Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case II

Abstract

We study Jacobi matrices with $N$-periodically modulated recurrence coefficients when the sequence of $N$-step transfer matrices is convergent to a non-trivial Jordan block. In particular, we describe asymptotic behavior of their generalized eigenvectors, we prove convergence of $N$-shifted Turán determinants as well as of the Christoffel--Darboux kernel on the diagonal. Finally, by means of subordinacy theory, we identify their absolutely continuous spectrum as well as their essential spectrum. By quantifying the speed of convergence of transfer matrices we were able to cover a large class of Jacobi matrices. In particular, those related to generators of birth-death processes.

arXiv comment50 pages

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Cited by in corpus (5)

• Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation II
• Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case II
• Asymptotic zeros' distribution of orthogonal polynomials with unbounded recurrence coefficients
• Nevai's condition for measures with unbounded supports
• Barrier nonsubordinacy and absolutely continuous spectrum of block Jacobi matrices