Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

ABSTRACT

In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattice.

KEYWORDS:

• Matrix orthogonal polynomials
• linear functional
• recurrence relation
• operator theory
• matrix Sylvester differential equations
• Toda-type systems
• Lax-type theorem

AMS SUBJECT CLASSIFICATION:

• 33C45
• 34K08
• 39B42
• 47N20
• 42C05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020, and also by CIDMA - Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (‘FCT - Fundação para a Ciência e a Tecnologia’) within project UID/MAT/04106/2013.