Browsing by Author "Denisov, Sergey A."

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    Jacobi matrices on trees generated by Angelesco systems: Asymptotics of coefficients and essential spectrum
    (EMS, 2021) Aptekarev, Alexander I.; Denisov, Sergey A.; Yattselev, Maxim L.; Mathematical Sciences, School of Science
    We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was recently discovered. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality.
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    Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials
    (AMS, 2020) Aptekarev, Alexander I.; Denisov, Sergey A.; Yattselev, Maxim L.; Mathematical Sciences, School of Science
    We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green’s functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on to a higher dimension. We illustrate the importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.
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    Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality
    (Elsevier, 2022-02-12) Denisov, Sergey A.; Yattselev, Maxim L.; Mathematical Sciences, School of Science
    We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.