Szegő asymptotics for matrix-valued measures with countably many bound states

Kozhan, Rostyslav

doi:10.1016/j.jat.2009.12.012

Cited by 3 publications

(9 citation statements)

References 13 publications

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“…To end this section, we get the following result for free as a corollary from Theorems 4.6, 4.4, and [17]. The scalar analogue is proven in Killip-Simon [15,Thm 9.14].…”

Section: Jost Asymptotics For Matrix-valued Orthogonal Polynomialsmentioning

confidence: 84%

“…By Theorem 4.4 u(z; J ) = (1 − z 2 )L(z), where L(z) = lim n→∞ z n p n (z + z −1 ). By the results from [17], L(z) is an H 2 (D) function with no singular inner part. Since 1 − z 2 is a bounded outer function, u is an H 2 (D) function with no singular inner part as well.…”

Section: Jost Asymptotics For Matrix-valued Orthogonal Polynomialsmentioning

confidence: 99%

“…2. O has an integral representation in terms of Potapov multiplicative integral, see [17] for the details.…”

Section: Jost Asymptotics For Matrix-valued Orthogonal Polynomialsmentioning

confidence: 99%

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Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

Kozhan

2012

Constr Approx

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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...

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“…To end this section, we get the following result for free as a corollary from Theorems 4.6, 4.4, and [17]. The scalar analogue is proven in Killip-Simon [15,Thm 9.14].…”

Section: Jost Asymptotics For Matrix-valued Orthogonal Polynomialsmentioning

confidence: 84%

Section: Jost Asymptotics For Matrix-valued Orthogonal Polynomialsmentioning

confidence: 99%

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Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

Kozhan

2012

Constr Approx

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“…Looking at (1.3), it is clear that any result on the asymptotics of p n (see e.g. [1], [4], [5]) would involve the limit lim n→∞ σ n . This explains the need for the following definition.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…This implies that σ n is Cauchy, and so converges. An alternative indirect way of proving that type 1 and type 2 are asymptotic to each other is as follows: it is proven in [4] that under condition (1.4) Szegő asymptotics for the type 2 block Jacobi matrix holds. In [5] the same fact is obtained for the type 1 Jacobi matrix.…”

Section: Proofs Of the Resultsmentioning

confidence: 99%