Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

Abstract

Abstract. We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L 1 -type condition on Jacobi coefficients, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of .The above results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters. MotivationThe main aim of this paper is to generalize some known properties from the theory of orthogonal polynomials on the real line to the matrix-valued case. The basic construction of the matrix-valued theory is almost identical to the scalar case. We give the details in the next section (see [4] for a more extensive review). This will lead us to considering the following question. We will be studying the l × l matrix-valued solutions (f n (E))where A n , B n are invertible l × l matrices with B n positive, 1 is the l × l identity matrix, and E a complex number.One of the possible solutions to this recurrence is the sequence of the (right) orthonormal polynomials f n (E) = p R n−1 (E, J ) associated with the block Jacobi matrix(see (2.3) below). Another natural choice, however, is the unique (up to a multiplicative constant) decaying Weyl solution, which exists for all E with Re E = 0. If the matrix J is reasonably close to the "free" block Jacobi matrix J 0 (which is, the block Jacobi matrix with A n ≡ 1, B n ≡ 0), then its (normalized) Weyl solution (u n ) ∞ n=0 converges to the Weyl solution of J 0 . In this case we call (u n ) ∞ n=0 the Jost solution (see Definition 3.2 below), and we say that Jost asymptotics holds. By the Jost function we will simply call the first element u 0 (see Definition 3.3 below).Jost solution and Jost function are natural objects of study for many various reasons. One of the most immediate ones is that Jost asymptotics is essentially equivalent to the existence of the limit z n p R n (z + z −1 ) (the so-called Szegő asymptotics). where µ is the (l×l matrix-valued) spectral measure of J . This is a meromorphic Herglotz function on C\ess supp µ. Recall (see more details in Section 2.3) that a Herglotz function is a function satisfying Im m(z) > 0 if Im z > 0. Conversely, any Herglotz function has the associated measure µ, and it could be of interest to study the correspondence between properties of m and of J . Jost asymptotics has been a very well studied topic for the scalar case (see [5,6, 11] and references therein), but the matrix-valued analogue still lacks the complete theory.The results of this paper can be divided into three parts. Part I of the results (Section 3.1) deals the direct problem: we prove that Jost asymptotics holds under an L 1 -type condition ((3.3)) on the Jacobi parameters A n , B n , and establish numerous properties of the Jost function and Jost solution.Part II of the results (Section 3.2) deals with the inverse problem: we characterize in an if...

Meromorphic Continuations of Finite Gap Herglotz Functions and Periodic Jacobi Matrices

“…As was explained in the Introduction, [17] established a connection between the rate of exponential convergence of Jacobi coefficients and meromorphic continuations of m.…”

“…The next three lemmas are taken from the author's [17]. (II) All of the following holds: (A) m has a meromorphic continuation to R R ; (B) m has no poles in π −1 (−2, 2), and at most simple poles at π −1 (±2); (C) (m − m ) −1 has no poles in π −1 (F R ), except at π −1 (±2), where they are at most simple;…”

“…A textbook exposition of the theory of periodic Jacobi matrices can be found there as well (in Chapter 5), along with an extensive historical discussion. Papers related to exponentially decaying perturbations of Jacobi matrices include (but are likely not limited to) [6,7,8,17,18].…”

“…1. We stated these lemmas in terms of m, rather than of M (see (1.5)) as it was in [17]. Note also that z −1 0 in [17, Thm 3.8(D)/3.9(D)] should be better thought of asz −1 0 .…”

Spectral theory of a class of block Jacobi matrices and applications

“…Operators of such a form are naturally related to matrix orthogonal polynomials theory, see for example [2,9,10,12,14,21,22] and references therein. Of course, the case where N = 1 corresponds to the usual scalar Jacobi operators that are well studied by different approaches, see e.g.…”