. Soc. Japan 22 (1967), no. 2, 431–436.
M. Toda, Wave propagation in anharmonic lattices, J Phys. Soc. Japan 23 (1967), no. 3, 501–506.
Walter Van Assche, Orthogonal polynomials and Painlevé equations, Australian Mathematical Society Lecture Series, vol. 27, Cambridge University Press, Cambridge, 2018. MR 3729446
Walter Van Assche, Orthogonal polynomials, Toda lattices and Painlevé equations, Phys. D 434 (2022), Paper No. 133214, 9. MR 4396612, DOI 10.1016/j.physd.2022.133214