Equivalence classes of block Jacobi matrices

Kozhan, Rostyslav

doi:10.1090/s0002-9939-2010-10582-8

Cited by 9 publications

(12 citation statements)

References 5 publications

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“…It was also shown in [18] that any equivalent Jacobi matrix, for which eventually each A n has real eigenvalues, is also asymptotic to type 1, 2, 3. Here Im T ≡ T −T * 2i .…”

Section: Equivalence Classes Of Block Jacobi Matricesmentioning

confidence: 97%

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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“…It was also shown in [18] that any equivalent Jacobi matrix, for which eventually each A n has real eigenvalues, is also asymptotic to type 1, 2, 3. Here Im T ≡ T −T * 2i .…”

Section: Equivalence Classes Of Block Jacobi Matricesmentioning

confidence: 97%

Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

Kozhan

2012

Constr Approx

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“…As in Theorem 1 this establishes Szegő asymptotics for the type 2 Jacobi matrix, and for any equivalent to it matrix for which the limit lim n→∞ σ n exists. In [10] it is shown that under (3.10) this limit does exist for matrices of type 1 and 3 (or more generally, for any J the A n -coefficients of which have eventually only real eigenvalues).…”

Section: The Equivalent Way Of Writing (38) Ismentioning

confidence: 99%

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

Kozhan

2010

Journal of Approximation Theory

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Let µ be a matrix-valued measure with the essential spectrum a single interval and countably many point masses outside of it. Under the assumption that the absolutely continuous part of µ satisfies Szegő's condition and the point masses satisfy a Blaschke-type condition, we obtain the asymptotic behavior of the orthonormal polynomials on and off the support of the measure.The result generalizes the scalar analogue of Peherstorfer-Yuditskii (2001) [12] and the matrix-valued result of Aptekarev-Nikishin (1983) [1], which handles only a finite number of mass points. Published by Elsevier Inc.

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“…So we consider discrete Schrödinger equations whose solutions are vector valued functions and extend some basic results of Jacobi and Schrödinger equations from one dimensional space. There has been some research work in higher dimension for example, see [3,4,7]. However, we did not find a detail presentations of the basic theory, required for further studies in this area, which we discussed in section 2.…”

Section: Introcuctionmentioning

confidence: 88%

A Note on Vector Valued Discrete Schrödinger Operators

Acharya

2016

Nepali Math. Sci. Rep.

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The main purpose of this paper is to extend some theory of Schr¨odinger operators from one dimension to higher dimension. In particular, we will give systematic operator theoretic analysis for the Schr¨odinger equations in multidimensional space. To this end, we will provide the detail proves of some basic results that are necessary for further studies in these areas. In addition, we will introduce Titchmarsh- Weyl m− function of these equations and express m− function in term of the resolvent operators.

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Equivalence classes of block Jacobi matrices

Cited by 9 publications

References 5 publications

Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

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

A Note on Vector Valued Discrete Schrödinger Operators

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