Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and hardy spaces of character-automorphic functions

Sodin, Mikhail; Yuditskii, Peter

doi:10.1007/bf02921627

Cited by 118 publications

(216 citation statements)

References 40 publications

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“…However, I have added some additional material. Complementary results to this sections, namely for the case of Cantor spectra, can be found in [12] - [14], and [212], [213].…”

Section: Notes On Literaturesupporting

confidence: 57%

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Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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“…However, I have added some additional material. Complementary results to this sections, namely for the case of Cantor spectra, can be found in [12] - [14], and [212], [213].…”

Section: Notes On Literaturesupporting

confidence: 57%

“…Additional references are [74], [169], [173], [230], and [175]. This technique can also be extended to surfaces with infinitely many branch points [12] - [14], [212] and [213].…”

Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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“…This task has been accomplished for so-called homogeneous sets K by Sodin-Yuditskii [14]; the special case of a finite gap set K is classical and has been studied in very great detail. Beyond the Sodin-Yuditskii result, very little is known at present.…”

Section: Introductionmentioning

confidence: 99%

“…The hypothesis of Theorem 1.5 will be satisfied by many sets K. For example, Theorem 1.5 applies to all weakly homogeneous sets K [10, Corollary 2.3]; see also [14].…”

Section: Introductionmentioning

confidence: 99%

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Approximation Results for Reflectionless Jacobi Matrices

Poltoratski

Remling

2010

International Mathematics Research Notices

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Trace formulas and a Borg‐type theorem for CMV operators with matrix‐valued coefficients

Zinchenko

2010

Mathematische Nachrichten

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Abstract. We prove a general Borg-type inverse spectral result for a reflectionless unitary CMV operator (CMV for Cantero, Moral, and Velázquez [13]) associated with matrix-valued Verblunsky coefficients. More precisely, we find an explicit formula for the Verblunsky coefficients of a reflectionless CMV matrix whose spectrum consists of a connected arc on the unit circle. This extends a recent result [39] for CMV operators with scalar-valued coefficients.In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory matrix-valued functions to prove an infinite sequence of trace formulas connected with CMV operators.

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Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and hardy spaces of character-automorphic functions

Cited by 118 publications

References 40 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

Approximation Results for Reflectionless Jacobi Matrices

Trace formulas and a Borg‐type theorem for CMV operators with matrix‐valued coefficients

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